Superlinear Lower Bounds Based on ETH
SStronger Lower Bounds for Polynomial TimeProblems
Andr´as Z. Salamon and Michael Wehar School of Computer Science, University of St Andrews, [email protected] Computer Science Department, Swarthmore College, [email protected] 18, 2020
Abstract
We reinvestigate the classical topic of limited nondeterminism, to provestronger conditional lower bounds for polynomial time problems. In par-ticular, we show that CircuitSAT for circuits with m gates and log( m )inputs (denoted by log-CircuitSAT) is not solvable in quasilinear time un-less the exponential time hypothesis (ETH) is false. In other words, apolynomial time improvement for log-CircuitSAT would lead to an expo-nential time improvement for NP-complete problems. Developing a deeper understanding of polynomial time problems is essential tothe fields of algorithm design and computational complexity theory. In thiswork, we build on prior concepts from the topic of limited nondeterminismto show a new kind of conditional lower bound for polynomial time problemswhere a polynomial time improvement for one problem leads to an exponentialtime improvement for another. We proceed by introducing basic notions andexplaining how they relate to existing research work.A polynomial time problem is a decision problem that can be decided in O ( n k ) time for some constant k , where n denotes the input length. As usual, P denotes the class of polynomial time problems. A decision problem has anunconditional time complexity lower bound t ( n ) if it cannot be decided in lessthan t ( n ) time. Polynomial time problems with non-trivial unconditional timecomplexity lower bounds do not commonly appear in complexity theory research(aside from lower bounds for one-tape Turing machines [21, 28]). Such problemsare known to exist by the time hierarchy theorem [20], but to the best of ourknowledge, there are few examples that appear in the literature. Most of the1 a r X i v : . [ c s . CC ] A ug nown examples are related to pebbling games [4] or intersection non-emptinessfor automata [33, 34]. For these examples, the unconditional lower bounds areproven by combining Turing machine simulations with classical diagonalizationarguments.Although unconditional lower bounds are rare, many polynomial time prob-lems have been shown to have conditional lower bounds in recent works onfine-grained complexity theory (see surveys [38, 7]). The primary goal of thiswork is to introduce stronger conditional lower bounds by applying new rela-tionships between deterministic and nondeterministic computations. Fine-grained complexity theory is a subject focused on exact runtime boundsand conditional lower bounds. A conditional lower bound for a polynomial timeproblem typically takes the following form. Polynomial time problem A is notsolvable in O ( n α − ε ) time for all ε > B is not solvablein O ( t ( n ) β − ε ) time for all ε > α and β are constants and t ( n ) is afunction (typically t ( n ) is either polynomial or exponential). This is referred toas a conditional lower bound because problem A has a lower bound under theassumption that B has a lower bound. Conditional lower bounds are known formany polynomial time problems including Triangle Finding, Orthogonal VectorsProblem (OVP), and 3SUM [38, 7].There are some known cases where conditional lower bounds can be strength-ened. In particular, it was shown that some polynomial problems have lowerbounds conditioned on a disjunction of assumptions [2]. A conditional lowerbound of this type takes the following form. Polynomial time problem A isnot solvable in O ( n α − ε ) time for all ε > { B i } i ∈ [ k ] satisfies the property that for some i ∈ [ k ], B i is not solvable in O ( t i ( n ) β − ε ) time for all ε >
0. In other words, problem A has a lower boundunder the assumption that at least one of the B i problems has a lower bound.In this paper, we demonstrate a much stronger conditional lower bound. Aquasilinear time problem is typically defined as a decision problem that can bedecided in O ( n · poly(log( n ))) time [32]. In this work, we relax this definitionto include problems that can be decided in O ( n ε ) time for all ε >
0. Weprovide an example of a polynomial time problem A such that A is not solvablein quasilinear time assuming that problem B is not solvable in O ( t ( n ) ε ) timefor some ε >
0. In other words, a polynomial improvement for the runtimeof A leads to an exponential improvement for the runtime of B . In order toprove this stronger lower bound, we must carefully explore relationships betweendeterministic and nondeterministic computations. By exponential improvement, we mean that when t ( n ) = 2 O ( n ) , the runtime improvesfrom exponential time 2 O ( n ) to subexponential time 2 o ( n ) . In other words, the improvementbreaks the barrier between a linear and sublinear exponent. .2 Exponential Time Hypothesis Many classic problems related to Boolean formulas are well known to be com-putationally hard [13, 27]. As a result, satisfiability of Boolean formulas (SAT)is a natural candidate for lower bound assumptions. In particular, it is commonto focus on satisfiability of Boolean formulas in conjunctive normal form withclause width at most k ( k -CNF-SAT) for a fixed k .The exponential time hypothesis (ETH) states that for some ε >
0, 3-CNF-SATcannot be solved in O (poly( n ) · ε · v ) time, where n denotes the input size and v denotes the number of variables [22]. The strong exponential time hypothesis(SETH) states that for every ε >
0, there is a sufficiently large k such that k -CNF-SAT cannot be solved in O (poly( n ) · (1 − ε ) · v ) time [22, 23, 11].Conditional lower bounds are frequently shown relative to the k -CNF-SATproblems. For example, it is well known that the Orthogonal Vectors Problem(OVP) on polylogarithmic length vectors is not solvable in O ( n − ε ) time for all ε > ε >
0, there exists k sufficiently large such that k -CNF-SAT is not solvable in O (poly( n ) · (1 − ε ) · v ) time [35, 38, 7]. Moreover,Orthogonal Vectors Problem (OVP) is not solvable in O ( n − ε ) time for all ε > Remark 1.
As far as we know, the current best reduction shows that an O ( n α ) time algorithm for OVP leads to a roughly O (poly( n ) · α · v ) time algorithm for k - CNF - SAT for all k [35, 38, 7]. Furthermore, we do not yet know whether theexistence of a quasilinear time algorithm for OVP would imply
ETH is false.
We also consider at a generalization of SAT called the satisfiability of Booleancircuits (CircuitSAT) decision problem. In particular, we focus on the satis-fiability of bounded fan-in Boolean circuits with m gates and log( m ) inputs(log-CircuitSAT) which is also considered in [8, 3]. A nondeterministic polynomial time problem is a decision problem that can bedecided in polynomial time by a nondeterministic algorithm or machine. Thereare multiple ways to introduce nondeterminism into a computation. For exam-ple, a machine could have nondeterministic bits written on a tape in advance orit could make nondeterministic guesses as the computation is carried out. Suchvariations don’t appear to make much difference when defining nondeterministicpolynomial time ( NP ). However, when considering limited nondeterminism, thedefinitions will require special care and attention.Limited nondeterminism is simply the restriction or bounding of the amountof nondeterminism in a computation. A survey by Goldsmith, Levy, and Mund-henk [18] provides a useful summary on this topic. We proceed by discussingsome key notions and results from prior works.Kintala and Fischer [25, 26] defined P f ( n ) as the class of languages thatcan be decided by a polynomial time bounded machine M such that M scansat most f ( n ) cells of a nondeterministic guess tape for each input of size n .3Alvarez, D´ıaz and Tor´an [5, 15], making explicit a concept of Xu, Doner, andBook [29], then defined β k as the class of languages that can be decided by apolynomial-time bounded machine M such that M uses at most O ((log( n )) k )bits of nondeterminism. Farr took a similar approach [16], defining f ( n )- NP asthe languages that can be decided by a polynomial-time bounded machine M with a binary guess tape alphabet such that M scans at most f ( q ( n )) cells ofthe guess tape for each input of size n . Here q is a polynomial that depends onlyon M . Note that f ( n )- NP is the union over all k of the classes P f ( n k ) . Anotherrelated approach was taken by Buss and Goldsmith [8] where N m P l is definedas the class of languages decided by nondeterministic machines in quasi- n l timemaking at most m · log( n ) nondeterministic guesses.Cai and Chen [10] then focused on machines that partition access to non-determinism, by first creating the contents of the guess tape, and then using adeterministic machine to check the guess. In this terminology, GC ( s ( n ) , C ) isthe class of languages that can be decided by a machine that guesses O ( s ( n ))bits and then uses the power of class C to verify. A related definition followedin Santhanam [31] where NTIGU ( t ( n ) , g ( n )) is the class of languages that canbe decided by a machine that makes g ( n ) guesses and runs for t ( n ) time.It follows from the definitions that P = P log( n ) = NTIGU ( P , log( n )) = N O (1) P O (1) = GC ( O (1) , P ) = β and NP = n - NP = NTIGU ( P , P ) = GC ( n O (1) , P ) = P n O (1) . Furthermore, the β k classes are meant to capture classes between P and NP .In this work, we focus on the log-CircuitSAT problem and the levels within P = β . It’s worth noting that a loosely related work [17] investigated thelog-Clique problem which is in β . By introducing a new notion of limited nondeterminism, we are able to provenew relationships between deterministic and nondeterministic computations. Inparticular, if more efficient deterministic algorithms exist, then there are eleganttrade-offs between time and nondeterminism.In order to prove these relationships we need to pay close attention to thedetails of our model for limited nondeterminism combining various elementsfrom the existing models from the literature and introducing new elements.In particular, for our model (like others) nondeterminism will be measured inbits and the coefficients on how many bits we are using will be very impor-tant. Unlike existing models, the nondeterministic guesses will be preallocatedas placeholders within an input string. This means that we can only fill in4laceholders with nondeterministic bits. This property is essential for provingstructural properties (see translation and padding lemmas in Subsection 3.3).With other models, proofs of structural properties are much more complex andlead to various overheads from tape reduction theorems.
Consider strings over a ternary alphabet Σ = { , , p } where p is referred to asthe placeholder character. For any string x ∈ Σ ∗ , we let | x | denote the lengthof x and p ( x ) denote the number of placeholder character occurrences in x . Definition 1.
Let a string r ∈ { , } ∗ be given. Define a function sub r : Σ ∗ → Σ ∗ such that for each string x ∈ Σ ∗ , sub r ( x ) is obtained from x by replacing place-holder characters with bits from r so that the i th placeholder character from x is replaced by the i th bit of r for all i satisfying ≤ i < min {| r | , p ( x ) } .Also, define SUB ( n ) such that SUB ( n ) := { sub r | | r | ≤ n } . We call sub r asubstitution operator and SUB ( n ) a set of substitution operators. Example 1.
Consider strings x = 11 p p p and r = 0110 . By applying thepreceding definition, we have that sub r ( x ) = 11001101 . A substitution operator sub r replaces the first | r | placeholder characterswith the bits of r in order. We will also consider the notion of partial filling ,which is any replacement of placeholders, without restricting replacements to aparticular order. Definition 2.
Let strings x and y ∈ Σ ∗ be given. We write x (cid:22) y if x can beobtained from y by replacing any number of placeholders in y with 0 or 1. Leta language L ∈ Σ ∗ be given. We define Clo ( L ) such thatClo ( L ) := { x ∈ Σ ∗ | ( ∃ y ∈ L ) x (cid:22) y } . Intuitively, Clo ( L ) is the closure of L under partial fillings. Example 2.
Consider a language L = { p p } . By applying the preceding defini-tion, we have that Clo ( L ) = { p p, p , p , p, p, , , , } . We now proceed by defining a complexity class for limited nondeterminism
DTIWI ( t ( n ) , w ( n )) where intuitively t ( n ) represents a time bound and w ( n )represents a bound on nondeterministic bits, in other words the witness length. Definition 3.
Consider a language L ⊆ Σ ∗ . We write L ∈ DTIWI ( t ( n ) , w ( n )) if there exist languages U and V ⊆ Σ ∗ satisfying the following properties. • U ∈ DTIME ( O ( n )) , • Clo ( U ) ∈ DTIME ( O ( n )) , V ∈ DTIME ( O ( t ( n ))) , and • for all x ∈ Σ ∗ , x ∈ L if and only if x ∈ U and there exists s ∈ SUB ( w ( | x | )) such that s ( x ) ∈ V .We refer to V as a verification language for L with input string universe U . Remark 2.
The language U represents the set of input strings that will beconsidered. The language Clo ( U ) represents any partial filling of a string in U .We generally won’t pay too close attention to languages U and Clo ( U ) , but theyare important for restricting input strings to a specific format or encoding. Remark 3.
We offer a generic definition for
DTIWI that can be adapted fordifferent machine models. Later it will be important to specify a particularmachine model. At that time, we will write
Turing - DTIWI (and
Turing - DTIME )to specify that the machines are multitape Turing machines that allow for readand write access to all tapes (including the input tape).
There are many different ways to encode mathematical structures as inputstrings over a fixed size alphabet. Therefore, common decision problems cantake many different forms as formal languages. Furthermore, in order to puta problem within
DTIWI ( t ( n ) , w ( n )), we need to provide an encoding for itsinput strings where we put placeholder characters at the appropriate positions.Consider the following examples. Example 3.
Sumset Sum is represented such that each input is a sequence ofplaceholder and binary number pairs. When a placeholder is nondeterministi-cally filled with 0 that means the number is not to be included in the sum and 1means that the number is to be included.
Example 4.
SAT is represented such that each input has placeholder charactersout front followed by an encoding of a Boolean formula. Each variable is repre-sented as a binary number representing an index to a specific placeholder. Theplaceholders will be nondeterministically filled to create a variable assignment.
Notice that in Example 3 the placeholders occur throughout an input whilein Example 4 the placeholders appear as a prefix at the beginning of an input.
The following two lemmas demonstrate elegant structural properties relatingtime and nondeterminism. These structural properties will be essential to prov-ing speed-up theorems in Subsection 3.4 that reveal new relationships betweendeterministic and nondeterministic computations.
Lemma 4 (Translation Lemma) . If DTIWI ( t ( n ) , w ( n )) ⊆ DTIME ( t (cid:48) ( n )) , thenfor all efficiently computable functions w (cid:48) , DTIWI ( t ( n ) , w ( n ) + w (cid:48) ( n )) ⊆ DTIWI ( t (cid:48) ( n ) , w (cid:48) ( n )) . roof. Suppose that
DTIWI ( t ( n ) , w ( n )) ⊆ DTIME ( t (cid:48) ( n )).Let a function w (cid:48) be given. Let L ∈ DTIWI ( t ( n ) , w ( n ) + w (cid:48) ( n )) be given.By definition, there exist an input string universe U and a verification language V ∈ DTIME ( O ( t ( n ))) satisfying for all x ∈ Σ ∗ , x ∈ L if and only if x ∈ U andthere exists s ∈ SUB( w ( | x | ) + w (cid:48) ( | x | )) such that s ( x ) ∈ V . Consider a newlanguage L (cid:48) := { x ∈ Clo( U ) | ( ∃ s ∈ SUB( w ( | x | ))) s ( x ) ∈ V } . By interpreting V as a verification language for L (cid:48) with input string universeClo( U ), we get L (cid:48) ∈ DTIWI ( t ( n ) , w ( n )). By assumption, it follows that L (cid:48) ∈ DTIME ( t (cid:48) ( n )) . Finally, by interpreting L (cid:48) as a verification language for L with input stringuniverse U , we get L ∈ DTIWI ( t (cid:48) ( n ) , w (cid:48) ( n )). Lemma 5 (Padding Lemma) . If DTIWI ( t ( n ) , w ( n )) ⊆ DTIME ( t (cid:48) ( n )) , then forall efficiently computable functions f , DTIWI ( t ( f ( n )) , w ( f ( n ))) ⊆ DTIME ( t (cid:48) ( f ( n ))) . Proof.
Suppose that
DTIWI ( t ( n ) , w ( n )) ⊆ DTIME ( t (cid:48) ( n )).Let a function f be given. Let L ∈ DTIWI ( t ( f ( n )) , w ( f ( n ))) be given. Bydefinition, there exist an input string universe U and a verification language V ∈ DTIME ( O ( t ( f ( n ))))satisfying for all x ∈ Σ ∗ , x ∈ L if and only if x ∈ U and there exists s ∈ SUB( w ( f ( | x | )))such that s ( x ) ∈ V . Consider new languages L (cid:48) , V (cid:48) , and U (cid:48) such that L (cid:48) := { k − · · x | k + | x | = f ( | x | ) ∧ x ∈ L } V (cid:48) := { k − · · x | k + | x | = f ( | x | ) ∧ x ∈ V } U (cid:48) := { k − · · x | k ≥ ∧ x ∈ U } . Since V ∈ DTIME ( O ( t ( f ( n )))), assuming that f can be computed efficientlyand that the machine model is capable of ignoring the input string’s prefix of1’s, we have that V (cid:48) ∈ DTIME ( O ( t ( n ))). By interpreting V (cid:48) as a verificationlanguage for L (cid:48) with input string universe U (cid:48) , we get L (cid:48) ∈ DTIWI ( t ( n ) , w ( n )).By assumption, it follows that L (cid:48) ∈ DTIME ( t (cid:48) ( n )). Again, assuming that f canbe computed efficiently and that the machine model can efficiently prepend apadding prefix to the input string, we get L ∈ DTIME ( t (cid:48) ( f ( n ))). Remark 4.
Initially, we tried to use existing notions of limited nondetermin-ism to prove the preceding lemmas. However, the proofs were messy with inputsize and time complexity blow-ups. In contrast, our model for limited nondeter-minism (
DTIWI ) offers straightforward proofs with no input size blow-ups. .4 Speed-up Theorems Unlike prior works that investigated speed-up theorems for time and space trade-offs [36, 9], we focus specifically on speed-up results for time and nondeterminismtrade-offs. Our speed-up results are also distinct from recent hardness magnifi-cation results from [12] which amplify circuit lower bounds rather than speed-upcomputations.We prove the first speed-up theorem by repeatedly applying the structuralproperties of limited nondeterminism from the preceding subsection. Theorem 6 (First Speed-up Theorem) . Let α such that ≤ α < be given. If DTIWI ( n, log( n )) ⊆ DTIME ( n α ) , then for all k ∈ N , DTIWI ( n, (Σ ki =0 α i ) log( n )) ⊆ DTIME ( n α k +1 ) .Proof. Let α such that 1 ≤ α < α = 1 some of thefollowing formulas can be simplified, but the proof still holds for this case.)Suppose that DTIWI ( n, log( n )) ⊆ DTIME ( n α ). We prove by induction on k that for all k ∈ N , DTIWI ( n, (Σ ki =0 α i ) log( n )) ⊆ DTIME ( n α k +1 ) . The base case ( k = 0) is true by assumption. Now, let’s consider the induc-tion step. Suppose that DTIWI ( n, (Σ ki =0 α i ) log( n )) ⊆ DTIME ( n α k +1 ) . By applying this assumption with Lemma 4, we get that
DTIWI ( n, (Σ k +1 i =0 α i ) log( n )) ⊆ DTIWI ( n α k +1 , α k +1 · log( n )) . Next, we apply our initial assumption and Lemma 5 with f ( n ) = n α k +1 , w ( n ) =log( n ), and t ( n ) = n . Therefore, DTIWI ( n α k +1 , α k +1 · log( n )) ⊆ DTIME ( n α k +2 ) . It follows that
DTIWI ( n, (Σ k +1 i =0 α i ) log( n )) ⊆ DTIME ( n α k +2 ) . Remark 5.
Theorem 6 is a speed-up result because when ≤ α < , theexponent from the runtime divided by the constant factor for the witness stringlength decreases as k increases. In particular, we have lim k →∞ α k +1 Σ ki =0 α i = ( α − · lim k →∞ α k +1 α k +1 − α − < . This repeated application of structural properties leads to a kind of iterated simulation. emark 6. The value k in Theorem 6 is a constant. Replacing k by a functionof n may result in several difficulties including compounding overheads on timecomplexity, formalizing machine encodings, and resolving how to algorithmicallyapply the translation and padding lemmas on arbitrary machine encodings. We prove the second speed-up theorem by combining the first speed-up the-orem with the padding lemma.
Theorem 7 (Second Speed-up Theorem) . Suppose that g is an efficiently com-putable non-decreasing function such that ( ∀ n ∈ N ) g ( n ) ≤ n and g ( n ) is ω (log( n )) . Let α such that < α < be given. If DTIWI ( n, log( n )) ⊆ DTIME ( n α ) , then ( ∀ ε > DTIWI (poly( n ) , g ( n )) ⊆ DTIME (2 (1+ ε ) · ( α − · g ( n ) ) . Proof.
Let α such that 1 < α < z ( α, k ) = Σ ki =0 α i . Note that z ( α, k ) = α k +1 − α − . Suppose that
DTIWI ( n, log( n )) ⊆ DTIME ( n α ). Let ε > k ∈ N , DTIWI ( n, z ( α, k ) log( n )) ⊆ DTIME ( n α k +1 ) . Next, we apply Lemma 5 with f ( n ) = 2 g ( n ) /z ( α,k ) , w ( n ) = z ( α, k ) log( n ),and t ( n ) = n . Therefore, DTIWI (2 g ( n ) /z ( α,k ) , g ( n )) ⊆ DTIME (2 ( α k +1 ) · g ( n ) /z ( α,k ) ) . Since g ( n ) is ω (log( n )) and z ( α, k ) > g ( n ) /z ( α,k ) is super-polynomial. Hence, DTIWI (poly( n ) , g ( n )) ⊆ DTIME (2 ( α k +1 ) · g ( n ) /z ( α,k ) ) . Since lim k →∞ α k +1 α k +1 − , there exists k sufficiently large such that α k +1 α k +1 − ≤ ε. Therefore, by choosing sufficiently large k , we have DTIWI (poly( n ) , g ( n )) ⊆ DTIME (2 (1+ ε ) · ( α − · g ( n ) ) . emark 7. In Theorem 7, the exponent from the first runtime is α and theexponent from the second runtime is a factor of α − . This is a speed-up resultbecause as α approaches , the second runtime improves exponentially while thefirst runtime only improves polynomially. Corollary 8.
Suppose that g is an efficiently computable non-decreasing func-tion such that ( ∀ n ∈ N ) g ( n ) ≤ n and g ( n ) is ω (log( n )) . If for all α > , DTIWI ( n, log( n )) ⊆ DTIME ( n α ) , then ( ∀ ε > DTIWI (poly( n ) , g ( n )) ⊆ DTIME (2 ε · g ( n ) ) . Proof.
Follows directly from Theorem 7.
Satisfiability of bounded fan-in Boolean circuits with m gates and log( m ) inputsis denoted by log-CircuitSAT. We encode this problem so that the placeholdercharacters are out front followed by a conventional encoding of a bounded fan-in Boolean circuit. Such an encoding can be naturally carried out so that if n denotes the total input length and m denotes the number of gates, then n isΘ( m · log( m )). Using brute force search, a robust random access machine modelcan be used to solve this problem in roughly O ( n + m ) time by preprocessing thecircuit into a graph data structure for efficient evaluation and then evaluatingthe circuit on every possible input assignment.Whether or not we can solve log-CircuitSAT in O ( n − ε ) time for some ε > NP -complete problems. It is a well known result of Pippenger and Fischer (1979) that any O ( t ( n ))time bounded Turing machine can be simulated by an O ( t ( n ) · log( t ( n ))) timeoblivious Turing machine [30]. Moreover, any O ( t ( n )) time bounded Turingmachine can be simulated by O ( t ( n ) · log( t ( n ))) size Boolean circuits and suchcircuits can be computed efficiently . Such circuits can even be computed efficiently by a Turing machine. heorem 9 ([30, 8, 24]) . If L ∈ Turing - DTIME ( t ( n )) , then in O ( t ( n ) · poly(log( t ( n )))) time, we can compute Boolean circuits for L of size at most O ( t ( n ) · log( t ( n ))) . Next, we apply Theorem 9 to show that any problem in Turing-
DTIWI ( n, log( n ))is efficiently reducible to log-CircuitSAT. Theorem 10.
For all L ∈ Turing - DTIWI ( n, log( n )) , L is reducible to loga-rithmically many instances of log - CircuitSAT in quasilinear time by a Turingmachine.Proof.
Let L ∈ Turing-
DTIWI ( n, log( n )) be given. Let V ∈ Turing-
DTIME ( n )denote a verification language for L with input string universe U ∈ Turing-
DTIME ( n ) . Let an input string x ∈ U of length n be given. By Theorem 9, we cancompute an n input Boolean circuit C for V with at most O ( n · log( n )) gatesin quasilinear time on a Turing machine. Now, we construct a family of circuits { C i } [log( n )] such that for each i ∈ [log( n )], C i is obtained by fixing the charactersof x into the circuit C so that only i input bits remain where these input bitsare associated with the first i placeholders within x . Therefore, the circuit C i has at most log( n ) inputs and at most O ( n · log( n )) gates. It follows that x ∈ L if and only if there exists i ∈ [log( n )] such that C i is satisfiable. Remark 8.
To the best of our knowledge, it’s an open problem to determinewhether O ( t ( n )) time bounded random access machines can be simulated by O ( t ( n ) · log( t ( n ))) size Boolean circuits. As a result, Theorem 10 is currentlylimited to Turing machines. Corollary 11. If ( ∀ α >
1) log - CircuitSAT ∈ Turing - DTIME ( n α ) , then ( ∀ α >
1) Turing - DTIWI ( n, log( n )) ⊆ Turing - DTIME ( n α ) . Proof.
Follows directly from Theorem 10.
Corollary 12.
Suppose that g is an efficiently computable non-decreasing func-tion such that ( ∀ n ∈ N ) g ( n ) ≤ n and g ( n ) is ω (log( n )) . If ( ∀ α >
1) log - CircuitSAT ∈ Turing - DTIME ( n α ) , then ( ∀ ε >
0) Turing - DTIWI (poly( n ) , g ( n )) ⊆ Turing - DTIME (2 ε · g ( n ) ) . Proof.
Follows by combining Corollary 11 with Corollary 8. Since L and V are over a ternary alphabet, the input strings are encoded into binarybefore being fed into the Boolean circuits. Therefore, C will actually have slightly more than n input bits to account for encoding x in binary. .3 ETH-hardness We apply results from the preceding subsection to prove strong conditionallower bounds for log-CircuitSAT. In particular, we show that the existence ofquasilinear time algorithms for log-CircuitSAT would imply that ETH is falseand that NP -complete problems would have exponentially faster algorithms. Theorem 13. If ( ∀ α >
1) log - CircuitSAT ∈ Turing - DTIME ( n α ) , then ( ∀ ε >
0) CircuitSAT ∈ Turing - DTIME (poly( n ) · ε · m ) where m is the number of gates.Proof. Suppose that for all α >
1, log-CircuitSAT ∈ Turing-
DTIME ( n α ). Byapplying Corollary 12 with g ( n ) = n log( n ) , we get that( ∀ ε >
0) Turing-
DTIWI (poly( n ) , n log( n ) ) ⊆ Turing-
DTIME (2 ε · n log( n ) ) . Recall that we encode Boolean circuits so that n is Θ( m · log( m )) where m is the number of gates. In fact, under reasonable encoding conventions, n log( n ) will actually be larger than the number of gates and inputs. Hence,CircuitSAT ∈ Turing-
DTIWI (poly( n ) , n log( n ) ) . Therefore, ( ∀ ε >
0) CircuitSAT ∈ Turing-
DTIME (2 ε · n log( n ) ). It follows that( ∀ ε >
0) CircuitSAT ∈ Turing-
DTIME (poly( n ) · ε · m ) . Corollary 14. If ( ∀ α >
1) log - CircuitSAT ∈ Turing - DTIME ( n α ) , then ETH isfalse.Proof. Because 3-CNF-SAT is a special case of CircuitSAT, Theorem 13 impliesthat ( ∀ ε >
0) 3-CNF-SAT ∈ Turing-
DTIME (poly( n ) · ε · m )where m is the number of bounded AND, OR, and NOT gates (which is ap-proximately three times the number of clauses). By applying the SparsificationLemma [23], we get that( ∀ ε >
0) 3-CNF-SAT ∈ Turing-
DTIME (poly( n ) · ε · v )where v is the number of variables. It follows that ETH is false. In the preceding subsection, we proved strong conditional lower bounds for thelog-CircuitSAT problem. In particular, we showed that quasilinear time algo-rithms for log-CircuitSAT would imply the existence of exponentially more effi-cient algorithms for NP -complete problems. Now, we consider how quasilinear12ime algorithms for log-CircuitSAT would imply the existence of quasilinear timealgorithms for polynomial time problems such as OVP, 3SUM, Triangle Finding,and even k -Clique for all k .Before proving these results, we briefly state the definitions for the OVP,3SUM, Triangle Finding, and k -Clique decision problems. • Orthogonal vectors decision problem (OVP) is defined as follows. Givena list of binary vectors, do there exist two vectors that are orthogonal? • • Triangle Finding decision problem is defined as follows. Given an undi-rected graph, do there exist three vertices that form a triangle? • k -Clique decision problem is defined as follows for any fixed number k .Given an undirected graph, do there exist k vertices that form a clique(i.e. a graph where every pair of vertices is adjacent)? Corollary 15.
OVP , , and Triangle Finding are reducible to log - CircuitSAT in quasilinear time. Moreover, log - CircuitSAT is not solvable in quasilinear timeunder the assumption that quasilinear time algorithms do not exist for at leastone of the problems
OVP , , or Triangle Finding .Proof.
We observe that OVP, 3SUM, and Triangle Finding (under appropriateencodings) are all in Turing-
DTIWI ( n · poly(log( n )) , log( n )) where n denotes thetotal input string length. • For OVP, we are given a list of bit vectors. We use nondeterministic bitsto guess one of the bit vectors. Then, we compute the dot product betweenit and all other bit vectors. • For 3SUM, we are given a list of binary numbers (appropriately paddedso that they all have the same length). We use nondeterministic bits toguess one of the binary numbers x . Then, for all other binary numbers,we multiply by 2 and add x . Finally, we solve the 2SUM problem on theresulting list of binary numbers. • For Triangle Finding, we are given a list of edges represented as a pair ofvertices. We use nondeterministic bits to guess one of the edges. Then,we check if this edge combined with an additional vertex forms a triangle.By applying a slight variation to Theorem 10 for quasilinear time verifiers withlog( n ) length witnesses, we get that OVP, 3SUM, and Triangle Finding areall quasilinear time reducible to log-CircuitSAT. Therefore, quasilinear timealgorithms for log-CircuitSAT would imply the existence of quasilinear timealgorithms for OVP, 3SUM, and Triangle Finding which is currently not known. Notice that Triangle Finding is equivalent to 3-Clique. heorem 16. If for all α > , DTIWI ( n, log( n )) ⊆ DTIME ( n α ) , then ( ∀ k ) ( ∀ α > DTIWI ( n, k · log( n )) ⊆ DTIME ( n α ) . Proof.
Suppose that for all α > DTIWI ( n, log( n )) ⊆ DTIME ( n α ). By Theo-rem 6, for all α such that 1 ≤ α < k ∈ N , DTIWI ( n, (Σ ki =0 α i ) log( n )) ⊆ DTIME ( n α k +1 ) . Notice that when α >
1, we have k ≤ (Σ ki =0 α i ). Therefore, for all α such that1 ≤ α < k ∈ N , DTIWI ( n, k · log( n )) ⊆ DTIME ( n α k +1 ) . Let k ∈ N and α > α > α k +12 ≤ α (just take α = α / ( k +1)1 ). It follows that DTIWI ( n, k · log( n )) ⊆ DTIME ( n α k +12 ) ⊆ DTIME ( n α ) . Corollary 17. If ( ∀ α >
1) log - CircuitSAT ∈ Turing - DTIME ( n α ) , then ( ∀ k ) ( ∀ α >
1) Turing - DTIWI ( n, k · log( n )) ⊆ Turing - DTIME ( n α ) . Proof.
Follows by combining Corollary 11 with Theorem 16.
Corollary 18. log - CircuitSAT is not solvable in quasilinear time under theassumption that quasilinear time algorithms do not exist for k - Clique for somesufficiently large fixed k .Proof. Follows by combining Corollary 17 with the observation that k -Clique ∈ Turing-
DTIWI ( n, k · log( n )) for all fixed k . Remark 9.
Although we do not focus on parameterized complexity theory here,the preceding arguments can also be used to show that if log - CircuitSAT is solv-able in quasilinear time, then W [ ] ⊂ non-uniform- FPT . Moreover, we suggestthat this implication could be extended to W [ P ] ⊂ non-uniform- FPT . We referthe reader to [3] for background information on W [ P ] and the W hierarchy. Although it may be more common for n to denote the number of vertices, we instead use n to denote the total length of the input string which encodes the graph. Conclusion
In this work, we have demonstrated strong conditional lower bounds for thelog-CircuitSAT decision problem by carefully investigating properties of limitednondeterminism. In particular, in Corollary 14 we showed that the existence ofquasilinear time Turing machines for log-CircuitSAT would imply that ETH isfalse. This means that a polynomial time improvement for log-CircuitSAT wouldlead to an exponential time improvement for NP -complete problems. Throughthis investigation we revealed new relationships between deterministic and non-deterministic computations.The main two open problems of this work are as follows. • Does Corollary 14 hold for random access machine models? This questionis related to the well known question of whether linear time for randomaccess machines can be simulated in subquadratic time by multitape Tur-ing machines [14]. It is also related to whether random access machinescan be made oblivious [19]. • Can the construction from the first speed up theorem (Theorem 6) becarried out for a non-constant number of iterations k . We speculate thatif it can, then DTIWI ( n, log( n )) ⊆ DTIME ( n · log( n )) would imply that NTIME ( n ) ⊆ DTIME (2 √ n ).Also, although this work doesn’t focus on circuit lower bounds, we suggestthat recent results connecting the existence of faster algorithms with circuitlower bounds [1, 37, 36, 6] could be applied to show that the existence of fasteralgorithms for log-CircuitSAT would imply new circuit lower bounds for E NP as well as other complexity classes.Finally, we leave the reader with the thought that the speed-up theorems forlimited nondeterminism (Theorems 6 and 7) might be a special case of a moregeneral speed-up result connecting nondeterminism, alternation, and time. Acknowledgements
We greatly appreciate all of the help and suggestions that we received. Weare especially grateful to Kenneth Regan and Jonathan Buss who shared anunpublished manuscript [9] on proposed speed-up results relating time and spacecomplexity. We also thank Michael Fischer, Mike Paterson, and Nick Pippenger,who tracked down and shared two unpublished manuscripts related to circuitsimulations. In addition, we thank Karl Bringmann whose advice helped us tobetter align this work with recent advances in fine-grained complexity. Likewise,we recognize helpful discussions with Henning Fernau and all of the participantsat the workshop on Modern Aspects of Complexity Within Formal Languages(sponsored by DFG). Finally, we very much appreciate all of the feedback fromJoseph Swernofsky. 15 eferences [1] Amir Abboud, Thomas Dueholm Hansen, Virginia Vassilevska Williams,and Ryan Williams. Simulating branching programs with edit distanceand friends: Or: a polylog shaved is a lower bound made. In
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