A polynomial time construction of a hitting set for read-once branching programs of width 3
aa r X i v : . [ c s . CC ] J a n Fundamenta Informaticae XX (2021) 1–46 DOI 10.3233/FI-2016-0000IOS Press
A Polynomial Time Construction of a Hitting Setfor Read-Once Branching Programs of Width 3
Jiˇr´ı ˇS´ıma * , Stanislav ˇZ ´ak † Institute of Computer Science of the Czech Academy of SciencesPrague, Czech [email protected], [email protected]
Abstract.
Recently, an interest in constructing pseudorandom or hitting set generators for re-stricted branching programs has increased, which is motivated by the fundamental issue of deran-domizing space-bounded computations. Such constructions have been known only in the case ofwidth 2 and in very restricted cases of bounded width. In this paper, we characterize the hittingsets for read-once branching programs of width 3 by a so-called richness condition. Namely,we show that such sets hit the class of read-once conjunctions of DNF and CNF (i.e. the weakrichness). Moreover, we prove that any rich set extended with all strings within Hamming dis-tance of 3 is a hitting set for read-once branching programs of width 3. Then, we show that anyalmost O (log n ) -wise independent set satisfies the richness condition. By using such a set due toAlon et al. (1992) our result provides an explicit polynomial time construction of a hitting set forread-once branching programs of width 3 with acceptance probability ε > / . We announcedthis result at conferences almost 10 years ago, including only proof sketches, which motivateda plenty of subsequent results on pseudorandom generators for restricted read-once branchingprograms. This paper contains our original detailed proof that has not been published yet. * Research was partially supported by project GA ˇCR P202/12/G061 and RVO: 67985807. The authors would like to thankPavel Pudl´ak for pointing out the problem of hitting sets for width-3 read-once branching programs. † Research was partially supported by project GA ˇCR P202/10/1333 and RVO: 67985807.Address for correspondence: Jiˇr´ı ˇS´ıma, Institute of Computer Science of the Czech Academy of Sciences, P.O. Box 5,182 07 Prague 8, Czech Republic
J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3
1. Introduction An ε -hitting set for a class of Boolean functions of n variables is a set H ⊆ { , } n such that forevery function f in the class, the following is satisfied: If a random input is accepted by f with prob-ability at least ε , then there is also an input in H that is accepted by f . An efficiently constructiblesequence of hitting sets for increasing n is a straightforward generalization of the hitting set gener-ator introduced in [1], which is a weaker (one-sided error) version of pseudorandom generator [2].Recall that an ε -pseudorandom generator for a class of Boolean functions of n variables is a function g : { , } s −→ { , } n which stretches a short uniformly random seed of length s bits into n bits( s ≪ n ) that cannot be distinguished from uniform ones. In particular, for every function f in theclass, condition | P r x ∼ U n [ f ( x ) = 1] − P r y ∼ U s [ f ( g ( y )) = 1] | ≤ ε holds where x ∼ U n means that x is uniformly distributed in { , } n .For the class of Boolean functions of polynomial complexity in any reasonable model, it is easyto prove the existence of ε -hitting set of polynomial size, if ε > /n c for a constant c where n is thenumber of variables. The proof is nonconstructive, since it uses a counting argument. An importantproblem in complexity theory is to find polynomial time constructible hitting sets for functions of poly-nomial complexity in different standard models like circuits, formulas, branching programs etc. Suchconstructions would have consequences for the relationship between deterministic and probabilisticcomputations in the respective models.Looking for polynomial time constructions of hitting sets for unrestricted models belongs to thehardest problems in computer science. Hence, restricted models are investigated. We consider read-once branching (1-branching) programs of polynomial size, which is a restricted model of space-bounded computations [3] for which pseudorandom generators with seed length O (log n ) have beenknown for a long time through the result of Nisan [4]. Note that an explicit pseudorandom generatorfor this model which is computable in logarithmic space and has seed length O (log n ) would sufficeto derandomize the complexity class BPL (Bounded-error Probabilistic Logarithmic-space). Recently,considerable attention has been paid to improving the seed length to O (log n ) in the constant-widthcase, which is a fundamental problem with many applications in circuit lower bounds and derandom-ization [5, 6]. The problem has been resolved for width 2 but the known techniques provably fail forwidth 3 [7, 8, 9, 10, 5, 6], which applies even to hitting set generators [8].In the case of width 3, we do not know of any significant improvement over Nisan’s result ex-cept for some recent progress in the severely restricted case of so-called regular oblivious read-oncebranching programs. Recall that an oblivious branching program queries the input variables in a fixedorder, which represents a provably weaker computational model [11]. For constant-width regular oblivious 1-branching programs which have the in-degree of all nodes equal to 2 (or 0), three indepen-dent constructions of ε -pseudorandom generator with seed length O (log n (log log n +log(1 /ε ))) wereachieved [12, 8, 9]. This seed length has later been improved to O (log n log(1 /ε )) for constant-width permutation oblivious 1-branching programs [13, 9] which are regular programs with the two edgesincoming to any node labeled 0 and 1, i.e. edges labeled with 0 respectively 1 create a permutation foreach level-to-level transition [5].In the constant-width regular 1-branching programs the fraction of inputs that are queried at anynode is always lower bounded by a positive constant. This excludes the fundamental capability of . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 general (non-regular) branching programs to recognize the inputs that contain a given substring ona non-constant number of selected positions. In our approach, we manage the analysis also for thisessential case. In particular, we identify two types of convergence of the number of inputs along acomputational path towards zero which implement read-once DNFs and CNFs, respectively. Thus, weachieve the construction of a hitting set generator for general width-3 1-branching programs whichneed not be regular nor oblivious. In our previous work [14], we constructed a hitting set for so-called simple width-3 1-branching programs which exclude one specific pattern of level-to-level transition intheir normalized form and cover the width-3 regular case.In the present paper, we provide a polynomial time construction of a hitting set for read-oncebranching programs of width 3 with acceptance probability ε > / , which need not be oblivious.This represents an important step in the effort of constructing hitting set generators for the model ofread-once branching programs of bounded width. For this purpose, we formulate a so-called richness condition which is independent of a rather technical definition of branching programs. In fact, the(full) richness condition implies its weaker version which is equivalent to the definition of hitting setsfor read-once conjunctions of DNF and CNF. Thus, a related line of study concerns pseudorandomgenerators for read-once formulas, such as read-once DNFs [15].We show that the richness property characterizes in a certain sense the hitting sets for width-31-branching programs. In particular, its weaker version proves to be necessary for such hitting sets,while the sufficiency of richness represents the main result of this paper. More precisely, we show thatany rich set extended with all strings within Hamming distance of 3 is a hitting set for 1-branchingprograms of width 3 with the acceptance probability greater than / . The proof is based on a detailedanalysis of structural properties of the width-3 1-branching programs that reject all the inputs fromthe candidate hitting set. Then, we prove that for a suitable constant C , any almost ( C log n ) -wiseindependent set which can be constructed in polynomial time by the result due to Alon et al. [16]satisfies the richness condition, which implies our result. In addition, it follows from the latter resultthat almost O (log n ) -wise independent sets are weakly rich and hence, they hit the class of read-onceconjunctions of DNF and CNF which is a generalization of the earlier result from [15].A preliminary version of this article appeared as extended abstracts [17, 18] including only proofsketches, where our result was formulated for acceptance probability ε > / . Since then a plentyof results on pseudorandom generators for restricted 1-branching programs [19, 20, 21, 22, 23, 24, 25,26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] have been achieved which were motivated and/or follows ourstudy referring to our result; see e.g. the paper [36] for a current survey of the newest achievementsalong this direction. This paper contains our original complete proof that has not been published yet.The paper is organized as follows. After a brief review of basic definitions regarding branchingprograms in Section 2 (see [3] for more information), the weak richness condition is formulated andproved to be necessary in Section 3. The richness condition and its sufficiency is presented in Section 4including the intuition behind the proof. The subsequent four Sections 5–8 are devoted to the technicalproof of this proposition. Furthermore, our theorem that any almost O (log n ) -wise independent set isrich is presented in Section 9 where also the main steps of the technical proof occupying the subsequentfour Sections 10–13 are introduced. Finally, our result is summarized in Section 14. J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3
2. Normalized Width- w A branching program P on the set of input Boolean variables X n = { x , . . . , x n } is a directedacyclic multi-graph G = ( V, E ) that has one source s ∈ V of zero in-degree and, except for sinks of zero out-degree, all the inner (non-sink) nodes have out-degree 2. In addition, the inner nodesget labels from X n and the sinks get labels from { , } . For each inner node, one of the outgo-ing edges gets the label 0 and the other one gets the label 1. The branching program P computesBoolean function P : { , } n −→ { , } as follows. The computational path of P for an input a = ( a , . . . , a n ) ∈ { , } n starts at source s . At any inner node labeled by x i ∈ X n , inputvariable x i is tested and this path continues with the outgoing edge labeled by a i to the next node,which is repeated until the path reaches the sink whose label gives the output value P ( a ) . Denote by P − ( a ) = { a ∈ { , } n | P ( a ) = a } the set of inputs for which P outputs a ∈ { , } . For inputs ofarbitrary lengths, infinite families { P n } of branching programs, each P n for one input length n ≥ ,are used.A branching program P is called read-once (or shortly program) if every input vari-able from X n is queried at most once along each computational path. Here we consider leveled branching programs in which each node belongs to a level, and edges lead from level k ≥ only tothe next level k + 1 . We assume that the source of P creates level 0, whereas the last level is com-posed of all sinks. The number of levels decreased by 1 equals the depth of P which is the length ofits longest path, and the maximum number of nodes on one level is called the width of P . In addition, P is called oblivious if all nodes at each level are labeled with the same variable.For a 1-branching program P of width w define a w × w transition matrix T k on level k ≥ suchthat t ( k ) ij ∈ { , , } is the half of the number of edges leading from node v ( k − j ( ≤ j ≤ w ) on level k − of P to node v ( k ) i ( ≤ i ≤ w ) on level k . For example, t ( k ) ij = 1 implies there is a doubleedge from v ( k − j to v ( k ) i . Clearly, P wi =1 t ( k ) ij = 1 since this sum equals the half of the out-degreeof inner node v ( k − j , and · P wj =1 t ( k ) ij is the in-degree of node v ( k ) i . Denote by a column vector p ( k ) = ( p ( k )1 , . . . , p ( k ) w ) T the distribution of inputs among w nodes on level k of P , that is, p ( k ) i is theprobability that a random input is tested at node v ( k ) i , which equals the ratio of the number of inputsfrom M ( v ( k ) i ) ⊆ { , } n that are tested at v ( k ) i to all n possible inputs. It follows S wi =1 M ( v ( k ) i ) = { , } n and P wi =1 p ( k ) i = 1 for every level k ≥ . Given the distribution p ( k − on level k − , thedistribution on the subsequent level k can be computed using the transition matrix T k as p ( k ) = T k · p ( k − . (1)It is because the ratio of inputs coming to node v ( k ) i from previous-level nodes equals p ( k ) i = P wj =1 t ( k ) ij p ( k − j since each of the two edges outgoing from node v ( k − j distributes exactly the half ofthe inputs tested at v ( k − j .We say that a 1-branching program P of width w is normalized if P has the minimum depthamong the programs computing the same function (e.g. P does not contain the identity transition T k )and P satisfies > p ( k )1 ≥ p ( k )2 ≥ · · · ≥ p ( k ) w > (2) . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 for every k ≥ log w (hereafter, log denotes the binary logarithm). Obviously, condition (2) can alwaysbe met by possible splitting (if p ( k ) w = 0 ) and permuting the nodes at each level of P : Lemma 2.1. ([14])
Any width- w w = 3 . Any such normalizedprogram P satisfies p ( k )1 + p ( k )2 + p ( k )3 = 1 and > p ( k )1 ≥ p ( k )2 ≥ p ( k )3 > , which implies p ( k )1 > , p ( k )2 < , p ( k )3 < (3)for every level ≤ k ≤ d where d ≤ n is the depth of P . Note that the strict inequalities for p ( k )1 and p ( k )3 in (3) hold since p ( k ) i = according to (1) and t ( k ) ij ∈ { , , } .
3. The Weak Richness Condition Is Necessary
Let P be a class of branching programs and ε > be a real constant. A set of input strings H ⊆ { , } ∗ is called an ε -hitting set for class P if for sufficiently large n , for every branching program P ∈ P with n input variables (cid:12)(cid:12) P − (1) (cid:12)(cid:12) n ≥ ε implies ( ∃ a ∈ H ∩ { , } n ) P ( a ) = 1 . (4)Furthermore, we say that a set A ⊆ { , } ∗ is weakly ε -rich if for sufficiently large n , for any indexset I ⊆ { , . . . , n } , and for any partition { Q , . . . , Q q , R , . . . , R r } of I where q ≥ and r ≥ , andfor any c ∈ { , } n the following implication holds: If − q Y j =1 (cid:18) − | Q j | (cid:19) × r Y j =1 (cid:18) − | R j | (cid:19) ≥ ε , (5)then there exists a ∈ A ∩ { , } n such that ( ∃ j ∈ { , . . . , q } ) ( ∀ i ∈ Q j ) a i = c i (6)and ( ∀ j ∈ { , . . . , r } ) ( ∃ i ∈ R j ) a i = c i . (7)Particularly for q = 0 inequality (5) reads r Y j =1 (cid:18) − | R j | (cid:19) ≥ ε (8)and conjunction (6) and (7) reduces to the second conjunct (7), while for r = 0 inequality (5) reads − q Y j =1 (cid:18) − | Q j | (cid:19) ≥ ε (9) J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 and conjunction (6) and (7) reduces to the first conjunct (6).Note that the product on the left-hand side of inequality (5) expresses the probability that a randomstring a ∈ { , } n (not necessarily in A ) satisfies the conjunction (6) and (7). Moreover, this formulacan be interpreted as a read-once conjunction of DNF and CNF (each variable occurs at most once) q _ j =1 ^ i ∈ Q j ℓ ( x i ) ∧ r ^ j =1 _ i ∈ R j ¬ ℓ ( x i ) , where ℓ ( x i ) = ( x i for c i = 1 ¬ x i for c i = 0 (10)which accepts a random input with probability at least ε according to (5). Hence, the weak richnesscondition is, in fact, equivalent to the definition of a hitting set for read-once conjunctions of DNF andCNF. The following theorem observes that the weak richness condition is necessary for any set to be ahitting set for width-3 1-branching programs. It based on the clear facts that the 1-branching programsof width 3 can implement any read-once conjunction of DNF and CNF, and any hitting set for a classof functions hits any of its subclass. Nevertheless, we provide a detailed proof for a reader to get usedto the introduced definitions and notations. Theorem 3.1.
Every ε -hitting set for the class of read-once branching programs of width 3 is weakly ε -rich. Proof:
We proceed by transposition. Assume a set H ⊆ { , } ∗ is not weakly ε -rich which means that forinfinitely many n there is an index set I ⊆ { , . . . , n } , a partition { Q , . . . , Q q , R , . . . , R r } of I satisfying (5), and a string c ∈ { , } n such that every a ∈ H ∩ { , } n meets ( ∀ j ∈ { , . . . , q } ) ( ∃ i ∈ Q j ) a i = c i (11)or ( ∃ j ∈ { , . . . , r } ) ( ∀ i ∈ R j ) a i = c i . (12)We will use this partition and c for constructing a (non-normalized oblivious) width-3 1-branchingprogram P such that (cid:12)(cid:12) P − (1) (cid:12)(cid:12) n ≥ ε and ( ∀ a ∈ H ∩ { , } n ) P ( a ) = 0 , (13)which negates that H is an ε -hitting set for 1-branching programs of width 3 according to (4). In fact, P implements the corresponding negated conjunction of DNF and CNF (10).We assume q ≥ , r ≥ , and | Q q | > , while the proof for q = 0 or r = 0 or | Q q | = 1 issimilar. As depicted in Figure 1, branching program P is composed of q + r consecutive blocks cor-responding to the partition classes Q , . . . , Q q , R , . . . , R r which determine the indices of variablesthat are queried within these blocks. The block associated with Q j for j ∈ { , . . . , q } starts on level k j = P j − ℓ =1 | Q ℓ | of P (e.g. k = 0 ) with a transition satisfying t ( k j +1)11 = t ( k j +1)21 = , followed by asequence of transitions that meet t ( k )11 = 1 and t ( k )12 = t ( k )22 = for every k = k j + 2 , . . . , k j + | Q j | ,except for the boundary level k q + | Q q | = k q +1 , which is defined below. In addition, there is a paralleldouble-edge path leading from the node v ( k +1)3 on level k + 1 up to node v ( k q +1 − , and thus t ( k )33 = 1 . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 Figure 1. The Necessary Condition.
J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 for every k = k + 2 , k + 3 , . . . , k q +1 − . This path is wired up by q − double edges comingfrom nodes v ( k j )2 , that is, t ( k j +1)32 = 1 for every j = 2 , . . . , q . Finally, a special boundary transition isdefined on level k q +1 as t ( k q +1 )31 = t ( k q +1 )13 = 1 and t ( k q +1 )12 = t ( k q +1 )32 = . Note that there are only twonodes v ( k q +1 )1 , v ( k q +1 )3 on the boundary level k q +1 .Furthermore, P continues analogously with blocks corresponding to R j for j = 1 , . . . , r , eachstarting on level k q + j = k q +1 + P j − ℓ =1 | R ℓ | (e.g. k q + r +1 = d is the depth of P ) with the transitionsatisfying t ( k q + j +1)11 = t ( k q + j +1)21 = , followed by t ( k )11 = 1 and t ( k )12 = t ( k )22 = for every k = k q + j + 2 , . . . , k q + j + | R j | , including the parallel double-edge path, that is, t ( k )33 = 1 for every k = k q +1 + 1 , . . . , d and t ( k q + j +1)32 = 1 for every j = 2 , . . . , r . The branching program P then queries thevalue of each variable x i such that i ∈ Q j for some j ∈ { , . . . , q } or i ∈ R j for some j ∈ { , . . . , r } only on one level k ∈ { k j , . . . , k j +1 − } or k ∈ { k q + j , . . . , k q + j +1 − } , respectively (i.e. the nodeson level k are labeled with x i ), while the single edge leading to v ( k +1)2 (or to v ( k q +1 )1 for k = k q +1 − )on the subsequent level k + 1 (indicated by a bold line in Figure 1) gets label c i . Finally, the sink v ( d )1 gets label , whereas the sinks v ( d )2 , v ( d )3 are labeled with the output , which completes theconstruction of P .Clearly, P is an (oblivious) read-once branching program of width 3. The probability that an inputreaches the node v ( k q +1 )3 on the boundary level k q +1 can simply be computed as p ( k q +1 )3 = q Y j =1 (cid:18) − | Q j | (cid:19) , (14)while the probability of the complementary event that an input reaches v ( k q +1 )1 equals p ( k q +1 )1 =1 − p ( k q +1 )3 . Therefore, the probability that P outputs can be expressed and lower bounded by (5): (cid:12)(cid:12) P − (1) (cid:12)(cid:12) n = p ( d )1 = − q Y j =1 (cid:18) − | Q j | (cid:19) × r Y j =1 (cid:18) − | R j | (cid:19) ≥ ε . (15)Furthermore, we split H ∩ { , } n = A ∪ A into two parts so that every a ∈ A satisfies thefirst term (11) of the underlying disjunction, whereas every a ∈ A = H \ A meets the secondterm (12). Thus, for any input a ∈ A and for every j ∈ { , . . . , q } the block of P correspond-ing to Q j contains a level k ∈ { k j , . . . , k j +1 − } where variable x i is tested such that a i = c i .This ensures that the computational path for a ∈ A reaches v ( k q +1 )3 and further continues through v ( k q +1 +1)3 , . . . , v ( d )3 , which gives P ( a ) = 0 for every a ∈ A . Similarly, for any input a ∈ A thereexists a block of P corresponding to R j for some j ∈ { , . . . , r } such that the computational pathfor a traverses nodes v ( k q + j )1 , v ( k q + j +1)2 , v ( k q + j +2)2 , . . . , v ( k q + j + | R j | )2 . For j < r this path continuesthrough v ( k q + j +1 +1)3 , . . . , v ( d )3 , whereas for j = r it terminates at v ( d )2 , which gives P ( a ) = 0 in bothcases. Hence, P satisfies (13), which completes the proof. ⊓⊔ . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3
4. The Richness Condition Is Sufficient
We say that a set A ⊆ { , } ∗ is ε -rich if for sufficiently large n , for any index set I ⊆ { , . . . , n } ,and for any partition { R , . . . , R r } of I ( r ≥ ) satisfying r Y j =1 (cid:18) − | R j | (cid:19) ≥ ε , (16)and for any Q ⊆ { , . . . , n } \ I such that | Q | ≤ log n , for any c ∈ { , } n there exists a ∈ A ∩ { , } n that meets ( ∀ i ∈ Q ) a i = c i and ( ∀ j ∈ { , . . . , r } ) ( ∃ i ∈ R j ) a i = c i . (17)One can observe that an ε -rich set is weakly ε -rich (see Section 3) since inequality (5) implies (16)and ensures that there is index j ∈ { , . . . , q } of Q j = Q such that | Q | ≤ log n . In particular, if | Q j | > log n for every j = 1 , . . . , q , then inequality (5) would give − ε ≥ q Y j =1 (cid:18) − | Q j | (cid:19) ≥ (cid:18) − log n (cid:19) n log n > − n · n log n = 1 − n (18)which is a contradiction for n > /ε . Thus, we have (17) which validates the conjunction of (6) and(7) completing the argument.It follows that any rich set is a hitting set for read-once conjunctions of DNF and CNF. Also notethat formula (17) can be interpreted as a read-once CNF (cf. 10) ^ i ∈ Q ℓ ( x i ) ∧ r ^ j =1 _ i ∈ R j ¬ ℓ ( x i ) , where ℓ ( x i ) = ( x i for c i = 1 ¬ x i for c i = 0 (19)which contains at most logarithmic number of single literals together with clauses whose sizes satisfy(16). Moreover, Theorem 9.1 in Section 9 proves that any almost O (log n ) -wise independent setsatisfies the richness condition.The following theorem shows that the richness condition is, in a certain sense, sufficient for a setto be a hitting set for 3-width 1-branching programs. For an input a ∈ { , } n and an integer constant c ≥ , denote by Ω c ( a ) = { a ′ ∈ { , } n | h ( a , a ′ ) ≤ c } the set of so-called h-neighbors of a , where h ( a , a ′ ) is the Hamming distance between a and a ′ (i.e. the number bits in which a and a ′ differ). Wealso define Ω c ( A ) = S a ∈ A Ω c ( a ) for a given set A ⊆ { , } ∗ . Theorem 4.1.
Let ε > . If A is ε ′ -rich for some ε ′ < ε , then H = Ω ( A ) is an ε -hitting set forthe class of read-once branching programs of width 3. Proof:
Suppose a read-once branching program P of width 3 with sufficiently many input variables n meets (cid:12)(cid:12) P − (1) (cid:12)(cid:12) n ≥ ε > . (20) J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3
We will prove that there exists a ∈ H such that P ( a ) = 1 . On the contrary, we assume that P ( a ) = 0 for every a ∈ H , (21)which will lead to a contradiction. Without loss of generality , we assume that P is normalizedaccording to Lemma 2.1. In this paragraph we will informally explain the main ideas of the proof with the pointers to thesubsequent paragraphs and sections where the precise and detailed argument is given. The assumptionthat branching program P accepts large fraction of inputs and rejects all the inputs from candidatehitting set H constrains the structure of P severely. In particular, we inspect the structure of P withrespect to (20) and (21) from its last level d (containing the sinks) and we proceed in the analysis stepby step backwards to lower levels . For this purpose, various parameters denoting certain levels in P are defined which are used to describe the structure of P . These definitions of levels, indicated inboldface, are scattered in the following proof since the definition of a level often builds on the previousanalysis of P .The underlying inspection reveals that the structure at the end of branching program P can be splitinto blocks whose typical shape is schematically depicted in Figure 2 while the subsequent Figures 3–8focus on particular parts of the block. Figure 2 also summarizes the definitions of levels in the blockhaving the form of “ a ≤ b ↑ ≤ c : C ( b ) ” which means b is the greatest level such that a ≤ b ≤ c and condition C ( b ) is satisfied (similarly, ↓ denotes the least such level). In addition, there are mainequations listed in Figure 2 concerning the distribution p ( k )1 , p ( k )2 , p ( k )3 of inputs among three nodes atimportant levels of the block.The last level of the block is denoted by m and this level m satisfies the following four so-called m -conditions :1. t ( m )11 = t ( m )21 = ,2. t ( m )32 > ,3. p ( m )3 < ,4. there is a ( m ) ∈ A such that if we put a ( m ) at node v ( m )1 or v ( m )2 , then its onward computationalpath arrives to the sink labeled with 1. More precisely, the logical argument goes as follows. The branching program P is transformed to an equivalent branchingprogram P which computes the same function as P (i.e. P preserves (20) and (21)) and has some additional property (e.g. P is normalized). In the following proof, several equivalent transformations are employed one after the other in order toachieve various extra properties, which generates a sequence of branching programs P, P , P , . . . , P c . After showing thatthe existence of the last program P c eventually leads to a contradiction one can conclude that the original program P cannotexist. Recall that we number the levels of P from the zero level containing the source up to the last level d which is composedof sinks. This means that, in figures, the lower levels are situated on the top of branching program whereas the upper levelsare located at the bottom. . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 Figure 2. The structure of a typical block J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3
Without loss of generality , these m -conditions can also be met for m = d (Paragraph 4.2) which isthe last level of P . In particular, the sinks v ( d )1 and v ( d )2 are labeled with 1 according to m -condition 4.Thus, the inspection of the structure of P starts with the analysis of the first block which constitutesthe tail of P .The first level of the block is denoted by m ′ (its formal definition can be found in Paragraph 6.2)which, in a typical case, proves also to satisfy the four m ′ -conditions 1–4. Thus the block is delimitedby levels m ′ and m . The shape of the block is being revealed step by step by the case analysis(Sections 5 and 6) which starts from m and proceeds towards lower levels down to m ′ . We will nowshortly outline the structure of a typical block as depicted in Figure 2 which results from this analysis.From the first level m ′ through µ , there is no edge between the first two columns on the one hand, andthe third column on the other hand, which means there is a double-edge path in the third column from m ′ through µ (Paragraph 6.2). Moreover, there is a double-edge path in the first column starting atlevel µ which leads up to level m − where it is split into vertices v ( m )1 and v ( m )2 at the next level m (cf. m -condition 1).On the top of Figure 2, a single-edge path from µ to ν is indicated in boldface which is used todefine the partition class R associated with this block (Paragraph 5.1). In particular, class R containsall the indices of the variables that are queried on this computational path up to level ν − . Moreover,the edge labels on this path define relevant bits of c ∈ { , } n so that any input passing through thispath that differs from c in at least one bit location from R turns to the double-edge path in the firstcolumn and consequently comes in node v ( m )1 or v ( m )2 . This implements one CNF clause ∨ i ∈ R ¬ ℓ ( x i ) from (19). Similarly, sets Q j for j = 1 , . . . , q associated with this block are defined (Paragraph 5.2)using the single-edge paths from λ j to κ j which are also highlighted in Figure 2 so that any input thatpasses through v ( λ j )3 and agrees with c on all the bit locations from Q j reaches the double-edge path inthe first column coming in v ( m )1 or v ( m )2 . This implements DNF monomials ∧ i ∈ Q j ℓ ( x i ) in (10) whichare candidates for the monomial ∧ i ∈ Q ℓ ( x i ) in (19).Under certain assumptions ((34) and (35)), one can show that level m ′ satisfies m ′ -condition 1–3(Paragraph 6.2). This opens the possibility that the first level m ′ = m r of the current r th block atthe same time represents the last level of the next lower-level ( r + 1) st block to which the struc-tural analysis could recursively be applied (Section 7). It suffices to show that level m ′ also meets m ′ -condition 4. For this purpose, the richness condition (17) is employed for Q = ∅ and for the par-tition classes R , . . . , R r associated with the first r blocks (that have been analyzed so far), providedthat this partition satisfies (16). This gives an input a ( m ′ ) ∈ A such that for every block j = 1 , . . . , r there is i ∈ R j such that a ( m ′ ) i = c i according to (17), that is, a ( m ′ ) satisfies ∧ rj =1 ∨ i ∈ R j ¬ ℓ ( x i ) from (19). Hence, if we put this a ( m ′ ) at node v ( m ′ )1 or v ( m ′ )2 ( m ′ = m r ), then the block structurein Figure 2 ensures that a ( m ′ ) also traverses v ( µ )1 or v ( µ )2 and reaches the double-edge path in the firstcolumn coming in v ( m )1 or v ( m )2 ( m = m r − ), by the definition of R and c i for i ∈ R . This argumentis applied recursively to each block j = r, r − , . . . , which implies that a ( m ′ ) eventually arrives tothe sink v ( d )1 or v ( d )2 ( m = d ) labeled with 1. This proves the m ′ -condition 4 also for level m ′ . Thusthe analysis including the definition of an associated partition class R = R r +1 and sets Q j = Q r +1 ,j The blocks are numbered from the bottom to the top of branching program P in the order reverse to that of levels. . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 is applied recursively to the next ( r + 1) st block for m replaced with m ′ etc.If, on the other hand, the underlying partition does not satisfy (16), then one can prove that thereis a set Q = Q bj associated with the b th block among the first r blocks (that have been analyzedso far) such that | Q | ≤ log n , and the recursive analysis ends (Section 8). In this case, the richnesscondition (17) for this set Q and for partition R , . . . , R b − provides a ∈ A such that a i = c i foreach i ∈ Q and for every block j = 1 , . . . , b − there is i ∈ R j such that a i = c i , that is, a ∈ A satisfies ∧ i ∈ Q ℓ ( x i ) ∧ ∧ b − j =1 ∨ i ∈ R j ¬ ℓ ( x i ) according to (19). Moreover, one can show (Lemma 8.1)that there is an h-neighbor a ′ ∈ Ω ( a ) ⊆ H that differs from this a in at most two bits so that thesebits guarantee that the computational path for a ′ in the b th block either reaches the double-edge pathin the first column, or comes in node v ( λ j )3 (see Figure 2). In the latter case, a ′ further traversesthe path corresponding to Q which reaches the double-edge path in the first column anyway by thedefinition of Q j and c i for i ∈ Q j . In both cases, input a ′ traverses node v ( m b − )1 or v ( m b − )2 , and bythe above-mentioned recursive argument it eventually arrives to the sink v ( d )1 or v ( d )2 labeled with 1.This provides the desired contradiction P ( a ′ ) = 1 for a ′ ∈ H . m = d We will first observe that the four m − conditions can be met for m = d . Clearly, both edges outgoingfrom v ( d − lead to the sink(s) labeled with 1 since p ( d − > due to (3) and | P − (0) | / n < ac-cording to (20). Hence, we will assume without loss of generality that t ( d )11 = t ( d )21 = ( m -condition 1)while the remaining edges that originally led to the sinks labeled with 1 or 0 are possibly redirected to v ( d )1 or v ( d )3 , respectively, so that the normalization condition p ( d )1 ≥ p ( d )2 > > p ( d )3 ( m -condition 3)is preserved by (20). Thus, sinks v ( d )1 and v ( d )2 are labeled with 1 ( m -condition 4) whereas sink v ( d )3 gets label 0. Finally, we show that t ( d )32 > ( m -condition 2). On the contrary, suppose t ( d )32 = 0 , whichimplies t ( d )33 > and H ⊆ P − (0) ⊆ M ( v ( d − ) due to t ( d )31 = 0 . In the case of t ( d )13 + t ( d )23 > ,the computational path for an h-neighbor a ′ ∈ Ω ( a ) of a ∈ A ⊆ H ⊆ M ( v ( d − ) that differs from a in the i th bit that is tested at node v ( d − (i.e. v ( d − is labeled with x i ), would reach the sink la-beled with 1, and hence P ( a ′ ) = 1 which contradicts the assumption H ⊆ P − (0) . For t ( d )33 = 1 ,on the other hand, we could shorten P by removing the last level d while preserving its function andcondition (20), which is in contradiction with the normalization of P . This completes the proof that m -conditions 1–4 can be assumed for m = d without loss of generality . Let level µ ′ be the least level of P such that ≤ µ ′ < m and t ( ℓ )11 = 1 for every ℓ = µ ′ + 1 , . . . , m − .We define level µ as µ = ( µ ′ − if t ( µ ′ )12 = 1 and t ( µ ′ )11 = t ( µ ′ )21 = µ ′ otherwise. (22) J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3
For the analysis of a single block structure (Sections 4–6, 8), we swap v ( µ )1 and v ( µ )2 if µ = µ ′ − for the notation simplicity so that t ( ℓ )11 = 1 for every ℓ = µ + 1 , . . . , m − at the cost of violatingcondition p ( µ )1 ≥ p ( µ )2 given by (2). Thus, for µ = µ ′ − , assume p ( µ )1 < p ( µ )2 , t ( µ +1)11 = 1 , and t ( µ +1)12 = t ( µ +1)22 = . For the recursion (Section 7) when the last level m in the next (lower-level)block may coincide with level µ of the current block we will nevertheless assume the original nodeorder and p ( µ )1 ≥ p ( µ )2 .The following lemma represents a technical tool which will be used for the analysis of the blockfrom level µ through m . For this purpose, define a so-called switching path starting from v ∈{ v ( k )2 , v ( k )3 } at level k , where µ ≤ k < m , to be a computational path of length at most 3 edgesleading from v to v ( ℓ )1 at level ℓ such that k < ℓ ≤ min( k + 3 , m ) or possibly to v ( m )2 for m ≤ k + 3 . Lemma 4.2. (i) µ > .(ii) There are no two parallel switching paths starting from v ( k )2 and v ( k )3 , respectively, at any level k such that µ ≤ k < m .(iii) If t ( k +1)12 > for some level k such that µ ≤ k < m , then t ( ℓ )11 = t ( ℓ )33 = 1 , t ( ℓ )12 = t ( ℓ )22 = forevery ℓ = µ + 1 , . . . , k , and t ( k +1)12 = (see Figure 3).(iv) If t ( k +1)13 > for some level k such that µ < k < m , then one of the following four casesoccurs:1. t ( k )11 = t ( k )23 = 1 and t ( k )12 = t ( k )32 = ,2. t ( k )11 = t ( k )23 = 1 and t ( k )22 = t ( k )32 = ,3. t ( k )11 = t ( k )22 = 1 and t ( k )13 = t ( k )33 = ,4. t ( k )11 = t ( k )22 = 1 and t ( k )23 = t ( k )33 = .In addition, if t ( k )23 = 1 (case 1 or 2), then t ( ℓ )11 = t ( ℓ )33 = 1 and t ( ℓ )12 = t ( ℓ )22 = for every ℓ = µ + 1 , . . . , k − (see Figure 3). Proof: (i) Suppose µ ≤ and let a ( m ) ∈ A be the input from m -condition 4. Then there exists anh-neighbor a ′ ∈ Ω ( a ( m ) ) of a ( m ) whose computational path starting from source v (0)1 reaches v ( µ )1 . Hence, P ( a ′ ) = 1 for a ′ ∈ H follows from M ( v ( µ )1 ) ⊆ M ( v ( m )1 ) ∪ M ( v ( m )2 ) and m -condition 4, which is a contradiction, and thus µ > . . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 (ii) Suppose there are two parallel switching paths starting from v ( k )2 and v ( k )3 , respectively, at somelevel k such that µ ≤ k < m , and let a ( m ) ∈ A be the input satisfying m -condition 4. Clearly, a ( m ) M ( v ( k )1 ) ⊆ M ( v ( m )1 ) ∪ M ( v ( m )2 ) since otherwise P ( a ( m ) ) = 1 for a ( m ) ∈ H . Thus,assume a ( m ) ∈ M ( v ) for v ∈ { v ( k )2 , v ( k )3 } . Then there is an h-neighbor a ′ ∈ Ω ( a ( m ) ) ∩ M ( v ) of a ( m ) whose computational path follows the switching path starting from v . Hence, a ′ ∈ M ( v ( m )1 ) ∪ M ( v ( m )2 ) implying P ( a ′ ) = 1 for a ′ ∈ H due to P is read-once. This completesthe proof of (ii). Figure 3. Lemma 4.2.iii and iv.
As depicted in Figure 3, at level k such that µ < k < m , denote by v ∈ { v ( k )2 , v ( k )3 } a node withthe edge outgoing to v ( k +1)1 , and let u be a node on level k − from which an edge leads to v , while v ′ ∈ { v ( k )2 , v ( k )3 } \ { v } and u ′ ∈ { v ( k − , v ( k − } \ { u } denote the other nodes. It follows from (ii)there is no edge from u ′ to v nor to v ( k )1 , which would establish two parallel switching paths startingfrom v ( k − and v ( k − , respectively. Hence, there must be a double edge from u ′ to v ′ . Since P isnormalized, u ′ = v ( k − and v ′ = v ( k )3 cannot happen simultaneously. Moreover, the second edgefrom u may lead either to v ( k )1 or to v ′ if v ′ = v ( k )3 . Now, the possible cases can be summarized:(iii) For t ( k +1)12 > we know v = v ( k )2 and v ′ = v ( k )3 , which implies t ( k )11 = t ( k )33 = 1 and t ( k )12 = t ( k )22 = . The proposition follows when this argument is applied recursively for k replacedwith k − etc. In addition, we will prove that t ( k +1)12 < for µ ≤ k < m . Clearly, t ( m )12 < from m -condition 2, and hence suppose k < m − . Also for k = µ = µ ′ − we know t ( µ +1)12 = and thus we further assume k ≥ µ ′ . On the contrary, suppose t ( k +1)12 = 1 whichimplies t ( k +1)23 = t ( k +1)33 = . For k > µ , one could shorten P by identifying level k with µ without changing its function, which is a contradiction with the normalization of P . J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3
For k = µ , on the other hand, we know that µ = µ ′ > from the definition of µ and we willfirst observe that there are at least two edges leading to v ( µ )3 . Suppose that only one edge leads to v ( µ )3 from u ∈ { v ( µ − , v ( µ − , v ( µ − } . If a ( m ) M ( u ) , then a ( m ) ∈ M ( v ( µ )1 ) ∪ M ( v ( µ )2 ) = M ( v ( µ +1)1 ) ⊆ M ( v ( m )1 ) ∪ M ( v ( m )2 ) implying P ( a ( m ) ) = 1 according to m -condition 4. If a ( m ) ∈ M ( u ) , then an h-neighbor a ′ ∈ Ω ( a ( m ) ) ∩ M ( u ) ⊆ H of a ( m ) exists which differs from a ( m ) in the variable that is tested at u and thus a ′ ∈ M ( v ( µ )1 ) ∪ M ( v ( µ )2 ) implying P ( a ′ ) = 1 .Now, with the two edges leading to v ( µ )3 , we could split v ( µ )3 into two nodes and merge v ( µ )1 and v ( µ )2 while preserving the function of P . Thus, for t ( µ +1)12 = 1 we can construct an equivalentbranching program with t ( µ +1)12 = 0 .(iv) For t ( k +1)13 > we know v = v ( k )3 and v ′ = v ( k )2 and the four cases listed in the propositionare obtained when the choice of u ∈ { v ( k − , v ( k − } is combined with whether the secondedge from u leads to v ( k )1 or v ′ . In addition, the remaining part for case 1 and 2 follows from(iii) when k + 1 is replaced with k . In particular, we know t ( k )12 > in case 1, while in case 2there is a switching path from v ( k − to v ( k +1)1 via v ( k )3 (substituting for t ( k )12 > ) and a similaranalysis applies to v = v ( k − excluding two switching paths starting from v ( k − and v ( k − ,respectively. ⊓⊔
5. Definition of Partition Class R and Sets Q , . . . , Q q µ to ν (Definition of R ) In the following corollary, we summarize the block structure from level µ through level ν by usingLemma 4.2, where ν is the greatest level such that µ ≤ ν ≤ m and t ( ℓ )12 + t ( ℓ )13 > for every ℓ = µ + 1 , . . . , ν . Note that ν = µ for t ( µ +1)12 = t ( µ +1)13 = 0 . In addition, let level γ be the greatestlevel such that µ ≤ γ ≤ ν and t ( γ )12 > if such γ exists, otherwise set γ = µ . Corollary 5.1. t ( ℓ )11 = t ( ℓ )33 = 1 and t ( ℓ )12 = t ( ℓ )22 = for ℓ = µ + 1 , . . . , γ − (Lemma 4.2.iii),2. t ( γ )11 = t ( γ )23 = 1 and t ( γ )12 = t ( γ )32 = if µ < γ < ν (case 1 of Lemma 4.2.iv),3. t ( ℓ )11 = t ( ℓ )22 = 1 and t ( ℓ )33 = for ℓ = γ + 1 , . . . , ν − (case 3 of Lemma 4.2.iv),4. if ν > µ , then t ( ν )12 < (Lemma 4.2.iii) and t ( ν )13 < for ν < m (similarly),5. t ( ℓ )12 = 0 for ℓ = ν + 1 , . . . , m (Lemma 4.2.iii). . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 Figure 4. The block structure from level µ through ν < m according to Corollary 5.1. Figure 4 shows a typical structure of the block from level µ through ν for the case of ν < m , whichcomes out of Corollary 5.1. In particular, there are two disjoint double-edge paths starting from level µ . One follows the first column from v ( µ )1 through v ( ν )1 . For ν < γ < ν , the other double-edge pathstarts from v ( µ )3 , follows the third column and turns to v ( γ )2 on level γ , and further continues throughthe second column up to v ( ν − . For γ = µ , this double-edge path follows only the second columnleading from v ( µ )2 through v ( ν − , whereas for γ = ν , it follows the third column from v ( µ )3 through v ( ν − . In addition, there is a node left on each level from µ through ν − that does not lay on theunderlying two disjoint double-edge paths. These remaining nodes are connected in a single-edge pathfrom level µ through ν − extended with an edge to v ( ν )2 or v ( ν )3 . For each node on this single-edgepath the other outgoing edge leads to the double-edge path in the first column.Furthermore, we shortly analyze level m for the special case of ν = m as depicted in Figure 5.Recall that t ( m )11 = t ( m )21 = and t ( m )32 > by m -condition 1 and 2, respectively. Moreover, either t ( m )12 = (i.e. ν = γ ) or t ( m )13 > (i.e. ν > γ ) by the definition of ν . It follows from Lemma 4.2.iithat either t ( m )33 = 1 for ν = γ or t ( m )32 = 1 for ν > γ . In the latter case of ν > γ , the other edgefrom v ( m − may lead either to v ( m )3 (i.e. t ( m )13 = t ( m )33 = ) or to v ( m )1 (i.e. t ( m )13 = 1 ) or v ( m )2 (i.e. t ( m )13 = t ( m )23 = ). This completes the analysis of level ν = m . We say that the underlying block is an empty block if ν = m and t ( m )33 = 0 (i.e. t ( m )13 + t ( m )23 = 1 and t ( m )32 = 1 ).Corollary 5.1 will be used for the definition of partition class R associated with the current block,if this block is not empty, which is illustrated in Figure 4. Moreover, class R is neither defined for ν = µ when only sets Q , . . . , Q q are associated with the block (see Paragraph 5.2 and Lemma 5.2in particular). Thus, for a non-empty block and ν > µ , we define the partition class R to be a set of J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3
Figure 5. Level m for ν = m . indices of the variables that are tested on the single-edge computational path v ( µ )2 , v ( µ +1)2 , . . . , v ( γ − ,v ( γ )3 , v ( γ +1)3 , . . . , v ( ν ′ − (or v ( µ )3 , v ( µ +1)3 , . . . , v ( ν ′ − if γ = µ or v ( µ )2 , v ( µ +1)2 , . . . , v ( ν ′ − if γ = ν )where level ν ′ is defined as ν ′ = min( ν, m − . (23)For the future use of condition (17) we also define relevant bits of string c ∈ { , } n . Thus, let c Ri be the corresponding labels of the edges creating this computational path (indicated by a bold line inFigure 4) including the edge outgoing from the last node v ( ν ′ − (or v ( ν ′ − if γ = ν ) that leads to v ( ν ′ )2 or v ( ν ′ )3 . ω to m (Definition of Q , . . . , Q q ) Furthermore, we define level ω to be the greatest level such that µ ≤ ω ≤ m and there is a double-edge path from µ through ω containing only nodes v ℓ ∈ { v ( ℓ )2 , v ( ℓ )3 } for every ℓ = µ, . . . , ω . Notethat this path possibly extends the double-edge path from Corollary 5.1 (see Figure 4) leading from v ( µ )2 to v ( ν − (for γ = µ < ν ) or from v ( µ )3 to v ( ν − (for µ < γ < ν ) or from v ( µ )3 to v ( ν − (for γ = ν > µ ). Hence, ω ≥ max( ν − , µ ) . (24)For the special case of ω = m (including the empty block) when this double-edge path reaches level m , no sets Q , . . . , Q q are associated with the current block and we set q = 0 . In this case, wewill observe in the following lemma that ν > µ , which ensures that at least class R is defined for anon-empty block (Paragraph 5.1) when ω = m . Lemma 5.2. If ω = m , then ν > µ . Proof:
On the contrary, suppose ω = m and ν = µ . It follows from Corollary 5.1.5 that t ( m )12 = 0 . Moreover, t ( m )22 = 0 since t ( m )22 > would require t ( m )13 > by the normalization of P , which contradicts . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 Lemma 4.2.ii, and hence, t ( m )32 = 1 . If t ( m − + t ( m − > , then p ( m )3 ≥ p ( m − ≥ p ( m − / > due to (3) which is in contradiction to m -condition 3. Hence, t ( m − = 1 which means µ ′ < m − .Furthermore, t ( µ +1)12 = t ( µ +1)13 = 0 by the definition of ν implying µ = µ ′ . Since P is normalized,we know t ( µ +1)22 > and either t ( µ +1)22 = 1 or t ( µ +1)23 = 1 due to ω = m , which implies t ( ℓ )22 = 1 for ℓ = µ + 2 , . . . , m − . It follows that p ( µ +1)2 ≤ p ( m )3 < according to m -condition 3.On the other hand, we know t ( µ )21 + t ( µ )31 > by the definition of µ ′ , which implies p ( µ )2 > due to (3). Hence, t ( µ +1)22 = t ( µ +1)32 = and t ( µ +1)23 = 1 because of p ( µ +1)2 < . This ensures t ( µ )11 = t ( µ )21 = since p ( µ − > . Thus, > p ( µ +1)2 > p ( µ − / which rewrites as p ( µ − < implying p ( µ − + p ( µ − > , and hence p ( µ − > due to p ( µ − ≥ p ( µ − . Clearly, t ( µ )32 = 0 sinceotherwise we get a contradiction > p ( µ +1)2 ≥ p ( µ − + p ( µ − > · + · = . Similarly, t ( µ )22 = 1 produces a contradiction > p ( µ +1)2 ≥ p ( µ − + p ( µ − > . It follows that t ( µ )12 > whereas t ( µ )12 = 1 contradicts µ = µ ′ according to (22), and hence t ( µ )12 = t ( µ )22 = and t ( µ )33 > . Thisgives a contradiction > p ( µ +1)2 ≥ ( p ( µ − + p ( µ − )+ p ( µ − > ( p ( µ − + p ( µ − + p ( µ − ) = . ⊓⊔ Thus, we will further assume ω < m throughout this Section 5. This implies t ( m )12 = 0 sinceotherwise t ( m )12 = t ( m )32 = ( m -condition 2) forces t ( m )33 = 1 by Lemma 4.2.ii which would prolongthe double-edge path from v ( µ )3 up to v ( m )3 according to Lemma 4.2.iii.We will show that one can assume t ( m )13 > without loss of generality . Suppose that t ( m )13 = 0 ,which implies t ( m )22 = t ( m )23 = 0 due to P is normalized, and hence t ( m )32 = t ( m )33 = 1 . More-over, we know t ( m )11 = t ( m )21 = by m -condition 1 and m -condition 3 ensures t ( m − = 1 . If t ( m − = t ( m − = 0 , then v ( m − and v ( m − can be merged and replaced by v ( m )3 , while v ( m − replaces v ( m − , which shortens P without changing its function. Hence, either t ( m − > or t ( m − > by Lemma 4.2.ii. In fact, t ( m − > contradicts ω < m according to Lemma 4.2.iii since t ( m − + t ( m − = t ( m )32 = t ( m )33 = 1 can, without loss of generality, prolong the double-edge pathfrom v ( µ )3 through v ( m − up to v ( m )3 . For t ( m − > , on the other hand, v ( m − and v ( m − can bemerged while v ( m − is split into two its copies, which produces t ( m − = t ( m − = , t ( m − = 1 ,and t ( m )11 = t ( m )21 = t ( m )12 = t ( m )22 = , t ( m )33 = 1 . After this modification, level m − satisfies thefour m -conditions 1–4 (see Paragraph 4.1) and thus, it can serve as a new level m while the originallevel m > d (for m = d program P could be shortened by removing its last level) is included in theprevious upper-level neighboring block, which is consistent with its structure (see Paragraph 6.2 andFigure 7 in particular).Thus, we assume t ( m )13 > without loss of generality, which implies t ( m )32 = 1 by Lemma 4.2.iiand t ( m − = 1 according to m -condition 3. Then Lemma 4.2.iv can be employed for k = m − where only case 3 and 4 may occur due to ω < m is assumed, which even implies ω < m − . Incase 3, t ( m − > and Lemma 4.2.iv can again be applied recursively to k = m − etc.In general, we start with level σ = m that meets t ( σ j )13 > for j = 1 . We proceed to lower levels J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3
Figure 6. The definition of Q , . . . , Q q . and inspect recursively the structure of subblocks indexed as j from level λ j through σ j where λ j isthe least level such that ω ≤ λ j < σ j − and the transitions from case 3 or 4 of Lemma 4.2.iv occurfor all levels ℓ = λ j + 1 , . . . , σ j − as depicted in Figure 6. This means t ( ℓ )11 = t ( ℓ )22 = 1 and t ( ℓ )33 = for every ℓ = λ j + 1 , . . . , σ j − . Note that λ j > µ because λ j = µ ensures t ( µ +1)22 = 1 implying ω > µ = λ j by the definition of ω , which contradicts ω ≤ λ j . In addition, we will observe thatcase 4 from Lemma 4.2.iv occurs at level λ j + 1 , that is t ( λ j +1)23 = . On the contrary, suppose that t ( λ j +1)13 = (case 3). For λ j > ω , this means case 1 or 2 occurs at level λ j < µ by the definition of λ j , which would be in contradiction to ω ≤ λ j according to Lemma 4.2.iv. For λ j = ω , on the otherhand, t ( ω +1)13 = contradicts the definition of ω by Lemma 4.2.iv. This completes the argument for t ( λ j +1)23 = .Furthermore, let level κ j be the least level such that λ j + 1 < κ j ≤ σ j and t ( κ j )13 > , which exists . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 since at least t ( σ j )13 > . Now we can define Q j associated with the current block (a candidate for Q inthe richness condition (17)) to be a set of indices of the variables that are tested on the computationalpath v ( λ j )3 , v ( λ j +1)3 , . . . , v ( κ j − , and let c Q j i be the corresponding labels of the edges creating thispath including the edge from v ( κ j − to v ( κ j )1 (indicated by a bold line in Figure 6). This extendsthe definition of c ∈ { , } n associated with R and Q k for ≤ k < j , which are usually pairwisedisjoint due to P is read-once. Nevertheless, the definition of c may not be unique for indices fromtheir nonempty intersections in some very special cases (including those corresponding to neighboringblocks) but the richness condition will only be used for provably disjoint sets (see Section 7).Finally, define next level σ j + to be the greatest level such that ω +1 < σ j +1 ≤ λ j and t ( σ j +1 )13 > ,if such σ j +1 exists, and continue in the recursive definition of λ j +1 , κ j +1 , Q j +1 with j replaced by j + 1 etc. If such σ j +1 does not exist, then set q = j and the definition of sets Q , . . . , Q q associatedwith the current block is complete. p ( m )1 + p ( m )2 in Terms of p ( ω +1)1 In this paragraph, we will upper bound p ( m )1 + p ( m )2 in terms of p ( ω +1)1 which will later be used forverifying the condition (16). For any ≤ j ≤ q , we know that t ( ℓ )11 = t ( ℓ )22 = 1 and t ( ℓ )23 = t ( ℓ )33 = forevery ℓ = λ j + 1 , . . . , κ j − (see Figure 6), which gives p ( κ j − + p ( κ j − = p ( λ j )2 + p ( λ j )3 , (25) p ( κ j − = p ( λ j )3 | Q j |− ≤ p ( λ j )2 + p ( λ j )3 | Q j | (26)because p ( λ j )3 ≤ ( p ( λ j )2 + p ( λ j )3 ) . We know t ( ℓ )12 = 0 for every ℓ = ω + 2 , . . . , m according toCorollary 5.1.5 where ν + 1 ≤ ω + 2 by (24). Moreover, it follows from the definition of σ j +1 >ω + 1 that t ( ℓ )13 = 0 for every ℓ = σ j +1 + 1 , . . . , λ j for any ≤ j < q , and t ( ℓ )13 = 0 for every ℓ = ω + 2 , . . . , λ q . Hence, p ( σ j +1 )2 + p ( σ j +1 )3 = p ( λ j )2 + p ( λ j )3 = p ( κ j − + p ( κ j − (27)for ≤ j < q and p ( ω +1)2 + p ( ω +1)3 = p ( λ q )2 + p ( λ q )3 = p ( κ q − + p ( κ q − (28)according to (25). Note that equation (28) holds trivially for λ q = ω + 1 and it is also valid for λ q = ω (recall λ q ≥ ω from the definition of λ j ) because t ( λ q +1)11 = t ( λ q +1)22 = 1 and t ( λ q +1)23 = t ( λ q +1)33 = (case 4 of Lemma 4.2.iv). Furthermore, we know t ( ℓ )22 = 1 for every ℓ = κ j , . . . , σ j − and t ( σ j )12 = 0 ,which implies p ( σ j )2 + p ( σ j )3 ≥ p ( κ j − + p ( κ j − − p ( κ j − ≥ p ( κ j − + p ( κ j − − p ( λ j )2 + p ( λ j )3 | Q j | = (cid:16) p ( σ j +1 )2 + p ( σ j +1 )3 (cid:17) (cid:18) − | Q j | (cid:19) (29) J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 for < j < q according to (26) and (27), while formula (29) reads p ( m )3 = p ( σ )3 ≥ (cid:16) p ( σ )2 + p ( σ )3 (cid:17) (cid:18) − | Q | (cid:19) (30)for j = 1 < q due to t ( m )32 = 1 , whereas (29) is rewritten as p ( σ q )2 + p ( σ q )3 ≥ (cid:16) p ( ω +1)2 + p ( ω +1)3 (cid:17) (cid:18) − | Q q | (cid:19) (31)for j = q > according to (28). Thus starting with (30), inequality (29) is applied recursively for j = 2 , . . . , q − , and, in the end, formula (31) is employed, leading to p ( m )3 ≥ (cid:16) p ( ω +1)2 + p ( ω +1)3 (cid:17) q Y j =1 (cid:18) − | Q j | (cid:19) (32)which is also obviously valid for the special case of q = 1 . This can be rewritten as p ( m )1 + p ( m )2 ≤ − (cid:16) − p ( ω +1)1 (cid:17) q Y j =1 (cid:18) − | Q j | (cid:19) (33)which represents the desired upper bound on p ( m )1 + p ( m )2 in terms of p ( ω +1)1 .
6. The Conditional Block Structure below µ µ + 1 Throughout this Section 6, we will assume p ( µ )3 < , (34) q Y j =1 (cid:18) − | Q j | (cid:19) > (35)where the product in (35) equals 1 for q = 0 . Based on these assumption, we will further analyzethe block structure below level µ in order to satisfy the m ′ -conditions 1–4 (see Paragraph 4.1) alsofor the first block level m ′ (the formal definition of m ′ appears at the beginning of Paragraph 6.2) sothat the analysis can be applied recursively when inequalities (34) and (35) hold (Section 7). For thispurpose, we still analyze level µ + 1 in the following lemma which implies ν > µ and thus guaranteesthat partition class R is defined for the underlying block if not empty. Lemma 6.1. t ( µ +1)12 = . . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 Proof:
For µ = µ ′ − , the proposition follows from the definition of µ , and thus assume µ = µ ′ . Considerfirst the special case of µ + 1 = m for which t ( m )12 = 0 implies t ( m )32 = 1 by using m -condition 2,Lemma 4.2.ii, and the normalization of P . Hence, we get a contradiction > p ( m )3 ≥ p ( m − ≥ p ( m − > by m -condition 3 and the definition of µ ′ . Thus further assume µ < m − . Clearly, t ( µ +1)32 < by the normalization of P whereas t ( µ +1)33 = 1 implies t ( µ +1)12 = , and thus, further con-sider the case when no double edge leads to v ( µ +1)3 . If t ( µ +1)12 > , then t ( µ +1)12 = by Lemma 4.2.iiifor k = µ . On the contrary, suppose t ( µ +1)12 = 0 , which gives t ( µ +1)22 > due to t ( µ +1)32 < . Assump-tion (34) ensures t ( µ )31 = 0 which implies t ( µ )21 > by the definition of µ ′ .We will first show that p ( µ +1)2 < . (36)For ω < m , assumption (35) together with m -condition 3 ensures p ( ω +1)2 + p ( ω +1)3 < (37)according to (32), which gives (36) for ω = µ . For ω > µ , we know by the definition of ω thatthere is a double-edge path starting from v ( µ )2 or v ( µ )3 and traversing v ( µ +1)2 as we assume no doubleedge to v ( µ +1)3 . For ω < m , we have t ( ℓ )22 = 1 for ℓ = µ + 2 , . . . , ω , and t ( ω +1)12 = 0 according toLemma 4.2.iii, and hence, p ( µ +1)2 ≤ p ( ω +1)2 + p ( ω +1)3 < due to (37). Similarly, p ( µ +1)2 ≤ p ( m )3 < for ω = m by m -condition 3, which completes the argument for (36).Suppose first that t ( µ )21 = 1 , which together with p ( µ − > implies t ( µ +1)22 = t ( µ +1)32 = according to (36). Obviously, < t ( µ )12 + t ( µ )13 < by the normalization of P . For t ( µ )12 + t ( µ )13 = 1 , either t ( µ )12 = t ( µ )33 = 1 or t ( µ )32 = t ( µ )13 = 1 when P could be shortened without changing its function, or t ( µ )12 = t ( µ )13 = t ( µ )32 = t ( µ )33 = implying p ( µ )1 = p ( µ )2 = p ( µ )3 = which contradicts (3). Hence, t ( µ )12 + t ( µ )13 = .Denote i ∈ { , } so that t ( µ )1 i = 1 whereas j ∈ { , } satisfies t ( µ )1 j = t ( µ )3 j = . If t ( µ +1)13 = 1 , then wecould shorten P while preserving its function, and hence t ( µ +1)23 > due to t ( µ +1)33 < . It follows that p ( µ +1)2 ≥ p ( µ − + p ( µ − j = (2 p ( µ − + p ( µ − j ) = (1 − p ( µ − i + p ( µ − ) ≥ which contradicts(36). Hence, t ( µ )11 = t ( µ )21 = due to t ( µ )11 < and t ( µ )31 = 0 , which implies t ( µ )12 < since µ = µ ′ .In addition, there are no ‘switching paths’ (cf. Lemma 4.2.ii) starting simultaneously from allthree vertices v ( µ − , v ( µ − , v ( µ − and leading to v ( µ )1 or v ( µ +1)1 since otherwise an h-neighbor a ′ ∈ Ω ( a ( m ) ) ∩ M ( v ( µ − i ) ⊆ H of a ( m ) ∈ M ( v ( µ − i ) from m -condition 4 would exist for some i ∈{ , , } such that a ′ ∈ M ( v ( µ +1)1 ) implying P ( a ′ ) = 1 . Recall we still need to contradict t ( µ +1)12 = 0 ,provided that t ( µ )11 = t ( µ )21 = , t ( µ )12 < , t ( µ +1)11 = 1 , t ( µ +1)22 > , and t ( µ +1)33 < .We will first consider the case of t ( µ )12 = which implies t ( µ )13 = 0 since three parallel switchingpaths starting from level µ − are excluded. Suppose that t ( µ )33 > which also gives t ( µ +1)13 = 0 becauseof ruling out the three switching paths, and hence t ( µ +1)23 > due to t ( µ +1)33 < . In addition, we know t ( µ )22 + t ( µ )32 = since we assume t ( µ )12 = . It follows that p ( µ +1)2 ≥ p ( µ − + p ( µ − + p ( µ − = J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 which contradicts (36). Hence, t ( µ )33 = 0 implying t ( µ )23 = 1 due to t ( µ )13 = 0 , which gives t ( µ )32 = . For t ( µ +1)23 > , we would again get a contradiction p ( µ +1)2 ≥ p ( µ − + p ( µ − + p ( µ − ≥ , and hencewe have t ( µ +1)23 = 0 and t ( µ +1)13 > because of t ( µ +1)33 < . We can assume without loss of generality that t ( µ +1)13 = since otherwise t ( µ )12 = t ( µ )32 = and t ( µ +1)13 = 1 (implying t ( µ +1)22 = t ( µ +1)32 = )could be replaced with t ( µ )12 = 1 while t ( µ )23 = 1 is replaced with t ( µ )22 = t ( µ )32 = t ( µ )23 = t ( µ )33 = and t ( µ +1)23 = t ( µ +1)33 = where v ( µ )3 is a copy of v ( µ )2 , which redefines level µ . Thus, it follows from t ( µ +1)13 = and t ( µ +1)23 = 0 that t ( µ +1)33 = and t ( µ +1)22 = 1 by the normalization of P .Recall once more we have t ( µ )11 = t ( µ )21 = t ( µ )12 = t ( µ )32 = , t ( µ )23 = 1 , t ( µ +1)11 = t ( µ +1)22 = 1 ,and t ( µ +1)13 = t ( µ +1)33 = . We know p ( µ − ≤ p ( µ +1)2 < due to (36) and p ( µ − = 2 p ( µ )3 < by (34), which implies p ( µ +1)1 = p ( µ − + p ( µ − < . This gives a contradiction p ( µ +1)2 ≥ ( p ( µ +1)2 + p ( µ +1)3 ) = (1 − p ( µ +1)1 ) > according to (36), which completes the argument for t ( µ )12 = .Further consider the case of t ( µ )13 > which ensures t ( µ )12 = 0 or equivalently t ( µ )22 + t ( µ )32 = 1 .We know t ( µ )22 < by the normalization of P , and hence t ( µ )32 > , which also ensures t ( µ +1)13 =0 since the three parallel switching paths starting from level µ − are excluded. It follows that t ( µ +1)23 > due to t ( µ +1)33 < . Thus, we get a contradiction p ( µ +1)2 ≥ p ( µ − + p ( µ − ≥ p ( µ − + p ( µ − + p ( µ − = according to (36).Similarly, for the remaining case of t ( µ )12 = t ( µ )13 = 0 we obtain t ( µ )32 = t ( µ )33 = 1 by thenormalization of P , which again ensures t ( µ +1)13 = 0 implying t ( µ +1)23 > . Hence, p ( µ +1)2 ≥ p ( µ − + p ( µ − + p ( µ − ≥ , which contradicts (36). This completes the proof of the lemma. ⊓⊔ m ′ to µ ( m ′ -Conditions 1–3) We define the first level m ′ of the underlying block to be the greatest level such that ≤ m ′ ≤ µ and t ( m ′ )32 > ( m ′ -condition 2), which exists since at least t (2)32 > . In the following lemma, we willanalyze the initial block structure from level m ′ through µ , which is illustrated in Figure 7 (where thedashed line shows that there is no edge from v ( k − or v ( k − to v ( k )3 for any m ′ < k ≤ µ ). Lemma 6.2. t ( k )31 = t ( k )32 = 0 and t ( k )33 = 1 for every k = m ′ + 1 , . . . , µ . Proof:
On the contrary, let k be the greatest level such that m ′ < k ≤ µ and t ( k )33 < , that is t ( ℓ )33 = 1 for ℓ = k + 1 , . . . , µ . Obviously, t ( k )33 > because t ( ℓ )32 = 0 for every ℓ = m ′ + 1 , . . . , k, . . . , µ bythe definition of m ′ , and t ( ℓ )31 = 0 for every ℓ = k, . . . , µ since otherwise p ( µ )3 ≥ p ( ℓ )3 > , whichcontradicts (34). Hence, t ( k )33 = and the edge from v ( k − to v ( k )3 is the only edge that leads to v ( k )3 due to t ( k )31 = t ( k )32 = 0 , while the other edge from v ( k − goes either to v ( k )1 or to v ( k )2 . Thus,either a ( m ) ∈ M ( v ( k )1 ) ∪ M ( v ( k )2 ) for a ( m ) satisfying m -condition 4 (Paragraph 4.1), or an h-neighbor a ′ ∈ Ω ( a ( m ) ) ∩ M ( v ( k − ) of a ( m ) exists that differs from a ( m ) in the variable that is tested at v ( k − . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 Figure 7. The block structure from m ′ to µ . so that also a ′ ∈ M ( v ( k )1 ) ∪ M ( v ( k )2 ) . Since M ( v ( k )1 ) ∪ M ( v ( k )2 ) = M ( v ( µ )1 ) ∪ M ( v ( µ )2 ) and t ( µ +1)12 = by Lemma 6.1, there is an h-neighbor a ′′ ∈ Ω ( a ( m ) ) ∩ M ( v ( µ +1)1 ) ⊆ H of a ( m ) such that P ( a ′′ ) = 1 by m -condition 4 since M ( v ( µ +1)1 ) ⊆ M ( v ( m )1 ) ∪ M ( v ( m )2 ) , which is a contradiction. Thus t ( k )33 = 1 for k = m ′ + 1 , . . . , µ . ⊓⊔ Lemma 6.2 together with assumption (34) gives p ( m ′ )1 + p ( m ′ )2 = p ( µ )1 + p ( µ )2 , (38) p ( m ′ )3 = p ( µ )3 < (39)which verifies m ′ -condition 3 for the first block level m ′ . Note that inequality (39) ensures m ′ ≥ due to p (2)3 ≥ . Finally, the following lemma shows m ′ -condition 1. Lemma 6.3. t ( m ′ )11 = t ( m ′ )21 = ( m ′ -condition 1). Proof:
Obviously, t ( m ′ )31 = 0 since otherwise p ( m ′ )3 > which contradicts (39). For t ( m ′ )11 = 1 or t ( m ′ )21 = 1 we obtain t ( m ′ )12 + t ( m ′ )22 > and t ( m ′ )13 + t ( m ′ )23 > by the normalization of P . Thus either a ( m ) ∈ M ( v ( m ′ − ) ⊆ M ( v ( m ′ )1 ) ∪ M ( v ( m ′ )2 ) or an h-neighbor a ′ ∈ Ω ( a ( m ) ) ∩ ( M ( v ( m ′ − ) ∪ M ( v ( m ′ − )) of a ( m ) exists such that a ′ ∈ M ( v ( m ′ )1 ) ∪ M ( v ( m ′ )2 ) . Since M ( v ( m ′ )1 ) ∪ M ( v ( m ′ )2 ) = M ( v ( µ )1 ) ∪ M ( v ( µ )2 ) and t ( µ +1)12 = by Lemma 6.1, there is an h-neighbor a ′′ ∈ Ω ( a ( m ) ) ∩ M ( v ( µ +1)1 ) ⊆ H of a ( m ) suchthat P ( a ′′ ) = 1 which is a contradiction. The last possibility t ( m ′ )11 = t ( m ′ )21 = follows. ⊓⊔ p ( ω +1)1 in Terms of p ( m ′ )1 + p ( m ′ )2 In Paragraph 5.3, we have upper bounded p ( m )1 + p ( m )2 at the last block level m in terms of p ( ω +1)1 provided that ω < m . In this paragraph, we will extend this estimate by upper bounding p ( ω +1)1 (or J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 p ( m )1 + p ( m )2 for ω = m ) in terms of p ( m ′ )1 + p ( m ′ )2 from the first block level m ′ . Putting these twobounds together, we will obtain a recursive formula for an upper bound on p ( m )1 + p ( m )2 in terms of p ( m ′ )1 + p ( m ′ )2 which will be used in Section 7 for verifying condition (16).We first resolve the case of the empty block when ν = m = ω , t ( m )33 = 0 , t ( m )13 + t ( m )23 = 1 ,and t ( m )32 = 1 (see Figure 5). It follows from Corollary 5.1 and Lemma 6.2 (see Figures 4 and 7,respectively) that M ( v ( m ′ )1 ) ∪ M ( v ( m ′ )2 ) = M ( v ( m )1 ) ∪ M ( v ( m )2 ) which ensures m ′ -condition 4 ( m ′ -conditions 1–3 have already been checked in Paragraph 6.2) and p ( m ′ )1 + p ( m ′ )2 = p ( m )1 + p ( m )2 . Hence,the empty block can be skipped in our analysis by replacing m ′ with m , and we will further consideronly the non-empty blocks.It follows from the definition of partition class R (see Figure 4) and Lemma 6.1 that p ( ν )1 = p ( µ )1 + p ( µ )2 (cid:18) − | R | (cid:19) for ν < m . (40)For ν = m when ν ′ = ν − , we know t ( m )33 > because we assume a non-empty block, and hence,either t ( m )12 = t ( m )32 = and t ( m )33 = 1 , or t ( m )13 = t ( m )33 = and t ( m )32 = 1 (see Figure 5) by the definitionof ν , Lemma 4.2.ii, and m -conditions 1 and 2, which also ensures ω = m in both cases. Thus, p ( m )1 + p ( m )2 = p ( µ )1 + p ( µ )2 (cid:18) − | R | +1 (cid:19) for ν = m = ω (41)according to (23). For ν = m − we know t ( m )12 = t ( m )13 = 0 leading to t ( m )32 = t ( m )33 = 1 , for which ω = m can be assumed without loss of generality .Further assume ν < m − , while the resulting formula for ν < m will also be verified for the caseof ν = m − (when ω = m ) below in (43). We know by the definition of ν that t ( ν +1)12 = t ( ν +1)13 = 0 ,which excludes t ( ν +1)32 = 1 and t ( ν +1)33 = 1 since P is normalized. First consider the case of ω > ν excluding ω = ν − ≥ µ and ω = ν for now, cf. (24). Then the double-edge path from the definitionof ω passes through a double edge from v ∈ { v ( ν )2 , v ( ν )3 } to v ( ν +1)2 , while the two edges from theother node v ′ ∈ { v ( ν )2 , v ( ν )3 } \ { v } lead to v ( ν +1)2 and v ( ν +1)3 , respectively, as depicted in Figure 8.For ℓ = ν + 2 , . . . , ω , we have either t ( ℓ )22 = 1 implying t ( ℓ )33 = if ℓ < m , or t ( ℓ )32 = 1 if ℓ = m .Moreover, t ( ω +1)12 = 0 for ω < m by Corollary 5.1.5. Hence, p ( ν +1)3 = p ( µ )2 / | R | +1 (cf. Figure 4and Lemma 6.1) upper bounds the fraction of all the inputs whose computational path traverses nodes v ′ , v ( ν +1)3 , v ( ν +2)3 , . . . , v ( ℓ )3 , v ( ℓ +1)1 for some ν + 1 ≤ ℓ ≤ min( ω, m − . It follows that p ( ω +1)1 ≤ p ( ν )1 + p ( µ )2 | R | +1 for ω < m (42)which is even valid for any max( ν − , µ ) ≤ ω < m since obviously p ( ω +1)1 = p ( ν )1 for ω = ν − ≥ µ as well as for ω = ν < m , while p ( m )1 + p ( m )2 ≤ p ( ν )1 + p ( µ )2 | R | +1 for ω = m (43) . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 Figure 8. The block structure from ν < ω to ω + 1 (or to m if ω = m ). which also holds for ν = m − because p ( m )1 + p ( m )2 = p ( ν )1 in this case.In addition, we will prove the following lemma: Lemma 6.4. p ( µ )1 + p ( µ )2 ≤ p ( µ )2 . (44) Proof:
First consider the case of µ > m ′ . Clearly, t ( µ )21 > follows from the definition of µ ′ for µ = µ ′ ,while for µ = µ ′ − , the case of t ( µ )21 = 0 translates to original t ( µ )11 = 0 (before v ( µ )1 and v ( µ )2 were swapped) which contradicts the normalization of P by Lemma 6.2. Hence, we have p ( µ )1 + p ( µ )2 = p ( µ − + p ( µ − ≤ p ( µ − ≤ p ( µ )2 according to Lemma 6.2. For µ = m ′ , on the otherhand, we will distinguish three cases. For t ( µ )32 = t ( µ )33 = 1 , we know p ( µ )1 = p ( µ )2 by Lemma 6.3,which implies (44). For t ( µ )12 + t ( µ )22 = , we have t ( µ )33 = 1 by Lemma 4.2.ii, which gives p ( µ )1 ≤ p ( µ − + p ( µ − < + = since p ( µ − < by m ′ -conditions 2 and (39). In addition, p ( µ − < p ( µ )3 < implying p ( µ − + p ( µ − < which means p ( µ )2 ≥ p ( µ − > by Lemma 6.3.It follows that p ( µ )1 < < < p ( µ )2 which gives (44). Similarly for t ( µ )13 + t ( µ )23 > , we have t ( µ )32 = 1 implying > p ( µ )3 ≥ p ( µ − ≥ p ( µ − due to (39), and hence p ( µ − > which ensures p ( µ )2 > byLemma 6.3, while p ( µ )1 ≤ p ( µ − + p ( µ − < + = < < p ( µ )2 which completes the proof ofthe lemma. ⊓⊔ For ν < m , equation (40) is plugged into (42) if ω < m or into (43) if ω = m , while equation(41) is considered for ν = m (implying ω = m ). Then Lemma 6.4 and equation (38) are employed, J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 which results in p ( ω +1)1 ≤ p ( µ )1 + p ( µ )2 (cid:18) − | R | (cid:19) + p ( µ )2 | R | +1 = p ( µ )1 + p ( µ )2 (cid:18) − | R | +1 (cid:19) ≤ (cid:16) p ( m ′ )1 + p ( m ′ )2 (cid:17) (cid:18) − | R | +3 (cid:19) for ω < m , (45) p ( m )1 + p ( m )2 ≤ (cid:16) p ( m ′ )1 + p ( m ′ )2 (cid:17) (cid:18) − | R | +3 (cid:19) for ω = m . (46)Formula (45) can further be plugged into (33) giving p ( m )1 + p ( m )2 ≤ − (cid:18) − (cid:16) p ( m ′ )1 + p ( m ′ )2 (cid:17) (cid:18) − | R | +3 (cid:19)(cid:19) q Y j =1 (cid:18) − | Q j | (cid:19) (47)which is even valid for ω = m (i.e. q = 0 ) since equation (47) coincides with (46) in this case.
7. The Recursion
In the previous Sections 4–6, we have analyzed the structure of the block of P from level m ′ through m (see Figure 2). We will now employ this block analysis recursively so that m = m r is replaced by m ′ = m r +1 . For this purpose, we introduce additional index b = 1 , . . . , r to the underlying objectsin order to differentiate among respective blocks. For example, the sets R, Q , . . . , Q q , defined inSection 5, corresponding to the b th block are denoted as R b , Q b , . . . , Q bq b , respectively.It follows from the definition of partition class in Paragraph 5.1 that, for any b > , the nodeslabeled with the variables whose indices are in R b are connected with the nodes corresponding to R b − through a computational path which traverses nodes v ( ν ′ b )1 , v ( ν ′ b +1)1 , . . . , v ( m b − since ν ′ b ≤ m b − according to (23). Hence, sets R , . . . , R r are pair-wise disjoint because P is read-once, and thus theycreate a partition. In particular, we will proceed by induction on r , starting with r = 0 and m = d . In the inductionstep for r + 1 , we assume that the four m r -conditions from Paragraph 4.1 are met for the last level m = m r of the ( r + 1) st block (see Paragraph 4.2 for r = 0 ), and let the assumption (34) be satisfiedfor the previous blocks, that is, p ( µ b )3 < (48)for every b = 1 , . . . , r . In addition, assume − Π r < δ = min (cid:18) ε − ε ′ , ε − (cid:19) < (49) . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 where ε > and ε ′ < ε are the parameters of Theorem 4.1 and denote Π k = k Y b =1 π b , π b = q b Y j =1 (cid:18) − | Q bj | (cid:19) , (50) ̺ k = k Y b =1 α b , α b = (cid:18) − | R b | +3 (cid:19) (51)for k = 1 , . . . , r , ̺ = Π = 1 , and π b = 1 for q b = 0 . It follows from (50) and (49) that π b ≥ Π r > − δ > (52)which verifies assumption (35) for every b = 1 , . . . , r . Hence, we can employ recursive inequality(47) from Section 6 which is rewritten as p b − ≤ − (1 − p b α b ) π b = 1 − π b + p b α b π b (53)for b = 1 , . . . , r where notation p b = p ( m b )1 + p ( m b )2 is introduced. Starting with p = p ( d )1 + p ( d )2 ≥ ε (54)which follows from (20), recurrence (53) can be solved as ε ≤ r X k =1 (1 − π k ) k − Y b =1 α b π b + p r r Y b =1 α b π b < r X k =1 (1 − π k )Π k − + p r ̺ r Π r = 1 − Π r + p r ̺ r Π r . (55)In addition, ̺ r > p r ̺ r Π r > ε − δ ≥ ε ′ (56)follows from (55) and (49). Throughout this paragraph, we will consider the case when − Π r +1 < δ (57)(cf. assumption (49)), while the case complementary to (57), which concludes the recursion, will beresolved below in Section 8. Assuming condition (57), we will prove that inductive assumptions (48)and (49) are met for r replaced with r + 1 together with the four m r +1 -conditions for the first level m r +1 of the ( r + 1) st block so that we can further proceed in the recursion.By analogy to (52), inequality (57) implies π r +1 > − δ > . (58) J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3
For ω r +1 < m r , we know p r ≤ − (cid:16) p ( ω r +1 +1)2 + p ( ω r +1 +1)3 (cid:17) π r +1 (59)according to (33), and p ( ω r +1 +1)2 + p ( ω r +1 +1)3 ≥ p ( µ r +1 )3 (60)by the definition of ω r +1 and Lemma 4.2.iii–iv (for k = ω r +1 ), which altogether gives ε < − Π r + (cid:16) − p ( µ r +1 )3 π r +1 (cid:17) ̺ r Π r (61)according to (55). Hence, ε − δ < (cid:16) − p ( µ r +1 )3 π r +1 (cid:17) ̺ r Π r < − p ( µ r +1 )3 π r +1 (62)follows from (49), which gives p ( µ r +1 )3 < − ε + δ − δ ≤ for ω r +1 < m r (63)by (58) and (49). Inequality (63) is even valid for ω r +1 = m r since p ( µ r +1 )3 ≤ p ( m r )3 < for ω r +1 = m r (64)according to m r -condition 3. Therefore, assumptions (34) and (35) of the analysis in Section 6 are alsomet for the ( r + 1) st block according to (63)–(64) and (58), respectively, which justifies recurrenceinequality (53) for b = r + 1 leading to the solution ε < − Π r +1 + p r +1 ̺ r +1 Π r +1 (65)by analogy to (55) where r is replaced with r + 1 . Similarly to (56), we obtain ̺ r +1 > ε ′ (66)by combining (65) with (57). Thus, inductive assumptions (48) and (49) are valid for r replaced by r + 1 according to (63)–(64) and (57), respectively.In order to proceed in the next induction step, we still need to verify the four m r +1 -conditions fromParagraph 4.1 for level m r +1 . In Paragraph 6.2, m r +1 -conditions 1–3 have been proven, and thus, itsuffices to validate m r +1 -condition 4. For this purpose, we exploit the fact that A is ε ′ -rich after weshow corresponding condition (16) for partition { R , . . . , R r +1 } of I = S r +1 b =1 R b . In particular, ε ′ < ̺ r +1 < r +1 Y b =1 (cid:18) − | R b | (cid:19) (67) . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 follows from (66) since for any ≤ b ≤ r + 1 , (cid:18) − | R b | +3 (cid:19) < − | R b | (68)for | R b | ≥ because f ( x ) = ln(1 − x ) / ln(1 − x ) is a decreasing function for x = 2 | R b | ≥ , and f (2) < . This provides required a ( m r +1 ) ∈ A such that for every b = 1 , . . . , r + 1 there exists i ∈ R b that meets a ( m r +1 ) i = c i according to (17) for Q = ∅ . Obviously, the computational path forthis a ( m r +1 ) ends up in sink v ( d )1 or v ( d )2 labeled with 1 when we put a ( m r +1 ) at node v ( m r +1 )1 or v ( m r +1 )2 by the definition of R b , c i and by the structure of branching program P (see Figure 4), which proves m r +1 -condition 4. Thus, the inductive assumptions are met for r + 1 and we can proceed recursivelyfor r replaced with r + 1 etc. until condition (57) is broken.
8. The End of Recursion
In this section, we will consider the case of − Π r +1 ≥ δ (69)complementary to (57), which concludes the recursion from Section 7 as follows. Suppose | Q bj | > log n for every b = 1 , . . . , r + 1 and j = 1 , . . . , q b , then we would have Π r +1 = r +1 Y b =1 q b Y j =1 (cid:18) − | Q bj | (cid:19) ≥ (cid:18) − log n (cid:19) n log n (70) > − n · n log n = 1 − n , (71)which breaks (69) for sufficiently large n . Hence, there must be ≤ b ∗ ≤ r + 1 and ≤ j ∗ ≤ q b ∗ such that | Q b ∗ j ∗ | ≤ log n , and we denote Q = Q b ∗ j ∗ . Clearly, Q ∩ R b = ∅ for b = 1 , . . . , b ∗ − dueto P is read-once while it may happen that Q ∩ R b ∗ − = ∅ for j ∗ = 1 , κ b ∗ = σ b ∗ = m b ∗ − , and t ( m b ∗− )23 = 0 . Thus, let r ∗ be the maximum of b ∗ − and b ∗ − such that Q ∩ R r ∗ = ∅ . We will againemploy the fact that A is ε ′ -rich. First condition (16) for partition { R , . . . , R r ∗ } of I = S r ∗ b =1 R b isverified as r ∗ Y b =1 (cid:18) − | R b | (cid:19) > ̺ r > ε ′ (72)according to (68) and (56). This provides a ∗ ∈ A such that a ∗ i = c Qi for every i ∈ Q and at the sametime, for every b = 1 , . . . , r ∗ there exists i ∈ R b that meets a ∗ i = c R b i according to (17). Lemma 8.1.
Denote λ = λ b ∗ j ∗ . There are two generalized ‘switching’ paths (cf. Lemma 4.2.ii)starting from v ( k )2 and v ( k )3 , respectively, at level k satisfying < max( λ − , µ b ∗ ) ≤ k < λ , whichmay lead to v ( λ )3 in addition to v ( λ − or v ( λ )1 . J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3
Proof:
For the notation simplicity, we will omit the block index b ∗ in this proof. We know ω < m due to q > , and λ > µ from Paragraph 5.2. Consider first the case when t ( λ )12 = t ( λ )13 = 0 . Obviously, t ( λ )22 < follows from the definition of λ for λ > ω and from the definition of ω for λ = ω , whichgives t ( λ )22 = t ( λ )32 = and t ( λ )23 > by the normalization of P . For t ( λ )33 = , we obtain two switchingpaths v ( λ − , v ( λ )3 and v ( λ − , v ( λ )3 . Thus assume t ( λ )33 = 0 which ensures t ( λ )23 = 1 and λ > µ + 1 since λ = µ + 1 would give ω > λ . Consider first the case when t ( λ − = t ( λ − = 0 , which implies t ( λ − > and t ( λ − > by t ( λ − = 1 and the normalization of P , providing two switching paths v ( λ − , v ( λ − , v ( λ )3 and v ( λ − , v ( λ − , v ( λ )3 . Two switching paths v ( λ − , v ( λ − and v ( λ − , v ( λ − are also guaranteed when t ( λ − > and t ( λ − > appear simultaneously. For t ( λ − = 0 and t ( λ − > , we have t ( λ − > by the normalization of P , which together with t ( λ )32 = producestwo switching paths v ( λ − , v ( λ − , v ( λ )3 and v ( λ − , v ( λ − . For t ( λ − > and t ( λ − = 0 , the case of t ( λ − > ensures two switching paths v ( λ − , v ( λ − and v ( λ − , v ( λ − , v ( λ )3 , while for t ( λ − = 0 we obtain t ( λ − = t ( λ − = and t ( λ − = 1 , which implies λ = ν + 1 and ω > λ by Lemma 4.2.iiicontradicting the definition of λ ≥ ω ≥ ν − . This completes the argument for t ( λ )12 = t ( λ )13 = 0 .The case of t ( λ )13 > and t ( λ )12 > produces two switching paths v ( λ − , v ( λ )1 and v ( λ − , v ( λ )1 .Further consider the case when t ( λ )13 > and t ( λ )12 = 0 . Obviously, t ( λ )22 < follows from the definitionof λ for λ > ω and from the definition of ω for λ = ω . Hence, t ( λ )32 > which provides two switchingpaths v ( λ − , v ( λ )3 and v ( λ − , v ( λ )1 . Finally, consider the case when t ( λ )12 > and t ( λ )13 = 0 , for which t ( λ )33 > generates two switching paths v ( λ − , v ( λ )1 and v ( λ − , v ( λ )3 , while for t ( λ )33 = 0 we obtain t ( λ )32 = and t ( λ )23 = 1 , which implies λ = ν and ω > λ by Lemma 4.2.iii contradicting the definitionof λ ≥ ω ≥ ν − . ⊓⊔ By a similar argument to Lemma 4.2.ii, Lemma 8.1 gives an h-neighbor a ′ ∈ Ω ( a ∗ ) ⊆ H of a ∗ ∈ A such that a ′ ∈ M ( v ( λ )1 ) ∪ M ( v ( λ )3 ) . Thus, either a ′ ∈ M ( v ( λ )1 ) ⊆ M ( v ( m b ∗− )1 ) ∪ M ( v ( m b ∗− )2 ) or a ′ ∈ M ( v ( λ )3 ) which implies a ′ ∈ M ( v ( κ b ∗ j ∗ )1 ) ⊆ M ( v ( m b ∗− )1 ) ∪ M ( v ( m b ∗− )2 ) since a ′ i = a ∗ i = c Qi for every i ∈ Q according to (17) (see Figure 2 and 6). Note that M ( v ( κ b ∗ j ∗ )1 ) = M ( v ( m b ∗− )1 ) for r ∗ = b ∗ − . Hence, P ( a ′ ) = 1 because for every b = 1 , . . . , r ∗ there exists i ∈ R b that meets a ′ i = a ∗ i = c R b i due to (17) (see Figure 2 and 4). This completes the proof of Theorem 4.1. ⊓⊔
9. The Richness of Almost k -wise Independent Sets In order to achieve an explicit polynomial time construction of a hitting set for read-once branchingprograms of width 3 we will combine Theorem 4.1 with the result due to Alon et al. [16] who providedsimple efficient constructions of almost k -wise independent sets. In particular, for β > and k = O (log n ) it is possible to construct a ( k, β ) -wise independent set A ⊆ { , } ∗ in time polynomial in nβ such that for sufficiently large n and any index set S ⊆ { , . . . , n } of size | S | ≤ k , the probabilitydistribution on S is almost uniform, i.e. the probability that a given c ∈ { , } n coincides with the . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 strings from A n = A ∩ { , } n on the bit locations from S can be approximated as (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) A Sn ( c ) (cid:12)(cid:12) |A n | − | S | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ β , (73)where A Sn ( c ) = { a ∈ A n | ( ∀ i ∈ S ) a i = c i } . We will prove that, for suitable k , any almost k -wiseindependent set is ε -rich. It follows that almost O (log n ) -wise independent sets are hitting sets for theclass of read-once conjunctions of DNF and CNF (cf. [15]). Theorem 9.1.
Let ε > , C be the least odd integer greater than ( ε ln ε ) , and < β < n C +3 . Thenany ( ⌈ ( C + 2) log n ⌉ , β ) -wise independent set is ε -rich. Proof:
Let
A ⊆ { , } ∗ be a ( ⌈ ( C + 2) log n ⌉ , β ) -wise independent set. We will show that A is ε -rich.Assume { R , . . . , R r } is a partition of index set I ⊆ { , . . . , n } satisfying condition (16), and Q ⊆{ , . . . , n } \ I such that | Q | ≤ log n . In order to show for a given c ∈ { , } n that there is a ∈ A n that meets (17) for Q and partition { R , . . . , R r } , we will prove that the probability p = p ( A n ) = (cid:12)(cid:12)(cid:12) A Qn ( c ) \ S rj =1 A R j n ( c ) (cid:12)(cid:12)(cid:12) |A n | (74)of the event that a ∈ A n chosen uniformly at random satisfies a ∈ A Qn ( c ) and a
6∈ A R j n ( c ) for every j = 1 , . . . , r , is strictly positive .The main idea of the proof lies in lower bounding the probability (74). By using the assumptionthat A is almost O (log n ) -wise independent this probability can be approximated by the probabilitythat any a ∈ { , } n (not necessarily in A n ) satisfies (17) which can be expressed and lower boundedas p ( { , } n ) = 12 | Q | r Y j =1 (cid:18) − | R j | (cid:19) ≥ εn > (75)according to (16) and | Q | ≤ log n . In particular, we briefly comment on the main steps of the proofwhich are schematically depicted in Figure 9 including references to corresponding sections, lemmas,and equations. In Section 10, we will first modify the partition classes R j so that their cardinalitiesare at most logarithmic whereas the classes of small constant cardinalities are merged with Q andalso c is adjusted correspondingly. Lemma 10.1 then ensures that the probability p from (74) islower bounded when using these modified classes. Furthermore, Bonferroni inequality (the inclusion-exclusion principle) and the assumption concerning the almost k -wise independence are employed inSection 11 where also the classes of the same cardinality are grouped. In Section 12, we will furtherreduce the underlying lower bound on p only to a sum over frequent cardinalities of partition classesto which Taylor’s theorem is applied in Section 13, whereas a corresponding Lagrange remainder isbounded using the assumption on constant C . J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3
Modifications of Partition Classes (Section 10) • superlogarithmic cardinalities: R ′ j ⊆ R j so that | R ′ j | ≤ log n (76) • small constant cardinalities: R ≤ = S | R ′ j |≤ σ R ′ j where σ is a constant (82) & (85) −→ Q ′ = Q ∪ R ≤ (89), c ′ i = 1 − c i for i ∈ R ≤ (92)Lemma 10.1: p ≥ (cid:12)(cid:12)(cid:12)(cid:12) A Q ′ n ( c ′ ) \ S r ′ j =1 A R ′ jn ( c ′ ) (cid:12)(cid:12)(cid:12)(cid:12) |A n | (93) Bonferroni inequality p ≥ C ′ X k =0 ( − k X ≤ j
10. Modifications of Partition Classes
We properly modify the underlying partition classes in order to further upper bound their cardinalitiesby the logarithmic function so that the assumption concerning almost ⌈ ( C + 2) log n ⌉ -wise inde-pendence of A can be applied in the following Section 11. Thus, we confine ourselves to at mostlogarithmic-size arbitrary subsets R ′ j of partition classes R j , that is R ′ j ( = R j if | R j | ≤ log n ⊂ R j so that | R ′ j | = ⌊ log n ⌋ otherwise , (76)which ensures R ′ j ⊆ R j and | R ′ j | ≤ log n for every j = 1 , . . . , r . For these new classes, assumption(16) can be rewritten as r Y j =1 (cid:18) − | R ′ j | (cid:19) > (cid:18) − log n (cid:19) n log n Y | R j |≤ log n (cid:18) − | R j | (cid:19) > (cid:18) − n · n log n (cid:19) ε = (cid:18) − n (cid:19) ε = ε ′ , (77)where ε ′ > is arbitrarily close to ε for sufficiently large n .Denote by { s , s , . . . , s m } = {| R ′ | , . . . , | R ′ r |} the set of all cardinalities ≤ s i ≤ log n ofclasses R ′ , . . . , R ′ r , and for every i = 1 , . . . , m , let r i = |{ j | | R ′ j | = s i }| be the number of classes R ′ j having cardinality s i , that is, r = P mi =1 r i . Furthermore, we define t i = r i s i > for i = 1 , . . . , m . (78)It follows from (77) and (78) that < ε ′ < r Y j =1 (cid:18) − | R ′ j | (cid:19) = m Y i =1 (cid:18) − s i (cid:19) r i = m Y i =1 (cid:18) − s i (cid:19) si ! t i < e − P mi =1 t i (79)implying m X i =1 t i < ln 1 ε ′ . (80)Moreover, we define constants ̺ = C − (cid:16) − ε ′ ε ′ ) (cid:17) C > C ≥ , (81) σ = log (cid:18) ̺ (1 + ε ′ ) ε ′ (cid:19) (82) J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 which are used for sorting the cardinalities s , . . . , s m so that r i > ̺ and s i > σ for i = 1 , . . . , m ′′ (83) r i ≤ ̺ and s i > σ for i = m ′′ + 1 , . . . , m ′ (84) s i ≤ σ for i = m ′ + 1 , . . . , m . (85)We will further confine ourselves to the first m ′ ≥ cardinalities satisfying s i > σ for i = 1 , . . . , m ′ .Without loss of generality, we can also sort the corresponding partition classes so that | R ′ j | > σ for j = 1 , . . . , r ′ (86) | R ′ j | ≤ σ for j = r ′ + 1 , . . . , r , (87)which implies r ′ = m ′ X i =1 r i = m ′ X i =1 t i s i > ̺ (1 + ε ′ ) ε ′ m ′ X i =1 t i (88)according to (78), (83)–(84), and (82). We include the remaining constant-size classes R ′ j for j = r ′ + 1 , . . . , r into Q , that is, Q ′ = Q ∪ r [ j = r ′ +1 R ′ j (89)whose size can be upper bounded as | Q ′ | ≤ log n + m X i = m ′ +1 r i log (cid:18) ̺ (1 + ε ′ ) ε ′ (cid:19) < n (90)for sufficiently large n , since m X i = m ′ +1 r i = m X i = m ′ +1 t i s i < ̺ (1 + ε ′ ) ε ′ ln 1 ε ′ (91)according to (78), (80), (85), and (82). This completes the definition of new classes Q ′ , R ′ , . . . , R ′ r ′ .In addition, we define c ′ ∈ { , } n that differs from c exactly on the constant number of bit locationsfrom R ′ r ′ +1 , . . . , R ′ r , e.g. c ′ i = ( − c i if i ∈ S rj = r ′ +1 R ′ j c i otherwise. (92)The modified Q ′ , R ′ , . . . , R ′ r ′ and c ′ are used in the following lemma for lower bounding the proba-bility (74). Lemma 10.1. p ≥ (cid:12)(cid:12)(cid:12)(cid:12) A Q ′ n ( c ′ ) \ S r ′ j =1 A R ′ j n ( c ′ ) (cid:12)(cid:12)(cid:12)(cid:12) |A n | = (cid:12)(cid:12)(cid:12) A Q ′ n ( c ′ ) (cid:12)(cid:12)(cid:12) |A n | − (cid:12)(cid:12)(cid:12)(cid:12)S r ′ j =1 A R ′ j ∪ Q ′ n ( c ′ ) (cid:12)(cid:12)(cid:12)(cid:12) |A n | . (93) . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 Proof:
For verifying the lower bound in (93) it suffices to show that A Q ′ n ( c ′ ) \ r ′ [ j =1 A R ′ j n ( c ′ ) ⊆ A Qn ( c ) \ r [ j =1 A R j n ( c ) (94)according to (74). Assume a ∈ A Q ′ n ( c ′ ) \ S r ′ j =1 A R ′ j n ( c ′ ) , which means a ∈ A Q ′ n ( c ′ ) ⊆ A Qn ( c ′ ) = A Qn ( c ) and a
6∈ A R ′ j n ( c ′ ) = A R ′ j n ( c ) ⊇ A R j n ( c ) for every j = 1 , . . . , r ′ by definitions (76), (89), (92),and the fact that S ⊆ S implies A S n ( c ) ⊆ A S n ( c ) . In addition, a ∈ A Q ′ n ( c ′ ) implies a
6∈ A R j n ( c ) for every j = r ′ + 1 , . . . , r according to (92), and hence, a ∈ A Qn ( c ) \ S rj =1 A R j n ( c ) . This completesthe proof of the lower bound, while the equality in (93) follows from A R ′ j ∪ Q ′ n ( c ′ ) ⊆ A Q ′ n ( c ′ ) for every j = 1 , . . . , r ′ . ⊓⊔
11. Almost k -Wise Independence Furthermore, we will upper bound the probability of the finite union of events appearing in formula(93) by using Bonferroni inequality for constant number C ′ = min( C, r ′ ) of terms, which gives p ≥ (cid:12)(cid:12)(cid:12) A Q ′ n ( c ′ ) (cid:12)(cid:12)(cid:12) |A n | − C ′ X k =1 ( − k +1 X ≤ j
For ≤ k ≤ C ′ , X ≤ j By grouping the classes of the same cardinality together, the left-hand side of inequality (101) can berewritten as X ≤ j 12. Frequent Cardinalities We sort out the terms with frequent cardinalities (83) from the sum in formula (104), that is, p > n C ′ X k =0 ( − k X k + ··· + k m ′′ = k ≤ k ≤ r ,..., ≤ k m ′′ ≤ r m ′′ m ′′ Y i =1 t k i i k i ! k i − Y j =1 (cid:18) − jr i (cid:19) − T − ε ′ , (105)where the inner sum in (105) equals zero for k > r ′′ = P m ′′ i =1 r i , and T = C ′ X k =0 ( − k +1 X k + ··· + k m ′ = k ≤ k ≤ r ,..., ≤ k m ′ ≤ r m ′ ( ∃ m ′′ +1 ≤ ℓ ≤ m ′ ) k ℓ > m ′ Y i =1 t k i i k i ! k i − Y j =1 (cid:18) − jr i (cid:19) (106)sums up the terms including rare cardinalities (84). In addition, we know ≥ m ′′ Y i =1 k i − Y j =1 (cid:18) − jr i (cid:19) ≥ (107) m ′′ Y i =1 (cid:18) − C − ̺ (cid:19) k i − > (cid:18) − C̺ (cid:19) C = 1 − ε ′ ε ′ ) (108)according to (83), (81), and k i ≤ k = P m ′′ i =1 k i ≤ C ′ ≤ C < ̺ . The upper bound (107) and lowerbound (108) on the underlying product are used to lower bound the negative terms of (105) for odd k and the positive ones for even k , respectively, that is, p > n C ′ X k =0 ( − k X k + ··· + k m ′′ = k ≤ k ≤ r ,..., ≤ k m ′′ ≤ r m ′′ m ′′ Y i =1 t k i i k i ! − ε ′ ε ′ ) T − T − ε ′ (109)where T = C ′ X k =0 , , ,... X k + ··· + k m ′′ = k ≤ k ≤ r ,..., ≤ k m ′′ ≤ r m ′′ m ′′ Y i =1 t k i i k i ! . (110)The following lemma upper bounds the above-introduced terms T and T . Lemma 12.1. (i) T < ε ′ .(ii) T < ε ′ ε ′ . J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 Proof: (i) We can only take the terms of (106) for odd k = 1 , , , . . . into account since those for even k are nonpositive (e.g. the term for k = 0 equals zero because there is no m ′′ + 1 ≤ ℓ ≤ m ′ suchthat k ℓ > in this case). Thus, T ≤ C ′ X k =1 , , ,... X k + ··· + k m ′ = k ≤ k ≤ r ,..., ≤ k m ′ ≤ r m ′ ( ∃ m ′′ +1 ≤ ℓ ≤ m ′ ) k ℓ > r ℓ s ℓ k ℓ t k ℓ − ℓ ( k ℓ − m ′ Y i =1 i = ℓ t k i i k i ! ≤ ̺ σ C ′ X k =1 , , ,... X k + ··· + k m ′ = k ≤ k ≤ r ,..., ≤ k m ′ ≤ r m ′ ( ∃ m ′′ +1 ≤ ℓ ≤ m ′ ) k ℓ > t k ℓ − ℓ ( k ℓ − m ′ Y i =1 i = ℓ t k i i k i ! (111)according to (78) and (84). Formula (111) is rewritten by replacing indices k ℓ − and k − with k ℓ and k , respectively, which is further upper bounded by removing the upper bounds thatare set on indices k , . . . , k m ′ and by omitting the condition concerning the existence of specialindex ℓ , as follows: T ≤ ̺ σ C ′ − X k =0 , , ,... X k + ··· + k m ′ = kk ≥ ,...,k m ′ ≥ m ′ Y i =1 t k i i k i ! = ̺ σ C ′ − X k =0 , , ,... (cid:16)P m ′ i =1 t i (cid:17) k k ! , (112)where the multinomial theorem is employed. Notice that the sum on the right-hand side ofequation (112) represents the first few terms of Taylor series of the hyperbolic cosine at point P m ′ i =1 t i ≥ , which implies T < ̺ σ cosh m ′ X i =1 t i ! < ε ′ ε ′ ) · ε ′ + ε ′ ε ′ (113)according to (80) and (82) since the hyperbolic cosine is an increasing function for nonnegativearguments.(ii) Similarly as in the proof of (i), we apply the multinomial theorem (cf. (112)) and the Taylorseries of the hyperbolic cosine (cf. (113)) to (110), which gives T ≤ C ′ X k =0 , , ,... X k + ··· + k m ′′ = kk ≥ ,...,k m ′′ ≥ m ′′ Y i =1 t k i i k i ! = C ′ X k =0 , , ,... (cid:16)P m ′′ i =1 t i (cid:17) k k ! (114) ≤ cosh m ′′ X i =1 t i ! < ε ′ ε ′ . (115) . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 ⊓⊔ We plug the bounds from Lemma 12.1 into (109) and obtain p > n C ′ X k =0 ( − k X k + ··· + k m ′′ = k ≤ k ≤ r ,..., ≤ k m ′′ ≤ r m ′′ m ′′ Y i =1 t k i i k i ! − ε ′ . (116) 13. Taylor’s Theorem In order to apply the multinomial theorem again, we remove the upper bounds that are set on indicesin the inner sum of formula (116), that is, p > n C ′ X k =0 ( − k X k + ··· + k m ′′ = kk ≥ ,...,k m ′′ ≥ m ′′ Y i =1 t k i i k i ! − T − ε ′ , (117)which is corrected by introducing additional term T = C ′ X k =0 ( − k X k + ··· + k m ′′ = kk ≥ ,...,k m ′′ ≥ ∃ ≤ ℓ ≤ m ′′ ) k ℓ >r ℓ m ′′ Y i =1 t k i i k i ! . (118)Thus, inequality (117) can be further rewritten as p > n C ′ X k =0 (cid:16) − P m ′′ i =1 t i (cid:17) k k ! − T − ε ′ (119) = 1 n e − P m ′′ i =1 t i − R C ′ +1 − m ′′ X i =1 t i ! − T − ε ′ ! , (120)where Taylor’s theorem is employed for the exponential function at point − P m ′′ i =1 t i producing theLagrange remainder R C ′ +1 − m ′′ X i =1 t i ! = (cid:16) − P m ′′ i =1 t i (cid:17) C ′ +1 ( C ′ + 1)! e − ϑ P m ′′ i =1 t i < P m ′′ i =1 t i √ C ′ ! C ′ +1 (121)with parameter < ϑ < . Note that the upper bound in (121) assumes C ′ > , whereas for C ′ = r ′ = 0 implying m ′′ = m ′ = 0 , we know R (0) = 0 . This remainder and term T are upperbounded in the following lemma. J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 Lemma 13.1. (i) T < ε ′ .(ii) R C ′ +1 (cid:16) − P m ′′ i =1 t i (cid:17) < ε ′ . Proof: (i) We take only the summands of (118) for even k ≥ into account since the summands for odd k are not positive, while for k = 0 there is no ≤ ℓ ≤ m ′′ such that k ≥ k ℓ > r ℓ ≥ , whichgives T ≤ C ′ X k =2 , , ,... X k + ··· + k m ′′ = kk ≥ ,...,k m ′′ ≥ ∃ ≤ ℓ ≤ m ′′ ) k ℓ >r ℓ s ℓ r ℓ k ℓ t k ℓ − ℓ ( k ℓ − m ′′ Y i =1 i = ℓ t k i i k i ! ≤ σ C ′ X k =2 , , ,... X k + ··· + k m ′′ = kk ≥ ,...,k m ′′ ≥ ∃ ≤ ℓ ≤ m ′′ ) k ℓ >r ℓ t k ℓ − ℓ ( k ℓ − m ′′ Y i =1 i = ℓ t k i i k i ! (122)using (78) and (83). Formula (122) is rewritten by replacing indices k ℓ − and k − with k ℓ and k , respectively, which is further upper bounded by omitting the condition concerning theexistence of special index ℓ , as follows: T ≤ σ C ′ − X k =1 , , ,... X k + ··· + k m ′′ = kk ≥ ,...,k m ′′ ≥ m ′′ Y i =1 t k i i k i ! = 12 σ C ′ − X k =1 , , ,... (cid:16)P m ′′ i =1 t i (cid:17) k k ! , (123)where the multinomial theorem is employed. Notice that the sum on the right-hand side ofequation (123) represents the first few terms of Taylor series of the hyperbolic sine at point P m ′′ i =1 t i , which implies T ≤ σ sinh m ′′ X i =1 t i ! < ε ′ ̺ (1 + ε ′ ) · ε ′ − ε ′ < ε ′ (124)according to (80) and (82) since the hyperbolic sine is an increasing function.(ii) For C ′ = C ≥ , Lagrange remainder (121) can further be upper bounded as R C ′ +1 − m ′′ X i =1 t i ! < ln ε ′ √ C ! C +1 < (cid:18) ε ′ (cid:19) C +1 < ε ′ (125) . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 for sufficiently large n by using (80) and the definition of C , while for C ′ = r ′ < C , theunderlying upper bound R C ′ +1 − m ′′ X i =1 t i ! ≤ P m ′ i =1 t i ̺ (1+ ε ′ ) ε ′ ! r ′ +12 < ln ε ′ ̺ (1+ ε ′ ) ε ′ < ε ′ (126)can be obtained from (88) and (80). ⊓⊔ Finally, inequality (79) together with the upper bounds from Lemma 13.1 are plugged into (120),which leads to p > ε ′ n = ε n (cid:18) − n (cid:19) > (127)according to (77). Thus, we have proven that for any c ∈ { , } n the probability that there is a ∈ A n satisfying the conjunction (17) for Q and partition { R , . . . , R r } is strictly positive, which means such a does exist. This completes the proof that A is ε -rich. ⊓⊔ 14. Conclusion In the present paper, we have made an important step in the effort of constructing hitting set generatorsfor the model of read-once branching programs of bounded width. Such constructions have so far beenknown only in the case of width 2 and in very restricted cases of bounded width (e.g. regular obliviousread-once branching programs). We have now provided an explicit polynomial-time constructionof a hitting set for read-once branching programs of width 3 with acceptance probability ε > .Although this model seems to be relatively weak, the presented proof is far from being trivial. Inparticular, we have formulated a so-called richness condition which is independent of the notion ofbranching programs. This condition characterizes the hitting sets for read-once branching programsof width 3. We have shown that such a hitting set hits read-once conjunctions of DNF and CNF,which corresponds to the weak richness condition. On the other hand, the richness condition provesto be sufficient for a set extended with all strings within Hamming distance of 3 to be a hitting set forwidth-3 1-branching programs. In addition, we have proven for a suitable constant C that any almost ( C log n ) -wise independent set which can be constructed in polynomial time due to Alon et al. [16],satisfies this richness condition, which implies our result. It also follows that almost O (log n ) -wiseindependent sets are hitting sets for read-once conjunctions of DNF and CNF.From the point of view of derandomization of unrestricted models, our result still appears to beunsatisfactory but it is the best we know so far. The issue of whether our technique based on therichness condition can be extended to the case of width 4 or to bounded width represents an openproblem for further research. Another challenge for improving our result is to optimize parameter ε ,e.g. to achieve the result for ε ≤ n , which would be important for practical derandomizations. J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 References [1] Goldreich O, Wigderson A. Improved Derandomization of BPP Using a Hitting Set Generator. In:Hochbaum DS, Jansen K, Rolim JDP, Sinclair A (eds.), Randomization, Approximation, and Combi-natorial Algorithms and Techniques, Third International Workshop on Randomization and ApproximationTechniques in Computer Science, and Second International Workshop on Approximation Algorithms forCombinatorial Optimization Problems RANDOM-APPROX’99, Berkeley, CA, USA, August 8-11, 1999,Proceedings, volume 1671 of Lecture Notes in Computer Science . Springer, 1999 pp. 131–137. doi:10.1007/978-3-540-48413-4 \ J. Comput. Syst. Sci. , 1994. (2):149–167. doi:10.1016/S0022-0000(05)80043-1.[3] Wegener I. Branching Programs and Binary Decision Diagrams. SIAM, 2000. ISBN 0-89871-458-3.[4] Nisan N. Pseudorandom generators for space-bounded computation. Comb. , 1992. (4):449–461. doi:10.1007/BF01305237.[5] Meka R, Zuckerman D. Pseudorandom Generators for Polynomial Threshold Functions. SIAM J. Comput. ,2013. (3):1275–1301. doi:10.1137/100811623.[6] Vadhan SP. Pseudorandomness. Found. Trends Theor. Comput. Sci. , 2012. (1-3):1–336. doi:10.1561/0400000010.[7] Bogdanov A, Dvir Z, Verbin E, Yehudayoff A. Pseudorandomness for Width-2 Branching Programs. Theory Comput. , 2013. :283–293. doi:10.4086/toc.2013.v009a007.[8] Brody J, Verbin E. The Coin Problem and Pseudorandomness for Branching Programs. In: 51th AnnualIEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas,Nevada, USA. IEEE Computer Society, 2010 pp. 30–39. doi:10.1109/FOCS.2010.10.[9] De A. Pseudorandomness for Permutation and Regular Branching Programs. In: Proceedings of the 26thAnnual IEEE Conference on Computational Complexity, CCC 2011, San Jose, California, USA, June8-10, 2011. IEEE Computer Society, 2011 pp. 221–231. doi:10.1109/CCC.2011.23.[10] Fefferman B, Shaltiel R, Umans C, Viola E. On Beating the Hybrid Argument. Theory Comput. , 2013. :809–843. doi:10.4086/toc.2013.v009a026.[11] Beame P, Machmouchi W. Making Branching Programs Oblivious Requires Superlogarithmic Overhead.In: Proceedings of the 26th Annual IEEE Conference on Computational Complexity, CCC 2011, San Jose,California, USA, June 8-10, 2011. IEEE Computer Society, 2011 pp. 12–22. doi:10.1109/CCC.2011.35.[12] Braverman M, Rao A, Raz R, Yehudayoff A. Pseudorandom Generators for Regular Branching Programs. SIAM J. Comput. , 2014. (3):973–986. doi:10.1137/120875673.[13] Kouck´y M, Nimbhorkar P, Pudl´ak P. Pseudorandom generators for group products: extended abstract. In:Fortnow L, Vadhan SP (eds.), Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC2011, San Jose, CA, USA, 6-8 June 2011. ACM, 2011 pp. 263–272. doi:10.1145/1993636.1993672.[14] ˇS´ıma J, ˇZ´ak S. A Polynomial Time Constructible Hitting Set for Restricted 1-Branching Programs ofWidth 3. In: van Leeuwen J, Italiano GF, van der Hoek W, Meinel C, Sack H, Plasil F (eds.), SOFSEM2007: Theory and Practice of Computer Science, 33rd Conference on Current Trends in Theory and Prac-tice of Computer Science, Harrachov, Czech Republic, January 20-26, 2007, Proceedings, volume 4362 of Lecture Notes in Computer Science . Springer, 2007 pp. 522–531. doi:10.1007/978-3-540-69507-3 \ . ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 [15] De A, Etesami O, Trevisan L, Tulsiani M. Improved Pseudorandom Generators for Depth 2 Circuits. In:Serna MJ, Shaltiel R, Jansen K, Rolim JDP (eds.), Approximation, Randomization, and CombinatorialOptimization. Algorithms and Techniques, 13th International Workshop, APPROX 2010, and 14th Inter-national Workshop, RANDOM 2010, Barcelona, Spain, September 1-3, 2010. Proceedings, volume 6302of Lecture Notes in Computer Science . Springer, 2010 pp. 504–517. doi:10.1007/978-3-642-15369-3 \ Random Struct. Algorithms , 1992. (3):289–304. doi:10.1002/rsa.3240030308.[17] ˇS´ıma J, ˇZ´ak S. A Sufficient Condition for Sets Hitting the Class of Read-Once Branching Programsof Width 3 - (Extended Abstract). In: Bielikov´a M, Friedrich G, Gottlob G, Katzenbeisser S, Tur´an G(eds.), SOFSEM 2012: Theory and Practice of Computer Science - 38th Conference on Current Trendsin Theory and Practice of Computer Science, ˇSpindler˚uv Ml´yn, Czech Republic, January 21-27, 2012.Proceedings, volume 7147 of Lecture Notes in Computer Science . Springer, 2012 pp. 406–418. doi:10.1007/978-3-642-27660-6 \ k -Wise Independent Sets Establish Hitting Sets for Width-3 1-Branching Pro-grams. In: Kulikov AS, Vereshchagin NK (eds.), Computer Science - Theory and Applications - 6thInternational Computer Science Symposium in Russia, CSR 2011, St. Petersburg, Russia, June 14-18,2011. Proceedings, volume 6651 of Lecture Notes in Computer Science . Springer, 2011 pp. 120–133.doi:10.1007/978-3-642-20712-9 \ Electron.Colloquium Comput. Complex. , 2012. :83. URL http://eccc.hpi-web.de/report/2012/083 .[21] Forbes MA, Shpilka A. Quasipolynomial-Time Identity Testing of Non-commutative and Read-OnceOblivious Algebraic Branching Programs. In: 54th Annual IEEE Symposium on Foundations of ComputerScience, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA. IEEE Computer Society, 2013 pp. 243–252. doi:10.1109/FOCS.2013.34.[22] Reingold O, Steinke T, Vadhan SP. Pseudorandomness for Regular Branching Programs via Fourier Anal-ysis. In: Raghavendra P, Raskhodnikova S, Jansen K, Rolim JDP (eds.), Approximation, Randomization,and Combinatorial Optimization. Algorithms and Techniques - 16th International Workshop, APPROX2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA, August 21-23, 2013.Proceedings, volume 8096 of Lecture Notes in Computer Science . Springer, 2013 pp. 655–670. doi:10.1007/978-3-642-40328-6 \ Comput. Complex. , 2013. (4):727–769. doi:10.1007/s00037-012-0036-6.[25] Forbes MA, Saptharishi R, Shpilka A. Hitting sets for multilinear read-once algebraic branching programs,in any order. In: Shmoys DB (ed.), Symposium on Theory of Computing, STOC 2014, New York, NY,USA, May 31 - June 03, 2014. ACM, 2014 pp. 867–875. doi:10.1145/2591796.2591816. J. ˇS´ıma, ˇZ´ak / Hitting Set for Read-Once Branching Programs of Width 3 [26] Bazzi L, Nahas N. Small-Bias is Not Enough to Hit Read-Once CNF. Theory Comput. Syst. , 2017. (2):324–345. doi:10.1007/s00224-016-9680-6.[27] Murtagh J, Reingold O, Sidford A, Vadhan SP. Derandomization Beyond Connectivity: Undirected Lapla-cian Systems in Nearly Logarithmic Space. In: Umans C (ed.), 58th IEEE Annual Symposium on Foun-dations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017. IEEE ComputerSociety, 2017 pp. 801–812. doi:10.1109/FOCS.2017.79.[28] Servedio RA, Tan L. Deterministic Search for CNF Satisfying Assignments in Almost Polynomial Time.In: Umans C (ed.), 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017,Berkeley, CA, USA, October 15-17, 2017. IEEE Computer Society, 2017 pp. 813–823. doi:10.1109/FOCS.2017.80.[29] Steinke T, Vadhan SP, Wan A. Pseudorandomness and Fourier-Growth Bounds for Width-3 BranchingPrograms. Theory Comput. , 2017. (1):1–50. doi:10.4086/toc.2017.v013a012.[30] Ahmadinejad A, Kelner JA, Murtagh J, Peebles J, Sidford A, Vadhan SP. High-precision Esti-mation of Random Walks in Small Space. CoRR , 2019. abs/1912.04524 . , URL http://arxiv.org/abs/1912.04524 .[31] Doron D, Hatami P, Hoza WM. Near-Optimal Pseudorandom Generators for Constant-Depth Read-OnceFormulas. In: Shpilka A (ed.), 34th Computational Complexity Conference, CCC 2019, July 18-20, 2019,New Brunswick, NJ, USA, volume 137 of LIPIcs . Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik,2019 pp. 16:1–16:34. doi:10.4230/LIPIcs.CCC.2019.16.[32] Meka R, Reingold O, Tal A. Pseudorandom generators for width-3 branching programs. In: CharikarM, Cohen E (eds.), Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing,STOC 2019, Phoenix, AZ, USA, June 23-26, 2019. ACM, 2019 pp. 626–637. doi:10.1145/3313276.3316319.[33] Servedio RA, Tan L. Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas.In: Achlioptas D, V´egh LA (eds.), Approximation, Randomization, and Combinatorial Optimization.Algorithms and Techniques, APPROX/RANDOM 2019, September 20-22, 2019, Massachusetts Instituteof Technology, Cambridge, MA, USA, volume 145 of LIPIcs . Schloss Dagstuhl - Leibniz-Zentrum f¨urInformatik, 2019 pp. 45:1–45:23. doi:10.4230/LIPIcs.APPROX-RANDOM.2019.45.[34] Braverman M, Cohen G, Garg S. Pseudorandom Pseudo-distributions with Near-Optimal Error for Read-Once Branching Programs. SIAM J. Comput. , 2020. (5). doi:10.1137/18M1197734.[35] Cheng K, Hoza WM. Hitting Sets Give Two-Sided Derandomization of Small Space. In: Saraf S (ed.), 35thComputational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbr¨ucken, Germany (VirtualConference), volume 169 of LIPIcs . Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik, 2020 pp. 10:1–10:25. doi:10.4230/LIPIcs.CCC.2020.10.[36] Hoza WM, Zuckerman D. Simple Optimal Hitting Sets for Small-Success RL. SIAM J. Comput. , 2020.49