Fractional Covers of Hypergraphs with Bounded Multi-Intersection
Georg Gottlob, Matthias Lanzinger, Reinhard Pichler, Igor Razgon
FFractional Covers of Hypergraphs with BoundedMulti-Intersection
Georg Gottlob
University of Oxford, UKTU Wien, [email protected]
Matthias Lanzinger
TU Wien, [email protected]
Reinhard Pichler
TU Wien, [email protected]
Igor Razgon
Birkbeck University of London, [email protected]
Abstract
Fractional (hyper-)graph theory is concerned with the specific problems that arise when fractionalanalogues of otherwise integer-valued (hyper-)graph invariants are considered. The focus of this paperis on fractional edge covers of hypergraphs. Our main technical result generalizes and unifies previousconditions under which the size of the support of fractional edge covers is bounded independently ofthe size of the hypergraph itself. This allows us to extend previous tractability results for checkingif the fractional hypertree width of a given hypergraph is ≤ k for some constant k . We also showhow our results translate to fractional vertex covers. Mathematics of computing → Hypergraphs; Mathematics ofcomputing → Enumeration; Information systems → Relational database query languages
Keywords and phrases
Fractional graph theory, fractional edge cover, fractional hypertree width,bounded multi-intersection, fractional cover, fractional vertex cover
Funding
This work was supported by the Austrian Science Fund (FWF):P30930.
Georg Gottlob : Work supported by the Royal Society Research Professorship grant “RAISON DATA”(Project reference: RP\R1\201074).
Fractional (hyper-)graph theory [10] has evolved into a mature discipline in graph theory –building upon early research efforts that date back to the 1970s [2]. The crucial observationin this field is that many integer-valued (hyper-)graph invariants have a meaningful fractionalanalogue. Frequently, the integer-valued invariants are defined in terms of some integer linearprogram (ILP) and the fractional analogue is obtained by the fractional relaxation. Examplesof problems which have been studied in fractional (hyper-)graph theory comprise matchingproblems, coloring problems, covering problems, and many more.Covering problems come in two principal flavors, namely vertex covers and edge covers.We shall concentrate on edge covers in the first place, but we will later also mention how ourresults translate to vertex covers.
Fractional edge covers have attracted a lot of attentionin recent times. On the one hand, this is due to a deep connection between informationtheory and database theory. Indeed, the famous “AGM bound” – named after the authorsof [1] – establishes a tight upper bound on the number of result tuples of relational joins in a r X i v : . [ c s . D M ] J u l Fractional Covers of Hypergraphs with Bounded Multi-Intersection terms of fractional edge covers. On the other hand, fractional hypertree width (fhw) is todate the most general width-notion that allows one to define tractable fragments of solvingConstraint Satisfaction Problems (CSPs), answering Conjunctive Queries (CQs), and solvingthe Homomorphism Problem [8]. The fractional hypertree width of a hypergraph is definedin terms of the size of fractional edge covers of the bags in a tree decomposition.Fractional (hyper-)graph invariants give rise to new challenges that do not exist in theintegral case. Intuitively, if a fractional (hyper-)graph invariant is obtained by the relaxationof a linear program (LP), one would expect things to become easier, since we move from theintractable problem of ILPs to the tractable problem of LPs. However, also the oppositemay happen, namely that the fractional relaxation introduces complications not present inthe integral case. To illustrate such an effect, we first recall some basic definitions. (cid:73)
Definition 1. A hypergraph H is a tuple H = ( V, E ) , consisting of a set of vertices V and a set of hyperedges (or simply “edges”), which are non-empty subsets of V .Let γ be a function of the form γ : E → [0 , . Then the set of vertices “covered” by γ isdefined as B ( γ ) = { v ∈ V | P e ∈ E,v ∈ e γ ( e ) ≥ } . Intuitively, γ assigns weights to the edgesand a vertex v is covered if the total weight of the edges containing v is at least 1.A fractional edge cover of H is a function γ with V ⊆ B ( γ ) . An integral edge cover is obtained by restricting the function values of γ to { , } . The support of γ is defined as support( γ ) = { e ∈ E | γ ( e ) = 0 } . The weight of γ is defined as weight( γ ) = P e ∈ E γ ( e ) .The minimum weight of a fractional (resp. integral) edge cover of a hypergraph H is referredto as the fractional (resp. integral) edge cover number of H . The following example adapted from [5] illustrates which complications may arise if we movefrom the integral to the fractional case. (cid:73)
Example 2.
Consider the family ( H n ) n ≥ of hypergraphs with H n = ( V n , E n ) defined as V n = { v , v , . . . , v n } E n = { e , e , . . . , e n } with e = { v , . . . , v n } and e i = { v , v i } for i ∈ { , . . . , n } .The integral edge cover number of each H n is 2 and an optimal integral edge cover can beobtained, e.g., by setting γ n ( e ) = γ n ( e ) = 1 and γ n ( e ) = 0 for all other edges. In contrast,the fractional edge cover number is 2 − n and the unique optimal fractional edge cover is γ n with γ n ( e ) = 1 − n and γ n ( e i ) = n for each i ∈ { , . . . , n } . For the support of thesetwo covers, we have | support( γ n ) | = 2 and | support( γ n ) | = n + 1. Hence, the support of theoptimal edge covers is bounded in the integral case but unbounded in the fractional case. (cid:5) As mentioned above, fractional hypertree width (fhw) is to date the most general width-notion that allows one to define tractable fragments of classical NP-complete problems,such as CSP solving and CQ answering. However, recognizing if a given hypergraph H hasfhw( H ) ≤ k for fixed k ≥ H ) ≤ k becomes tractable if we can efficiently enumeratethe fractional edge covers of size ≤ k [7]. This fact can be exploited to show that, for classesof hypergraphs with bounded rank (i.e., max. size of edges), bounded degree (i.e., max.number of edges containing a particular vertex), or bounded intersection (i.e., max. numberof vertices in the intersection of two edges), checking fhw( H ) ≤ k becomes tractable. Thesize of the support has been recently [7] identified as a crucial parameter for the efficientenumeration of fractional edge covers of weight ≤ k for given k ≥ overarching goal of this work is to further extend and provide a uniform view ofpreviously known structural properties of hypergraphs that guarantee a bound on the sizeof the support of fractional edge covers of a given weight. In particular, when looking at ottlob, Lanzinger, Pichler, Razgon 3 Example 2, we want to avoid the situation that the support of fractional edge covers increaseswith the size of the hypergraph. Our main combinatorial result (Theorem 5) will be that thesize of the support of a fractional edge cover does not depend on the number of vertices oredges of a hypergraph but instead only on the weight of the cover as well as the structure ofits edge intersections.Formally, the structure of the edge intersections is captured by the so-called
Bounded-Multi-Intersection-Propery (BMIP) [5]: a class C of hypergraphs has this property, if in everyhypergraph H ∈ C , the intersection of c edges of H has at most d elements, for constants c ≥ d ≥
0. The BMIP thus generalizes all of the above mentioned hypergraphproperties that ensure bounded support of fractional edge covers of given weight and, hence,also guarantee tractability of checking fhw( H ) ≤ k , namely bounded rank, bounded degree,and bounded intersection. Moreover, when considering the incidence graph G of H , theBMIP corresponds to G not having large complete bipartite graphs. A notable result in thearea of parameterized complexity [9] is the polynomial kernelizability of the DominatingSet Problem for graphs without K c,d . A minor tweaking of the results yields a polynomialkernelization for the Set Cover Problem if the corresponding incidence graph does not contain K c,d . Our result thus makes an interesting connection: it shows that a condition that enablesefficient solving of the Set Cover problem also enables efficient checking of bounded fractionalhypertree width.In summary, the main results of this paper are as follows:First of all, we show that the size of the support of a fractional edge cover only dependson the weight of the cover and of the structure of its edge intersections (Theorem 5).More specifically, if the intersection of c edges of a hypergraph H has at most d elements,and H has a fractional edge cover of weight ≤ k , then H also has a fractional edge coverof weight ≤ k with a support whose size only depends on c, d , and k .As an important consequence of this result, we show that the problem of checking if agiven hypergraph H has fhw( H ) ≤ k is tractable for hypergraph classes satisfying theBMIP (Theorem 25). In particular, BMIP generalizes all previously known hypergraphclasses with tractable fhw-checking, namely bounded rank, bounded degree, and boundedintersection.We transfer our results on fractional edge covers to fractional vertex covers, where we againvastly generalize previously known hypergraph classes (such as hypergraphs of boundedrank [6]) that guarantee a bound on the size of the support of fractional vertex covers(Theorem 28).The paper is organized as follows: after recalling some basic notions and results in Section 2,we will present our main technical result on fractional edge covers in Section 3. The detailedproof of a crucial lemma is separated in Section 4. In Section 5, we apply our result on thebounded support of fractional edge covers to fractional hypertree width and fractional vertexcovers. Finally, in Section 6, we summarize our results and discuss some directions for futureresearch. Due to space limitations, some proofs are given in the appendix. Some general notation.
It is convenient to use the following short-hand notation forvarious kinds of sets: we write [ n ] for the set { , . . . , n } of natural numbers. Let S be a setof sets. Then we write T S and S S for the intersection and union, respectively, of the setsin S , i.e., T S = { x | x ∈ s for all s ∈ S } and S S = { x | x ∈ s for some s ∈ S } . Fractional Covers of Hypergraphs with Bounded Multi-Intersection
Hypergraphs.
We recall some basic notions on hypergraphs next. We have alreadyintroduced in Section 1 hypergraphs as pairs (
V, E ) consisting of a set V of vertices and a set E of edges. Without loss of generality, we assume throughout this paper that a hypergraphneither contains isolated vertices (i.e., vertices that do not occur in any edge) nor emptyedges. We call a hypergraph H = ( V, E ) reduced if, in addition to these restrictions, itcontains no two vertices of the same type, i.e., there do not exist v = v in V such that { e ∈ E | v ∈ e } = { e ∈ E | v ∈ e } . Note that, for computing (edge or vertex) covers, wemay always assume that a hypergraph is reduced. It is sometimes convenient to identify ahypergraph with its set of edges E with the understanding that V = S E . A subhypergraph of a hypergraph H is obtained by taking a subset of the edges of H . By slight abuse ofnotation, we may thus write H ⊆ H for a subhypergraph H of H .Given a hypergraph H = ( V, E ), the dual hypergraph H d = ( W, F ) is defined as W = E and F = {{ e ∈ E | v ∈ e } | v ∈ V } . If H is reduced, then we have ( H d ) d = H , i.e., the dualof the dual of H is H itself. The incidence graph of a hypergraph H = ( V, E ) is a bipartitegraph (
W, F ) with W = V ∪ E , such that, for every v ∈ V and e ∈ E , there is an edge { v, e } in F iff v ∈ e . Note that a hypergraph H and its dual hypergraph H d have the sameincidence graph.In this work, we are particularly interested in the structure of the edge intersections of ahypergraph. To this end, recall the notion of ( c, d )-hypergraphs for integers c ≥ d ≥ H = ( V, E ) is a ( c, d )-hypergraph if the intersection of any c distinct edges in E has at most d elements, i.e., for every subset E ⊆ E with | E | = c , we have | T E | ≤ d . Aclass C of hypergraphs is said to satisfy the bounded multi-intersection property (BMIP) [5],if there exist c ≥ d ≥
0, such that every H in C is a ( c, d )-hypergraph. As a specialcase studied in [4, 5], a class C of hypergraphs is said to satisfy the bounded intersectionproperty (BIP) , if there exists d ≥
0, such that every H in C is a (2 , d )-hypergraph.We now recall tree decompositions, which form the basis of various notions of width. Atuple ( T, ( B u ) u ∈ T ) is a tree decomposition (TD) of a hypergraph H = ( V, E ), if T is a tree,every B u is a subset of V and the following two conditions are satisfied: (a) For every edge e ∈ E there is a node u in T , such that e ⊆ B u , and (b) for every vertex v ∈ V , { u ∈ T | v ∈ B u } is connected in T .The vertex sets B u are usually referred to as the bags of the TD. Note that, by slight abuseof notation, we write u ∈ T to express that u is a node in T .For a function f : 2 V → R + , the f -width of a TD ( T, ( B u ) u ∈ T ) is defined as sup { f ( B u ) | u ∈ T } and the f -width of a hypergraph is the minimal f -width over all its TDs.An edge weight function is a function γ : E → [0 , γ a fractional edge cover of a set X ⊆ V by edges in E , if for every v ∈ X , we have P { e | v ∈ e } γ ( e ) ≥
1. The weightof a fractional edge cover is defined as weight( γ ) = P e ∈ E γ ( e ). For a set S ⊆ E , we define γ ( S ) = P e ∈ S γ ( e ), i.e., the total weight of the edges in S . For X ⊆ V , we write ρ ∗ H ( X ) todenote the minimal weight over all fractional edge covers of X . The fractional hypertreewidth (fhw) of a hypergraph H , denoted fhw( H ), is then defined as the f -width for f = ρ ∗ H .Likewise, the fhw of a TD of H is its ρ ∗ H -width.We state an important technical lemma for weight-functions of ( c, d )-hypergraphs. (cid:73) Lemma 3.
There is a function f ( c, d, k ) with the following property: let H be a ( c, d ) -hypergraph and let γ be an edge weight function with weight( γ ) ≤ k . Moreover, let < (cid:15) ≤ and assume that, for each e ∈ E , γ ( e ) ≤ (cid:15) c . Let B (cid:15) ( γ ) be the set of all vertices of weight atleast (cid:15) . Then | B (cid:15) ( γ ) | ≤ f ( c, d, k ) holds. The above lemma is essentially an extract of Lemma 7.3 in [7]. For convenience, we haveincluded a proof in the appendix. ottlob, Lanzinger, Pichler, Razgon 5
Linear Programs.
We assume some familiarity with Linear Programs (LPs). Formally,we are dealing here with minimization problems of the form c T x = min subject to Ax ≥ b and x ≥
0, where x is a vector of n variables, c is a vector of n constants, A is an m × n matrix, b is a vector of m constants, and 0 stands for the n -dimensional zero-vector. Morespecifically, for a hypergraph H = ( V, E ) and vertices X ⊆ V , the fractional edge covernumber ρ ∗ H ( X ) of X is obtained as the optimal value of the following LP: let E = { e , . . . , e n } and X = { x , . . . , x m } , then c is the n -dimensional vector (1 , . . . , b is the m -dimensionalvector (1 , . . . , A ∈ { , } [ m ] × [ n ] , such that A ij = 1 if x i ∈ e j and A ij = 0 otherwise.In the sequel, we will refer to such LPs with c ∈ { } n , b ∈ { } m and A ∈ { , } [ m ] × [ n ] as unary linear programs.For given number n of edges, there are at most 2 n possible different inequalities of theform A i x ≥
1. We thus get the following property of unary LPs, which intuitively states thatif the optimum is bigger than some threshold k , then it exceeds k by some distance. (cid:73) Lemma 4.
For every positive integers n and k , there is an integer D ( n, k ) such that forany unary LP Z of at most n variables if OPT( Z ) > k then OPT( Z ) − k > D ( n,k ) , where OPT( Z ) denotes the minimum of the LP. In this section we establish our main combinatorial result, Theorem 5. Every set of verticesin a ( c, d )-hypergraph can be covered in a way such that the size of the support dependsonly on c , d , and the set’s fractional edge cover number. Due to space constraints proofs ofsome statements have to be omitted and we refer to the appendix for additional details. (cid:73) Theorem 5.
There is a function h ( c, d, k ) such that the following is true. Let c, d, k be constants. Let H = ( V, E ) be a ( c, d ) -hypergraph and let γ : E → [0 , Assume that weight( γ ) ≤ k . Then there exists an assignment ν : E → [0 , such that weight ( ν ) ≤ k , B ( γ ) ⊆ B ( ν ) and | support( ν ) | ≤ h ( c, d, k ) . The first step of our reasoning is to consider the situation where | B ( γ ) | is bounded. Inthis case it is easy to transform γ into the desired ν . Partition all the hyperedges of H into equivalence classes corresponding to non-empty subsets of B ( γ ) such that two edges e and e are equivalent if and only if e ∩ B ( γ ) = e ∩ B ( γ ). Then let s X be the totalweight (under γ ) of all the edges from the equivalence class where e ∩ B ( γ ) = X . Identify onerepresentative of each (non-empty) equivalence class and let e X be the representative of theequivalence class corresponding to X . Then define ν as follows. For each X correspondingto a non-empty equivalence class, set ν ( e X ) = s X . For each edge e whose weight has notbeen assigned in this way, set ν ( e ) = 0. It is clear that B ( γ ) ⊆ B ( ν ) and that the support of ν is at most 2 | B ( γ ) | , which is bounded by assumption.Of course, in general we cannot assume that | B ( γ ) | is bounded. Therefore, as the nextstep of our reasoning, we consider a more general situation where we have a bounded set S = { S , . . . , S r } where each S i is a set of at most c hyperedges such that the following conditionshold regarding S : (i) for each 1 ≤ i ≤ r , γ ( S i ) ≥ U = B ( γ ) \ S i ∈ [ r ] T S i is of bounded size. Then the assignment ν as in Theorem 5 can be defined as follows. Foreach e ∈ S S , set ν ( e ) = γ ( e ). Next, we observe that for the subhypergraph H = H − S S , | B H ( γ ) | is bounded, where subscript H means that we consider B for hypergraph H and γ is restricted accordingly. Therefore, we define ν on the remaining edges as in the paragraphabove. It is not hard to see that the support of the resulting ν is of size at most c · r + 2 | U | . Fractional Covers of Hypergraphs with Bounded Multi-Intersection
We are going to show that such a family of sets of edges can always be found for ( c, d )hypergraphs (after a possible modification of γ ). (cid:73) Definition 6 (Well-formed pair) . Let H = ( V, E ) be a hypergraph and let γ : E → [0 , bean edge weight function. We say ( S , U ) is a well-formed pair (with regard to γ ) if it satisfiesthe following conditions: U ⊆ B ( γ ) S = { S , . . . , S r } where each S i is a set of at most c hyperedges of H . B ( γ ) \ U ⊆ S i ∈ [ r ] T S i .We denote P i ∈ [ r ] | S i | + 2 | U | by n ( S , U ) and refer to it as the size of ( S , U ) . (cid:73) Definition 7 (Perfect well-formed pairs) . A well-formed pair ( S , U ) is perfect if there isan assignment ν : E → [0 , with weight( ν ) ≤ k and | support( ν ) | ≤ n ( S , U ) such that S i ∈ [ r ] T S i ∪ U ⊆ B ( ν ) . Our aim now is to demonstrate the existence of a perfect pair ( S , U ) of size bounded bya function depending on c , d , and k . Clearly, this will imply Theorem 5.In particular, we will define the initial pair which is a well-formed pair but not necessarilyperfect. Then we will define two transformations from one well-formed pair into anotherand prove existence of a function transf so that if ( S , U ) is transformed into ( S , U ), then n ( S , U ) ≤ transf( n ( S , U )). We will then prove that if we form a sequence of well-formedpairs starting from the initial pair and obtain every next element by a transformation of thelast one then, after a bounded number of steps we obtain a perfect well-formed pair. Westart by defining the initial pair. (cid:73) Definition 8 (Initial pair) . The initial pair is ( S , U ) where S = {{ e } | γ ( e ) ≥ / (2 c ) } and U = B ( γ ) \ S { e }∈ S e . (cid:73) Lemma 9.
There is a function init such that n ( S , U ) ≤ init( c, d, k ) . Proof. | U | ≤ f ( c, d, k ) where f is as in Lemma 3 (for (cid:15) = 1) and | S S | ≤ ck byconstruction. (cid:74) We now introduce two kinds of transformations, folding and extension . A folding removesa set S ∗ of c edges from S and adds to U the vertices in the intersection of the edges of S ∗ .In the resulting well-ordered pair ( S , U ), S has one less element than S and U , comparedto U , has a bounded size increase of at most d vertices. Thus the action of folding gets theresulting well-formed pair closer to one with empty first component, which is a perfect pairaccording to the paragraph immediately after the statement of Theorem 5. (cid:73) Definition 10 (Folding) . Let ( S , U ) be a well-formed pair such that S contains elementsof size c . Let S ∗ ∈ S such that | S ∗ | = c . Let S = S \ { S ∗ } and U = U ∪ ( T S ∗ ∩ B ( γ )) . Wecall ( S , U ) a folding of ( S , U ) . The folding, however, is possible only if S has an element of size c . Otherwise, we need amore complicated transformation called an extension . The extension takes an element S ∈ S of size r < c and expands it by replacing S with several subsets of E each containing allthe edges of S plus one extra edge. This replacement may miss some of the elements v of B ( γ ) ∩ T S simply because v is not contained in any of these extra edges. This excess ofmissed elements is added to U and thus all the conditions of a well-formed pair are satisfied. ottlob, Lanzinger, Pichler, Razgon 7 (cid:73) Definition 11 (Extension) . Let ( S , U ) be a well-formed pair with S = ∅ such that everyelement of S is of size at most c − . For the extension, we identify S ∈ S be an elementcalled the extended element and a set S of hyperedges called the extending set . We refer to L = ( T S ∩ B ( γ )) \ S S as the set of light vertices . An extension of ( S , U ) is ( S , U ) where S = ( S \ { S } ) ∪ { S ∪ { e } | e ∈ S } and U = U ∪ L . (cid:73) Proposition 12.
With data as in Definition 11, ( S , U ) is a well-formed pair. At the first glance the transformation performed by the extension is radically opposite tothe one done by the folding: the first component grows rather than shrinks. Note, however,that the new sets replacing the removed one contain a larger number of edges and thus theyare closer to being of size c at which stage the folding can be applied to them. Our claimis that after a sufficiently large number of foldings and extensions, a well-formed pair withempty first component is eventually obtained.For our overall goal we then need to show that the size of the resulting perfect pairis indeed bounded by a function of c , d , and k . To that end, the following lemma firstestablishes that a single step in this process increases the size of the well-formed pair in acontrolled manner. To streamline our path to the main result, the proof of the lemma isdeferred to Section 4. (cid:73) Lemma 13.
There is a function ext : N → N such that the following holds. Let ( S , U ) bea well-formed pair with S = ∅ such that every element of S is of size at most c − . Thenone of the following two statements is true. ( S , U ) is a perfect pair. There is an extension ( S , U ) of ( S , U ) such that n ( S , U ) ≤ ext( n ( S , U )) . We refer to ( S , U ) as a bounded extension of ( S , U ) . For the sake of syntactical convenience, we unify the notions of folding and boundedextension into a single notion of transformation and prove the related statement followingfrom Lemma 13 and the definition of folding. (cid:73)
Definition 14 (Transformation) . Let ( S , U ) and ( S , U ) be well-formed pairs. We say that ( S , U ) is a transformation of ( S , U ) if it is either a folding or a bounded extension of ( S , U ) . (cid:73) Lemma 15.
There is a monotone function transf : N → N with transf( x ) ≥ x for anynatural number x such that the following holds. If ( S , U ) be a well-formed pair, then one ofthe following two statements is true. ( S , U ) is a perfect pair. There is a transformation ( S , U ) of ( S , U ) such that n ( S , U ) ≤ transf( n ( S , U )) . Proof.
Assume that ( S , U ) is not a perfect pair. Then | S | is not empty (see the discussion atthe beginning of this section). Suppose that an element of S is of size c . Then we set ( S , U )to be a folding of ( S , U ). By definition of the folding and of ( c, d )-hypergraphs, ( S , U ) isobtained from ( S , U ) by removal of an element from S and adding at most d vertices to U .Hence the size of ( S , U ) is clearly bounded in the size of ( S , U ). If all elements of S are ofsize at most c − S , U ) of ( S , U ).Clearly, we can specify a function transf so that in both cases n ( S , U ) ≤ transf ( n ( S , U )).To satisfy the requirement for transf, set transf( x ) = max( x, max i ∈ [ x ] transf ( x )) for eachnatural number x . (cid:74) Now that we know that each individual step on our path to a perfect pair increases thesize only in a bounded fashion, we need to establish that the number of steps is also bounded
Fractional Covers of Hypergraphs with Bounded Multi-Intersection by a function of c , d , and k . The following auxiliary theorem states that such a bound exists.A full proof of Theorem 17 is available in the appendix. (cid:73) Definition 16.
A sequence of ( S , U ) , . . . , ( S q , U q ) is a sequence of transformations if foreach i ∈ [ q − the following two statements hold ( S i , U i ) is not a perfect pair. ( S i + , U i +1 ) is a transformation of ( S i , U i ) . (cid:73) Theorem 17.
There is a monotone function sl : N → N such that the following is true.Let ( S , U ) , . . . , ( S q , U q ) be a sequence of transformations. Then q ≤ sl( n ( S , U )) . In summary, we have shown that we can reach a perfect pair in a bounded number oftransformations. Moreover, each transformation increases the size of a pair in a controlledmanner. We are now ready to prove our main result.
Proof of Theorem 5.
Consider the following algorithm. Let ( S , U ) be the initial pair (see Definition 8). q ← While ( S q , U q ) is not a perfect pair a. q ← q + 1 b. Let ( S q , U q ) be a transformation of ( S q − , U q − ) existing by Lemma 15By Theorem 17, the above algorithm stops with the final q being at most sl( n ( S , U )). Itfollows from the description of the algorithm that ( S q , U q ) is a perfect pair. It remains toshow that its size is bounded by a function of c, d, k . q ≤ sl( n ( S , U )) ≤ sl(init( c, d, k )) (1)the second inequality follows from Lemma 9 and the monotonicity of sl. Next, by theproperties of transf, an inductive application of Lemma 15 and Lemma 9 yields n ( S q , U q ) ≤ transf q (init( c, d, k )) (2)where superscript q means that the function transf is applied q times.Let h ( c, d, k ) = transf sl(init( c,d,k )) (init( c, d, k )). It follows from combination of (1) and (2)that n ( S q , U q ) ≤ h ( c, d, k ). (cid:74) The first step of the proof is to define a unary linear program of bounded size associated with( S , U ). Then we will demonstrate that if the optimal value of this linear program is at most k , then ( S , U ) is perfect. Otherwise, we show that a bounded extension can be constructed.In order to define the linear program, we first formally define equivalence classes of edgescovering U (see the informal discussion at the beginning of Section 3). (cid:73) Definition 18 (Working subset, witnessing edge) . A set of vertices U ⊆ U is called workingset (for ( S , U ) ) if there is e ∈ E \ S S such that e ∩ U = U . This e is called a witnessingedge of U and the set of all witnessing edges of U is denoted by W U . Continuing on the previous definition, it is not hard to see that the sets W U partitionthe set of edges of E \ S S having a non-empty intersection with U . Choose an arbitrarybut fixed representative of each W U and let A U be the set of these representatives which wealso refer to as the set of witnessing representatives . Now, we are ready to define the linearprogram. ottlob, Lanzinger, Pichler, Razgon 9 (cid:73) Definition 19 ( LP ( S , U ) ) . The linear program LP ( S , U ) of ( S , U ) has the set of variables X = { x e | e ∈ S S ∪ A U } . The objective function is the minimization of P x e ∈ X x e . Theconstraints are of the following two kinds. { One S | S ∈ S } where One S is P e ∈ S x e ≥ . { One u | u ∈ U } where One u is P e ∈ E u x e ≥ where E u is the subset of S S ∪ A U consisting of all the edges containing u . (cid:73) Lemma 20.
Assume that the optimal solution of LP ( S , U ) is at most k . Then ( S , U ) is aperfect pair. Proof.
Each variable x e of LP ( S , U ) corresponds to an edge e and this correspondence isinjective. For each x e , let ν ( e ) be the value of x e in the optimal solution. For each edge e not having a corresponding edge, set ν ( e ) = 0. It follows from a direct inspection that U ∪ S i ∈ [ r ] T S i ⊆ B ( ν ) and the size of support of ν is at most n ( S , U ). (cid:74) As stated above, in case the optimal value of LP ( S , U ) is greater than k we are going todemonstrate existence of a bounded extension of ( S , U ). The first step towards identifyingsuch an extension is to identify the extending element of S . Combining Lemma 4 fromSection 2 with Lemma 21 below, we observe that S has an element S ∗ such that γ ( S ∗ ) is much smaller than 1. This S ∗ will be the extended element. (cid:73) Lemma 21.
Let ( S , U ) be a well-formed pair. Let S ∗ be the subset of S consisting ofall S such that γ ( S ) < . Let α be an optimal solution for LP ( S , U ) . Then weight( α ) ≤ weight( γ ) + P S ∈ S ∗ (1 − γ ( S )) . Proof.
Let β be an arbitrary assignment of weights to the hyperedges of H . We say that β satisfies a constraint One S for S ∈ S if β ( S ) ≥ β satisfies the constraint One u for u ∈ U if β ( E u ) ≥ γ by at most P S ∈ S ∗ (1 − γ ( S )) and that satisfies all the constraints One S and One v . Clearly,this will imply correctness of this theorem.For each S ∈ S ∗ choose an arbitrary edge e S and let INCR be the set of all such edges.For each e ∈ INCR , let incr e = max { − γ ( S ) | e = e S } .Let γ be obtained from γ as follows. If e ∈ INCR then γ ( e ) = γ ( e ) + incr e . Otherwise, γ ( e ) = γ ( e ). It is not hard to see that γ satisfies the constraints One S for each S ∈ S , thatweight( γ ) ≤ weight( γ ) + P S ∈ S ∗ (1 − γ ( S )), and that, since γ does not decrease the weightof any edge, U ⊆ B ( γ ).Let { U , . . . , U a } be all the working subsets of U and let e , . . . , e a be the respectivewitnessing representatives. Then the assignment γ of weights is defined as follows. If there is 1 ≤ i ≤ a such that e ∈ W U i then γ ( e ) = γ ( W u i ) = γ ( W U i ) if e = e i and γ ( e ) = 0 otherwise. Otherwise, γ ( e ) = γ ( e ).Let W = S i ∈ [ a ] W U i . Note that, by construction, γ ( W ) = γ ( W ) and the weights ofedges outside W are the same under γ and γ and thus, weight( γ ) = weight( γ ). Moreoversince S S does not intersect with W , γ satisfies the constraints One S for all S ∈ S .It remains to show that γ satisfies the constraints One u for each u ∈ U . Let e , . . . , e r be the edges of S S containing u , let { U , . . . , U b } be the working subsets of U containing u ,and let e , . . . , e b be the respective witnessing representatives. As u ∈ B ( γ ), it follows that P i ∈ [ r ] γ ( e i ) + P i ∈ [ b ] γ ( W U i ) ≥
1. By construction, γ ( e i ) = γ ( e i ) for each 1 ≤ i ≤ r and γ ( e i ) = γ ( W U i ) for each 1 ≤ i ≤ b . Consequently, P i ∈ [ r ] γ ( e i ) + P i ∈ [ b ] γ ( e i ) ≥
1. Weconclude that γ satisfies One u . (cid:74) Lemma 4 and Lemma 21 imply the following corollary. (cid:73)
Corollary 22.
Let ( S , U ) be a well-formed pair. Assume that weight ( γ ) ≤ k while OP T ( LP ( S , U )) > k . Let n = n ( S , U ) . Then there is an S ∗ ∈ S with − γ ( S ∗ ) > / ( D ( n, k ) ∗ | S | ) . In particular this means that S ∗ is not empty where S ∗ is as in Lemma 21. Proof.
Note that the number of variables of LP ( S , U ) is at most n . It follows from thecombination of Lemma 4 and Lemma 21 that weight( γ ) + P S ∈ S ∗ (1 − γ ( S )) > k + 1 /D ( n, k )and, since weight( γ ) ≤ k , P S ∈ S ∗ (1 − γ ( S )) > /D ( n, k ) and hence there is S ∗ ∈ S ∗ with(1 − γ ( S ∗ )) > D ( n,k ) ·| S ∗ | ≥ D ( n,k ) ·| S | . (cid:74) Proof of Lemma 13.
If the value of the optimal solution of LP ( S , U ) is at most k , we aredone by Proposition 20.Otherwise, let S ∗ ∈ S be as in Corollary 22. Let (cid:15) = ( D ( n, k ) · | S | ) − . It followsfrom Corollary 22 that vertices of B ( γ ) ∩ T S ∗ need weight contribution of at least (cid:15) fromhyperedges of H other than S ∗ . We define the extending set S to be the set of all hyperedgesof H other than S ∗ whose weight is at least (cid:15)/ c and therefore | S | ≤ ck/(cid:15) . Accordingly, wedefine the set L of light vertices to be the subset of B ( γ ) ∩ T S ∗ consisting of all vertices x that, besides S ∗ are contained only in hyperedges of weight smaller than (cid:15)/ c . By Lemma 3, | L | ≤ f ( c, d, k ) and the size of S ∗ is clearly bounded by a function on n and c, d, k . It is nothard to see that the size of the resulting extension is bounded as well. (cid:74) Now that our main combinatorial result has been established we move our attention to analgorithmic application of the support bound. In particular, we are interested in the problemof deciding whether for an input hypergraph H and constant k we have fhw( H ) ≤ k . Theproblem is known to be NP-hard even for k = 2 [5]. However, as noted in the introduction,in recently published research we were able to show that for hypergraph classes which enjoybounded intersection or bounded degree, it is indeed tractable to check fhw( H ) ≤ k forconstant k [7].Due to limited space we will recall the main components of the framework for tractablewidth checking developed in [7] and use them in a black-box fashion. (cid:73) Definition 23 ( q -limited fractional hypertree width) . Let ρ ∗ q ( U ) be the minimal weight of anassignment γ such that U ⊆ B ( γ ) and | support( γ ) | ≤ q . We define the q - limited fractionalhypertree width of a hypergraph H as its ρ ∗ q -width. (cid:73) Lemma 24 (Theorem 4.5 & Lemma 6.2 in [7]) . Fix c , d , and q as constant integers. Thereis a polynomial time algorithm testing whether a given ( c, d ) -hypergraph has a q -limitedfractional hypertree width at most k . The underlying intuition of q -limited fhw is that the bounded support allows for apolynomial time enumeration of all the (inclusion) maximal covers of sufficient weight. For( c, d )-hypergraphs it is then possible to compute a set of candidate bags such that a fittingtree decomposition, if one exists, uses bags only from this set. Deciding whether a treedecomposition can be created from a given set of candidate bags is tractable under someminor restrictions to the structure of the resulting decomposition (not of any concern to thecase discussed here). ottlob, Lanzinger, Pichler, Razgon 11 We now apply our main result and show that, under BMIP, there exists a constant q suchthat the q -limited fractional hypertree width always equals fractional hypertree width. Fromthe previous lemma is then straightforward to arrive at the desired tractability result. (cid:73) Theorem 25.
There is a polynomial time algorithm for testing whether the fhw of thegiven ( c, d ) -hypergraph H is at most k (the degree of the polynomial is upper bounded by afixed function depending on c, d, k ). Proof.
It follows from Theorem 5 that if fhw( H ) ≤ k for a ( c, d )-hypergraph H then the h ( c, d, k )-limited fhw of H is also at most k .Indeed, let ( T, ( B u ) u ∈ T ) be a tree decomposition with fhw at most k . Then, accordingto Theorem 5, for each node u in T there is an edge weight function γ with | support( γ ) | ≤ h ( c, d, k ) such that B u ⊆ B ( γ ). In other words, it follows that ( T, ( B u ) u ∈ T ) has ρ ∗ q -width atmost k where q is h ( c, d, k ). Thus, H also has h ( c, d, k )-limited fractional hypertree width atmost k .Thus to test whether fhw( H ) ≤ k , it is enough to test whether the h ( c, d, k )-limited fhwof H is at most k . This can be done in polynomial time according to Lemma 24. (cid:74) There are two natural dual concepts of fractional edge cover. One is the fractional vertexcover problem which is the dual in the sense that it is equivalent to the fractional edge coveron the dual hypergraph. The other, the fractional independent set problem, corresponds tothe dual linear program of a linear programming formulation of finding an optimal fractionalcover. Here we discuss how our results extend to vertex covers and discuss how the resultingstatement generalizes, in a particular sense, a well-known statement of Füredi [6]. Somenotes on connections to fractional independent sets are given in our discussion of future workin Section 6.We start by giving a formal definition of the fractional vertex cover problem. Let H = ( V, E ) be a hypergraph and β : V → [0 ,
1] be an assignment of weights to the verticesof H . Analogous to the definition of fractional edge covers we define B v ( β ) = { e ∈ E | P v ∈ e β ( v ) ≥ } ,vsupport( β ) = { v ∈ V | β ( v ) > } ,and weight( β ) = P v ∈ V β ( v ).A fractional vertex cover is also called a transversal in some contexts (cf. [10]). For aset of edges E we denote the weight of the minimal fractional vertex cover β such that E ⊆ B v ( β ) as τ ∗ ( E ). For hypergraph H = ( V, E ), we say τ ∗ ( H ) = τ ∗ ( E ). Recall, that weassume reduced hypergraphs and therefore there is a one-to-one correspondence of vertices in H and edges in H d . We will make use of the following well-known fact about the connectionof what we will call dual weight assignments . (cid:73) Proposition 26.
Let H = ( V, E ) be a (reduced) hypergraph and H d = ( W, F ) its dual. Wewrite f v to identify the edge in F that corresponds to the vertex v in V . The following twostatements hold:For every γ : E → [0 , and the function β : W → [0 , with β ( e ) = γ ( e ) it holds that B v ( β ) = { f v | v ∈ B ( γ ) } .For every β : V → [0 , and the function γ : F → [0 , with γ ( f v ) = β ( v ) it holds that B ( γ ) = { v | f v ∈ B v ( β ) } . In the following we extend Theorem 5 to an analogous statement for fractional vertexcovers thereby generalizing the previous proposition significantly. To derive the resultwe need a final observation about ( c, d )-hypergraphs. In a sense, we show that boundedmulti-intersection is its own dual property. (cid:73)
Lemma 27.
Let H be a ( c, d ) -hypergraph. Then the dual hypergraph H d is a ( d + 1 , c ) -hypergraph. Proof.
Let v , v , . . . , v d +1 be d + 1 distinct arbitrary vertices of a ( c, d )-hypergraph H =( V, E ). We write I ( v ) = { e ∈ E | v ∈ e } for the set of edges incident to a vertex v .Since H is a ( c, d )-hypergraph, it must hold that X = T j ∈ [ d +1] I ( v j ) has no more than c elements. Otherwise, there would be at least c + 1 edges in X that share d + 1 vertices, i.e.,a contradiction to the assumption that H is a ( c, d )-hypergraph.Now, consider the edges f , f . . . , f d +1 in H d = ( W, F ) that correspond to the vertices v , v , . . . , v d +1 in H . It follows from the definition of the dual hypergraph that | T j ∈ [ d +1] f j | = | X | since any two edges in H d share exactly one vertex for each edge in H that they areboth incident to. We know from above that | X | ≤ c . As this applies to any choice of verticesin H , and thus also to any choice of d + 1 edges in H d , we see that any intersection of d + 1edges in H d has cardinality less or equal c . (cid:74)(cid:73) Theorem 28.
There is a function h ( c, d, k ) such that the following is true. Let c, d, k be constants. Let H be a ( c, d ) -hypergraph and β be an assignment of weights to V ( H ) .Assume that weight ( β ) ≤ k . Then there is an assignment ν of weights to V ( H ) such that weight ( ν ) ≤ k , B v ( β ) ⊆ B v ( ν ) and | vsupport( ν ) | ≤ h ( c, d, k ) . Proof.
Let γ be the dual weight assignment of β as in Proposition 26. That is, γ : F → [0 , H d = ( W, F ) with | support( γ ) | = | vsupport( β ) | and weight( γ ) = weight( β ).From Lemma 27 we have that H d is a ( d + 1 , c )-hypergraph and thus by Theorem 5there is an edge weight function ν with B ( γ ) ⊆ B ( ν ) and | support( ν ) | ≤ h ( d + 1 , c, k ).Let ν now be the dual weight assignment of ν . By Proposition 26 we then see that also B v ( β ) ⊆ B v ( ν ) and | vsupport( ν ) | = | support( ν ) | ≤ h ( d + 1 , c, k ). (cid:74) To conclude this section we wish to highlight the connection of Theorem 28 to a classicalresult on fractional edge covers. The following result is due to Füredi [6], who extendedearlier results by Chung et al. [3]. (cid:73)
Proposition 29 ([6], page 152, Proposition 5.11.(iii)) . For every hypergraph H of rank (i.e.,maximal edge size) r , and every fractional vertex cover w for H satisfying weight( w ) = τ ∗ ( H ) ,the property | vsupport( w ) | ≤ r · τ ∗ ( H ) holds. Recall that a hypergraph H with rank r is also a (1 , r )-hypergraph. Hence, the aboveproposition means that, for a (1 , r )-hypergraph H , there is a fractional vertex cover of optimalweight whose support is bounded by a function of the weight and r . Theorem 28 generalizesProposition 29 in two aspects. First, Theorem 28 considers ( c, d )-hypergraphs with c ≥ Note that the superscript of H d only signifies that it is the dual of H . It is not connected to the integerconstant d used for the multi-intersection size of H . ottlob, Lanzinger, Pichler, Razgon 13 We have proved novel upper bounds on the size of the support of fractional edge covers andvertex covers. These bounds have then been fruitfully applied to the problem of checkingfhw( H ) ≤ k for given hypergraph H . Recall that, without imposing any restrictions onthe hypergraph H , this problem is NP-complete even for k = 2 [5], thus ruling out XP-membership. In contrast, for hypergraph classes that exhibit bounded multi-intersection,we have actually managed to establish XP-membership, that is, checking fhw( H ) ≤ k forhypergraphs in such a class is feasible in polynomial time for any constant k .However, there is still room for improvement: first, our tractability result depends on abig constant h ( c, d, k ). Hence, an important next step for future research will be a deeperinvestigation of algorithms for checking fhw( H ) ≤ k in case of the BMIP and either furtherimprove the runtime or prove a matching lower bound. Moreover, XP-membership is only“second prize” in terms of a parameterized complexity result. It will be interesting to searchfor further restrictions on the hypergraphs to achieve fixed-parameter tractability (FPT).Another major challenge for future research is the computation of fhw( H ). Note that ourtractability result refers to the decision problem of checking fhw( H ) ≤ k . However, at itsheart, dealing with fractional hypertree width is an optimization problem , namely computingthe minimum possible width of all fractional hypertree decompositions of H . The difficultyhere is that our bound h ( c, d, k ) tends to infinity as k approaches the actual value of fhw( H ).Substantial new ideas are required to overcome this problem.Our bound on the support of fractional vertex covers generalizes a classical result byFüredi in two aspects. In future work, we plan to explore how this generalization canbe applied to known consequences (cf. [6]) of Füredi’s result. Finally, we have left openthe extension to fractional independent sets. By use of the complementary slackness oflinear programs our main result also implies structural restrictions for optimal fractionalindependent sets since there can only be a bounded number of constraints that have slack.We believe that an in-depth study of the connections to independent sets is merited. References Albert Atserias, Martin Grohe, and Dániel Marx. Size bounds and query plans for relationaljoins.
SIAM J. Comput. , 42(4):1737–1767, 2013. doi:10.1137/110859440 . Claude Berge.
Fractional Graph Theory . ISI Lecture Notes 1, Macmillan of India, 1978. Fan R. K. Chung, Zoltán Füredi, M. R. Garey, and Ronald L. Graham. On the fractionalcovering number of hypergraphs.
SIAM J. Discret. Math. , 1(1):45–49, 1988. doi:10.1137/0401005 . Wolfgang Fischl, Georg Gottlob, Davide Mario Longo, and Reinhard Pichler. Hyperbench:A benchmark and tool for hypergraphs and empirical findings. In
Proceedings of the 38thACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2019,Amsterdam, The Netherlands, June 30 - July 5, 2019 , pages 464–480, 2019. doi:10.1145/3294052.3319683 . Wolfgang Fischl, Georg Gottlob, and Reinhard Pichler. General and fractional hypertreedecompositions: Hard and easy cases. In
Proceedings of the 37th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2020, Houston, TX, USA, June10-15, 2018 , pages 17–32, 2018. doi:10.1145/3196959.3196962 . Zoltán Füredi. Matchings and covers in hypergraphs.
Graphs Comb. , 4(1):115–206, 1988. doi:10.1007/BF01864160 . Georg Gottlob, Matthias Lanzinger, Reinhard Pichler, and Igor Razgon. Complexity analysisof generalized and fractional hypertree decompositions.
CoRR , abs/2002.05239, 2020. URL: https://arxiv.org/abs/2002.05239 , arXiv:2002.05239 . Martin Grohe and Dániel Marx. Constraint solving via fractional edge covers.
ACM Trans.Algorithms , 11(1):4:1–4:20, 2014. doi:10.1145/2636918 . Geevarghese Philip, Venkatesh Raman, and Somnath Sikdar. Polynomial kernels for dominatingset in graphs of bounded degeneracy and beyond.
ACM Trans. Algorithms , 9(1):11:1–11:23,2012. doi:10.1145/2390176.2390187 . Edward Scheinerman and Daniel Ullman.
Fractional Graph Theory: A Rational Approach tothe Theory of Graphs . Dover Publications, Inc., 2011. ottlob, Lanzinger, Pichler, Razgon 15
AppendixA Full ProofsA.1 Proof of Lemma 4
Proof.
A constraint of a unary linear program can be associated with a subset of the set of n variables and a unary linear program is just a set of constraints. Hence there are at most2 n non-isomorphic unary linear programs with n variables. Clearly, the number of optimalvalues of these programs that are greater than k are at most that many and (by assumption)at least 1. Therefore, these optimal values have a minimum. Denote this minimum by Opt ( n, k ). Set D ( n, k ) = 2 / ( Opt ( n, k ) − k ) (cid:74) A.2 Proof of Lemma 3
Recall our claim from Lemma 3. There is a function f ( c, d, k ) with the following property:let H be a ( c, d )-hypergraph and let γ be an edge weight function with weight( γ ) ≤ k .Moreover, let 0 < (cid:15) ≤ e ∈ E , γ ( e ) ≤ (cid:15) c . Let B (cid:15) ( γ ) be the set ofall vertices of weight at least (cid:15) . Then | B (cid:15) ( γ ) | ≤ f ( c, d, k ) holds. Proof of Lemma 3.
The proof is based on the following claim (which is Lemma 7.2 in [7]).
Claim A.
Fix an integer c ≥
1. Let X = { x , . . . , x n } be a set of positive numbers ≤ δ andfix w such that P nj =1 x j ≥ w ≥ δc . Then we have P x i · x i · · · · · x i c ≥ ( w − δc ) c , wherethe sum is over all c -tuples ( i , . . . , i c ) of distinct integers from [ n ].We proceed with a counting argument. Imagine a bipartite graph G = ( B (cid:15) ( γ ) , T, E ( G ))where T is the set of all c -tuples of distinct edges from H . In G , there is an edge from v ∈ B (cid:15) ( γ ) to ( e , . . . , e c ) ∈ T iff v is in e ∩ · · · ∩ e c . Furthermore, we assign weight Q cj =1 γ ( e j )to every edge in E ( G ) incident to a tuple ( e , . . . , e c ) ∈ T . To avoid confusion, in this proof,we write E ( G ) and E ( H ) to refer to the set of edges in the graph G and in the hypergraph H , respectively.We now count the total weight in G from both sides. First observe that on the T side, we have degree at most d because H is a ( c, d )-hypergraph. Therefore, the totalweight in G is at most d · P ( e ,...,e c ) ∈ T Q cj =1 γ ( e j ). Observe that P ( e ,...,e c ) ∈ T Q cj =1 γ ( e j ) ≤ (cid:16)P e ∈ E ( H ) γ ( e ) (cid:17) · · · · · (cid:16)P e c ∈ E ( H ) γ ( e c ) (cid:17) as, by distributivity, all the terms of the sum onthe left-hand side are also present on the right-hand side of the inequality. Furthermore, wehave P e ∈ E ( H ) γ ( e ) ≤ k and thus, by putting it all together, we see that the total weight in G is at most k c d .From the B (cid:15) ( γ ) side, consider an arbitrary vertex v ∈ B (cid:15) ( γ ) and let e , . . . , e n be theedges in E ( H ) containing v . We have P nj =1 γ ( e j ) ≥ (cid:15) and γ ( e j ) ≤ (cid:15) c for each j ∈ [ n ].We can apply the above claim for X = { γ ( e ) , . . . , γ ( e n ) } , δ = (cid:15) c , and w = (cid:15) to get theinequality P γ ( e j ) · · · · · γ ( e j c ) ≥ ( (cid:15) − (cid:15) c · c ) c = ( (cid:15) ) c , where the sum ranges over all c -tuples( e j , . . . , e j c ) of distinct edges in E ( H ) containing v .We conclude that v (now considered as a vertex in G ) is incident to edges whose totalweight is ≥ ( (cid:15) ) c in E ( G ). Since we have seen above that the total weight of all edges in E ( G ) is ≤ k c d , there can be no more than d ( k(cid:15) ) c vertices in B (cid:15) ( γ ). (cid:74) A.3 Proof of Theorem 17
For this theorem, rather than considering a well-formed pair ( S , U ) itself we consider the pair( A, b ) where A is the multiset of sizes of the sets of S and | U | = b . We call ( A, b ) a bare bones c -pair ( c -BBP). A transformation of ( S , U ) is translated into a bounded size transformationof ( A , b ). In the next five definitions we formalize this intuition. Then we state Theorem 35claiming that a sufficiently long sequence of bounded transformations of c -BBPs results inone where the first component is empty. This will imply Theorem 17 because a c -BBP withthe empty first component is translated back into a well-formed pair with the empty firstcomponent which is perfect. Finally, we prove Theorem 35. (cid:73) Definition 30. A bare bones c -pair , abbreviated as c -BBP is a pair ( A, b ) where A ismultisets of integers in the range [1 , c ] and b is just a non-negative integer. We denote b + P x ∈ A x by n ( A, b ) . Note that the number of occurrences in the sum of each x ∈ A is itsmultiplicity in A . (cid:73) Definition 31.
Let ( A, b ) be a c -BBP and assume that c ∈ A . Let A = A \ { c } (that is,the multiplicity of c in A is reduced by one) and let b = b + d where d is a non-negativeinteger. Clearly ( A , b ) is a c -BBP, we refer to it as a folding of ( A, b ) . (cid:73) Definition 32.
Let ( A, b ) be a c -BBP and let x ∈ A such that x < c . Let A be obtainedfrom A by removal of one occurrence of x and adding d occurrences of x + 1 for somenon-negative integer d . Let b = b + d for some non-negative integer d . Clearly ( A , b ) isa c -BBP, we refer to it as a extension of ( A, b ) (cid:73) Definition 33.
Let ( A, b ) and ( A , b ) be c -BBPs such that ( A , b ) is either a folding or anextension of ( A.b ) . We then say that ( A , b ) is a transformation of ( A, b ) . Let n = n ( A, b ) and n = n ( A , b ) and suppose that n ≤ g ( n ) for some function g . We then say that ( A , b ) is a g - transformation of ( A, b ) . (cid:73) Definition 34.
Let g be a function of one argument and let ( A , b ) , . . . , ( A r , b r ) be asequence of c -BBPs such that for each ≤ i ≤ r , ( A i , b i ) is a g -transformation of ( A i − , b i − ) .We call ( A , b ) , . . . , ( A r , b r ) a g -transformation sequence. Note that for each ≤ i < i , A i is not empty for otherwise, it is impossible to apply a transformation to ( A i , b i ) . (cid:73) Theorem 35.
Let g be a function of one argument. Then there is a function h [ g ] such thatif ( A , b ) , . . . , ( A r , b r ) is a g -transformation sequence then r ≤ h [ g ]( n ) where n = n ( A , b ) . We first show how to prove Theorem 17 using Theorem 35 and then we will proveTheorem 35 itself.
Proof of Theorem 17.
Let ( S , U ) be a well-formed pair. Let bbp ( S , U ) be ( A, b ) where A is the multiset of sizes of elements of S (each x occurs in A exactly the number of timesas there are sets of size x in S ) and b = | U | . It is not hard to see that ( A, b ) is a c -BBP.Moreover, n ( S , U ) = n ( A, b ) (3)Let ( S , U ) , . . . , ( S r , U r ) be a transformation sequence. Let ( A , b ) , . . . , ( A r , b r ) be a se-quence of c -BBPs such that ( A i , b i ) = bbp ( S i , U i ) for each 1 ≤ i ≤ r .We are going to show that ( A , b ) , . . . , ( A r , b r ) is a transf-transformation sequence. ByTheorem 35, this will imply that r ≤ h [transf]( n ) where n = n ( A , b ) = n ( S , U ) by (3)thus implying the theorem. ottlob, Lanzinger, Pichler, Razgon 17 So, consider two arbitrary consecutive elements ( A i , b i ) and ( A i +1 , b i +1 ).Assume first that ( S i + , U i +1 ) is obtained from ( S i , U i ) by folding. It is not hard tosee that ( A i +1 , b i +1 ) is obtained from ( A i , b i ) by removal one occurrence of c and adding b i +1 = b i +( | U i +1 |−| U i | ). That is ( A i +1 , b i +1 ) is obtained from ( A i , b i ) as result of folding. As n ( S i + , U i +1 ) ≤ transf( n ( S i , U i )). It follows from (3) that n ( A i + , b i +1 ) ≤ transf( n ( A i , b i )).We conclude that ( A i +1 , b i +1 ) is obtained from ( A i , b i ) as a result of transf-transformation.Assume now that ( S i + , U i +1 ) is obtained from ( S i , U i ) by extension. This means that S i + is obtained from S i by removal of some S ∗ of size less than c and replacing it with d sets of size | S ∗ | + 1 for some integer d ≥
0. Also U i +1 is obtained from U i by adding d newelements for some integer d ≥
0. It follows by construction that ( A i +1 , b i +1 ) is an extensionof ( A i , b i ). By the same argumentation as in the end of the previous paragraph, we concludethat ( A i +1 , b i +1 ) is obtained from ( A i , b i ) by transf-transformation. (cid:74) Proof of Theorem 35.
We assume without loss of generality that n ≤ g ( n ) and that g ismonotone, that is, for n < n , we have g ( n ) ≤ g ( n ). Indeed, otherwise, since g is definedover the non-negative integers, we can define g ∗ ( n ) as the maximum over n, g (0) , . . . , g ( n )and use g ∗ instead of g . This monotonicity allows us to derive the following inequality.Suppose that ( A , b ) , . . . , ( A x , b x ) is a g -transformation sequence and x ≤ y . Then n ( A x , b x ) ≤ g ( y ) ( n ( A , b )) (4)where g ( y ) signifies the y -fold application of function g .For i ∈ { , . . . , c − } a ( g, i )-transformation is a subset of g -transformations with anadditional property recursively defined as follows. ( A , b ) is a ( g, A, b ) if ( A , b ) is obtained from ( A, b ) by folding. Suppose i > q, i − A , b ) is a ( q, i )-transformation of ( A, b ) if it is either a ( q, i − A is of size c − i .A ( g, i )-transformation sequence is a sequence of the form ( A , b ) , . . . , ( A r , b r ) where foreach 2 ≤ j ≤ r , ( A j , b j ) is obtained from ( A j − , b j − ) by a ( g, i )-transformation.We prove by induction that for each i ∈ { , . . . , c − } , there is a function h i [ g ] such that r as above is at most h i [ g ]( n ( A , b )). Then h c − [ g ] will be the desired function h [ g ]. Forthe sake of simplicity we will omit g in the square brackets and refer to these functions as h , . . . , h c − .The existence of function h is easy to observe. Indeed, the number of consecutive foldingsis at most the multiplicity of c in A . So, we can put h = n ( A , b ).Assume now that i > A , b ) , . . . , ( A r , b r ) is a ( g, i )-transformation sequence.If it is in fact a ( g, i − r ≤ h i − ( A , b ) by the inductionassumption. Otherwise, let 1 < x < . . . x a ≤ r be all the indices such that for each 1 ≤ j ≤ a ,( A x j , b x j ) is obtained from ( A x j − , b x j − ) by extension removing an element c − i .For the sake of succinctness, denote n ( A , b ) by n and for each 1 ≤ j ≤ a , we denote n ( A x j , b x j ) by n j .For each integer j ≥
1, define function f j as follows. f ( x ) = h i − ( x ) + 1. Suppose that j > f j − has been defined. Then f j ( x ) = f j − ( x ) + h i − ( g ( f j − ( n )) ( x )).We show that for each 1 ≤ j ≤ a , x j ≤ f j ( n ). Note that ( A , b ) , . . . , ( A x − , b x − ) is a( g, i − x − ≤ h i − ( n ),hence x ≤ f ( n ).Furthermore, let j >
1. Then ( A x j − , b x j − ) , . . . , ( A x j − , b x j − ) is also a ( q, i − x j ≤ x j − + h i − ( n j − ). By the induction assumption, x j − ≤ f j − ( n ) and, by (4), n j − ≤ g ( f j − ( n )) ( n ). Therefore, x j ≤ f j − ( n ) + h j − ( g ( f j − ( n )) ( n )) = f j ( n ) as required.Applying the same argumentation to the sequence following ( A x a , b x a ), we conclude that r ≤ f a +1 ( n ). Clearly a is at most the number of occurrences of c − i in ( A , b ) which is atmost n . Hence, we conclude that r ≤ f n +1 ( n ). Hence, we can set h i = f n +1 ..