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A Survey of Lower Bounds for Satisfiability and Related Problems
 Foundations and Trends in Theoretical Computer Science
, 2007
"... Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a lineartime, logarithmicspace algorithm for satisfiability was not ruled out. In 1997 Fortnow ..."
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Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a lineartime, logarithmicspace algorithm for satisfiability was not ruled out. In 1997 Fortnow, building on earlier work by Kannan, ruled out such an algorithm. Since then there has been a significant amount of progress giving nontrivial lower bounds on the computational complexity of satisfiability. In this article we survey the known lower bounds for the time and space complexity of satisfiability and closely related problems on deterministic, randomized, and quantum models with random access. We discuss the stateoftheart results and present the underlying arguments in a unified framework. 1
TimeSpace Tradeoffs for Counting NP Solutions Modulo Integers
 In Proceedings of the 22nd IEEE Conference on Computational Complexity
, 2007
"... We prove the first timespace tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known timespace tradeoffs for Sat. Let m> 0 be an integer, and define MODmSat to be the problem of determining if a given Boolean formula has exactly km satisf ..."
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We prove the first timespace tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known timespace tradeoffs for Sat. Let m> 0 be an integer, and define MODmSat to be the problem of determining if a given Boolean formula has exactly km satisfying assignments, for some integer k. We show for all primes p except for possibly one of them, and for all c < 2cos(π/7) ≈ 1.801, there is a d> 0 such that MODpSat is not solvable in n c time and n d space by general algorithms. That is, there is at most one prime p that does not satisfy the tradeoff. We prove that the same limitation holds for Sat and MOD6Sat, as well as MODmSat for any composite m that is not a prime power. Our main tool is a general method for rapidly simulating deterministic computations with restricted space, by counting the number of solutions to NP predicates modulo integers. The simulation converts an ordinary algorithm into a “canonical ” one that consumes roughly the same amount of time and space, yet canonical algorithms have nice properties suitable for counting.
On the possibility of faster SAT algorithms
"... We describe reductions from the problem of determining the satisfiability of Boolean CNF formulas (CNFSAT) to several natural algorithmic problems. We show that attaining any of the following bounds would improve the state of the art in algorithms for SAT: • an O(n k−ε) algorithm for kDominating S ..."
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We describe reductions from the problem of determining the satisfiability of Boolean CNF formulas (CNFSAT) to several natural algorithmic problems. We show that attaining any of the following bounds would improve the state of the art in algorithms for SAT: • an O(n k−ε) algorithm for kDominating Set, for any k ≥ 3, • a (computationally efficient) protocol for 3party set disjointness with o(m) bits of communication, • an n o(d) algorithm for dSUM, • an O(n 2−ε) algorithm for 2SAT with m = n 1+o(1) clauses, where two clauses may have unrestricted length, and • an O((n + m) k−ε) algorithm for HornSat with k unrestricted length clauses. One may interpret our reductions as new attacks on the complexity of SAT, or sharp lower bounds conditional on exponential hardness of SAT.
Automated proofs of time lower bounds
, 2007
"... A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6SAT, MajorityofMajoritySAT, and Tautologies, to name a few. The ..."
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A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6SAT, MajorityofMajoritySAT, and Tautologies, to name a few. These lower bound proofs all follow a certain diagonalizationbased proofbycontradiction strategy. A pressing open problem has been to determine how powerful such proofs can possibly be. We propose an automated theoremproving methodology for studying these lower bound problems. In particular, we prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. We describe an implementation of a smallscale theorem prover and discover surprising experimental results. In some settings, our program provides strong evidence that the best known lower bound proofs are already optimal for the current framework, contradicting the consensus intuition; in others, the program guides us to improved lower bounds where none had been known for years.
NonLinear Time Lower Bound for (Succinct) Quantified Boolean Formulas
"... Abstract. We give a reduction from arbitrary languages in alternating time t(n) to quantified Boolean formulas (QBF) describable in O(t(n)) bits. The reduction works for a reasonable succinct encoding of Boolean formulas and for several reasonable machine models, including multitape Turing machines ..."
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Abstract. We give a reduction from arbitrary languages in alternating time t(n) to quantified Boolean formulas (QBF) describable in O(t(n)) bits. The reduction works for a reasonable succinct encoding of Boolean formulas and for several reasonable machine models, including multitape Turing machines and logarithmiccost RAMs. By a simple diagonalization, it follows that our succinct QBF problem requires superlinear time on those models. To our knowledge this is the first known instance of a nonlinear time lower bound (with no space restriction) for solving a natural linear space problem on a variety of computational models.
The Complexity of Linear Dependence Problems in Vector Spaces
"... We study extensions of the natural kSUM problem to vector spaces over finite fields. Given a subset S ⊆ F n q of size r ≤ q n, an integer k, 2 ≤ k ≤ n, and a vector v ∈ (F k q \ {0}) k, we define the TargetSum problem to be the problem of finding k elements xi1,..., xik ∈ S for which ∑k vjxij j=1 = ..."
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We study extensions of the natural kSUM problem to vector spaces over finite fields. Given a subset S ⊆ F n q of size r ≤ q n, an integer k, 2 ≤ k ≤ n, and a vector v ∈ (F k q \ {0}) k, we define the TargetSum problem to be the problem of finding k elements xi1,..., xik ∈ S for which ∑k vjxij j=1 = z, where z may either be an input or a fixed vector. We also study a variant of this, where instead of finding xi1,..., xik ∈ S for which ∑k vjxij = z, we require that z be in j=1 span(xi1,..., xik), which we call the (k, r)LinDependenceq problem. These problems are natural generalizations of wellstudied problems that occur in coding theory and property testing. Indeed, the (k, r)LinDependenceq problem is just the Maximum Likelihood Decoding problem. Also, in the TargetSum problem, if instead of general z we require z = 0n, then this is the Weight Distribution problem. In property testing, these problems have been implicitly studied in the context of testing linearinvariant properties. We give nearly optimal bounds for TargetSum and (k, r)LinDependenceq for every r, k, and constant q. Namely, assuming 3SAT requires exponential time, we show that any algorithm for these problems must run in min(2Θ(n) , rΘ(k) ) time. Moreover, we give deterministic upper bounds that match this complexity, up to small factors. Our lower bound strengthens and simplifies an earlier min(2Θ(n) , rΩ(k1/4)) lower bound for both the Maximum Likelihood Decoding and Weight Distribution problems. We also give upper and lower bounds for variants of these problems, e.g., for the problem for which we must find xi1,..., xik for which z ∈ span(xi1,..., xik) but z is not spanned by any proper subset of these vectors, and for the counting version of this problem. Part of this work was done while the author was an intern at IBM Almaden.
AT&T Labs
"... We describe reductions from the problem of determining the satisfiability of Boolean CNF formulas (CNFSAT) to several natural algorithmic problems. We show that attaining any of the following bounds would improve the state of the art in algorithms for SAT: • an O(n k−ε) algorithm for kDominating S ..."
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We describe reductions from the problem of determining the satisfiability of Boolean CNF formulas (CNFSAT) to several natural algorithmic problems. We show that attaining any of the following bounds would improve the state of the art in algorithms for SAT: • an O(n k−ε) algorithm for kDominating Set, for any k ≥ 3, • a (computationally efficient) protocol for 3party set disjointness with o(m) bits of communication, • an n o(d) algorithm for dSUM, • an O(n 2−ε) algorithm for 2SAT formulas with m = n 1+o(1) clauses, where two clauses may have unrestricted length, and • an O((n + m) k−ε) algorithm for HornSat with k unrestricted length clauses. One may interpret our reductions as new attacks on the complexity of SAT, or sharp lower bounds conditional on exponential hardness of SAT. ∗This work was done while the author was a Raviv Postdoctoral
www.stacsconf.org ALTERNATIONTRADING PROOFS, LINEAR PROGRAMMING, AND
, 2010
"... Abstract. A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6SAT, MajorityofMajoritySAT, and Tautologies, to name ..."
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Abstract. A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6SAT, MajorityofMajoritySAT, and Tautologies, to name a few. The proofs of these lower bounds follow a certain proofbycontradiction strategy that we call alternationtrading. An important open problem is to determine how powerful such proofs can possibly be. We propose a methodology for studying these proofs that makes them amenable to both formal analysis and automated theorem proving. We prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. Implementing a smallscale theorem prover based on this result, we extract new humanreadable time lower bounds for several problems. This framework can also be used to prove concrete limitations on the current techniques. 1.
unknown title
"... interest in STACS has remained at a high level over the past years. The STACS 2010 call for papers led to over 238 submissions from 40 countries. Each paper was assigned to three program committee members. The committee selected 54 papers during a two week electronic meeting held in November. As co ..."
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interest in STACS has remained at a high level over the past years. The STACS 2010 call for papers led to over 238 submissions from 40 countries. Each paper was assigned to three program committee members. The committee selected 54 papers during a two week electronic meeting held in November. As cochairs of the program committee, we would like to sincerely thank its members and the many external referees for their valuable work. In particular, there were intense and interesting discussions. The overall very high quality of the submissions made the selection a difficult task. We would like to express our thanks to the three invited speakers, Mikołaj Bojańczyk, Rolf Niedermeier, and Jacques Stern. Special thanks go to Andrei Voronkov for his EasyChair software (www.easychair.org). Moreover, we would like to warmly thank Wadie Guizani for preparing the conference proceedings and continuous help throughout the conference organization. For the third time, this year’s STACS proceedings are published in electronic form. A printed version was also available at the conference, with ISBN. The electronic proceedings are available through several portals, and in particular through HAL and LIPIcs series. The proceedings of the Symposium, which are published electronically in