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  • 1. Atserias, A.
    et al.
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Nordström, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Narrow proofs may be maximally long2014In: Proceedings of the Annual IEEE Conference on Computational Complexity, 2014, p. 286-297Conference paper (Refereed)
    Abstract [en]

    We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size nω(w). This shows that the simple counting argument that any formula refutable in width w must have a proof in size nO(ω) is essentially tight. Moreover, our lower bounds can be generalized to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. Our results do not extend all the way to Lasserre, however-the formulas we study have Lasserre proofs of constant rank and size polynomial in both n and w.

  • 2. Beyersdorff, Olaf
    et al.
    Galesi, Nicola
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    A characterization of tree-like Resolution size2013In: Information Processing Letters, ISSN 0020-0190, E-ISSN 1872-6119, Vol. 113, no 18, p. 666-671Article in journal (Refereed)
    Abstract [en]

    We explain an asymmetric Prover-Delayer game which precisely characterizes proof size in tree-like Resolution. This game was previously described in a parameterized complexity context to show lower bounds for parameterized formulas (Beyersdorff et al. (2013) [2]) and for the classical pigeonhole principle (Beyersdorff et al. (2010) [1]). The main point of this note is to show that the asymmetric game in fact characterizes tree-like Resolution proof size, i.e. in principle our proof method allows to always achieve the optimal lower bounds. This is in contrast with previous techniques described in the literature. We also provide a very intuitive information-theoretic interpretation of the game.

  • 3. Beyersdorff, Olaf
    et al.
    Galesi, Nicola
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    A Characterization of Tree-Like Resolution Size2012In: Electronic Colloquium on Computational Complexity (ECCC), ISSN 1433-8092, no 161Article in journal (Refereed)
    Abstract [en]

    We explain an asymmetric Prover-Delayer game which precisely characterizes proof size in tree-like Resolution. This game was previously described in a parameterized complexity context to show lower bounds for parameterized formulas [BGL11] and for the classical pigeonhole principle [BGL10]. The main point of this note is to show that the asymmetric game in fact characterizes tree-like Resolution proof size, i. e. in principle our proof method allows to always achieve the optimal lower bounds. This is in contrast with previous techniques described in the literature. We also provide a very intuitive information-theoretic interpretation of the game.

  • 4. Beyersdorff, Olaf
    et al.
    Galesi, Nicola
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Parameterized Complexity of DPLL Search Procedures2013In: ACM Transactions on Computational Logic, ISSN 1529-3785, E-ISSN 1557-945X, Vol. 14, no 3, p. 20-Article in journal (Refereed)
    Abstract [en]

    We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game that models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k-clique requires n(Omega(k)) steps for a nontrivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked by Beyersdorff et al. [2012] of understanding the Resolution complexity of this family of formulas.

  • 5. Beyersdorff, Olaf
    et al.
    Galesi, Nicola
    Lauria, Massimo
    Razborov, Alexander A.
    Parameterized Bounded-Depth Frege Is not Optimal2012In: ACM Transactions on Computation Theory, ISSN 1942-3454, E-ISSN 1942-3462, Vol. 4, no 3, p. 7-Article in journal (Refereed)
    Abstract [en]

    A general framework for parameterized proof complexity was introduced by Dantchev et al. [2007]. There, the authors show important results on tree-like Parameterized Resolution-a parameterized version of classical Resolution-and their gap complexity theorem implies lower bounds for that system. The main result of this article significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n Ω(k) in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in Dantchev et al. [2007]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNFs

  • 6. Chan, S. M.
    et al.
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Nordstrom, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Vinyals, Marc
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Hardness of Approximation in PSPACE and Separation Results for Pebble Games2015In: Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS, Institute of Electrical and Electronics Engineers (IEEE), 2015, p. 466-485Conference paper (Refereed)
    Abstract [en]

    We consider the pebble game on DAGs with bounded fan-in introduced in [Paterson and Hewitt '70] and the reversible version of this game in [Bennett '89], and study the question of how hard it is to decide exactly or approximately the number of pebbles needed for a given DAG in these games. We prove that the problem of deciding whether s pebbles suffice to reversibly pebble a DAG G is PSPACE-complete, as was previously shown for the standard pebble game in [Gilbert, Lengauer and Tarjan '80]. Via two different graph product constructions we then strengthen these results to establish that both standard and reversible pebbling space are PSPACE-hard to approximate to within any additive constant. To the best of our knowledge, these are the first hardness of approximation results for pebble games in an unrestricted setting (even for polynomial time). Also, since [Chan '13] proved that reversible pebbling is equivalent to the games in [Dymond and Tompa '85] and [Raz and McKenzie '99], our results apply to the Dymond - Tompa and Raz - McKenzie games as well, and from the same paper it follows that resolution depth is PSPACE-hard to determine up to any additive constant. We also obtain a multiplicative logarithmic separation between reversible and standard pebbling space. This improves on the additive logarithmic separation previously known and could plausibly be tight, although we are not able to prove this. We leave as an interesting open problem whether our additive hardness of approximation result could be strengthened to a multiplicative bound if the computational resources are decreased from polynomial space to the more common setting of polynomial time. © 2015 IEEE.

  • 7. Filmus, Y.
    et al.
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Mikša, Mladen
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Nordström, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Vinyals, M.
    From small space to small width in resolution2015In: ACM Transactions on Computational Logic, ISSN 1529-3785, E-ISSN 1557-945X, Vol. 16, no 4, article id 28Article in journal (Refereed)
    Abstract [en]

    In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of a Conjunctive Normal Form (CNF) formula is always an upper bound on the width needed to refute the formula. Their proof is beautiful but uses a nonconstructive argument based on Ehrenfeucht-Fraïssé games. We give an alternative, more explicit, proof that works by simple syntactic manipulations of resolution refutations. As a by-product, we develop a "black-box" technique for proving space lower bounds via a "static" complexitymeasure that works against any resolution refutation-previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similarmethods.

  • 8. Filmus, Y.
    et al.
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Mikša, Mladen
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Nordström, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Vinyals, Marc
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    From small space to small width in resolution2014In: 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014), Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing , 2014, Vol. 25, p. 300-311Conference paper (Refereed)
    Abstract [en]

    In 2003, Atserias and Dalmau resolved a major open question about the resolution proof system by establishing that the space complexity of formulas is always an upper bound on the width needed to refute them. Their proof is beautiful but somewhat mysterious in that it relies heavily on tools from finite model theory. We give an alternative, completely elementary, proof that works by simple syntactic manipulations of resolution refutations. As a by-product, we develop a "black-box" technique for proving space lower bounds via a "static" complexity measure that works against any resolution refutation-previous techniques have been inherently adaptive. We conclude by showing that the related question for polynomial calculus (i.e., whether space is an upper bound on degree) seems unlikely to be resolvable by similar methods.

  • 9. Filmus, Yuval
    et al.
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Mikša, Mladen
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Nordström, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Vinyals, Marc
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Towards an understanding of polynomial calculus: New separations and lower bounds (extended abstract)2013In: Automata, Languages, and Programming: 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, Springer, 2013, no PART 1, p. 437-448Conference paper (Refereed)
    Abstract [en]

    During the last decade, an active line of research in proof complexity has been into the space complexity of proofs and how space is related to other measures. By now these aspects of resolution are fairly well understood, but many open problems remain for the related but stronger polynomial calculus (PC/PCR) proof system. For instance, the space complexity of many standard "benchmark formulas" is still open, as well as the relation of space to size and degree in PC/PCR. We prove that if a formula requires large resolution width, then making XOR substitution yields a formula requiring large PCR space, providing some circumstantial evidence that degree might be a lower bound for space. More importantly, this immediately yields formulas that are very hard for space but very easy for size, exhibiting a size-space separation similar to what is known for resolution. Using related ideas, we show that if a graph has good expansion and in addition its edge set can be partitioned into short cycles, then the Tseitin formula over this graph requires large PCR space. In particular, Tseitin formulas over random 4-regular graphs almost surely require space at least Ω(√n). Our proofs use techniques recently introduced in [Bonacina-Galesi '13]. Our final contribution, however, is to show that these techniques provably cannot yield non-constant space lower bounds for the functional pigeonhole principle, delineating the limitations of this framework and suggesting that we are still far from characterizing PC/PCR space.

  • 10. Filmus, Yuval
    et al.
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Nordstrom, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Ron-Zewi, Noga
    Thapen, Neil
    Space complexity in polynomial calculus2015In: SIAM journal on computing (Print), ISSN 0097-5397, E-ISSN 1095-7111, Vol. 44, no 4, p. 1119-1153Article in journal (Refereed)
    Abstract [en]

    During the last 10 to 15 years, an active line of research in proof complexity has been to study space complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important concern in SAT solving, and so research has mostly focused on weak systems that are used by SAT solvers. There has been a relatively long sequence of papers on space in resolution, which is now reasonably well-understood from this point of view. For other proof systems of interest, however, such as polynomial calculus or cutting planes, progress has been more limited. Essentially nothing has been known about space complexity in cutting planes, and for polynomial calculus the only lower bound has been for conjunctive normal form (CNF) formulas of unbounded width in [Alekhnovich et al., SIAM J. Comput., 31 (2002), pp. 1184-1211], where the space lower bound is smaller than the initial width of the clauses in the formulas. Thus, in particular, it has been consistent with current knowledge that polynomial calculus could be able to refute any k-CNF formula in constant space. In this paper, we prove several new results on space in polynomial calculus (PC) and in the extended proof system polynomial calculus resolution (PCR) studied by Alekhnovich et al.: (1) We prove an omega(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole principle formulas PHPmn with m pigeons and n holes, and show that this is tight. (2) For PCR, we prove an omega(n) space lower bound for a bitwise encoding of the functional pigeonhole principle. These formulas have width O(log n), and hence this is an exponential improvement over Alekhnovich et al. measured in the width of the formulas. (3) We then present another encoding of the pigeonhole principle that has constant width, and prove an omega(n) space lower bound in PCR for these formulas as well. (4) Finally, we prove that any k-CNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not obviously the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3-CNF in the canonical way, something that we believe can be useful when proving PCR space lower bounds for other well-studied formula families in proof complexity.

  • 11. Filmus, Yuval
    et al.
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Nordström, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Ron-Zewi, Noga
    Thapen, Neil
    Space Complexity in Polynomial Calculus2012In: Electronic Colloquium on Computational Complexity (ECCC), ISSN 1433-8092, no 132Article in journal (Refereed)
    Abstract [en]

    During the last decade, an active line of research in proof complexity has been to study space complexity and time-space trade-offs for proofs. Besides being a natural complexity measure of intrinsic interest, space is also an important issue in SAT solving, and so research has mostly focused on weak systems that are used by SAT solvers.

    There has been a relatively long sequence of papers on space in resolution, which is now reasonably well understood from this point of view.  For other natural candidates to study, however, such as polynomial calculus or cutting planes, very little has been known. We are not aware of any nontrivial space lower bounds for cutting planes, and for polynomial calculus the only lower bound has been for CNF formulas of unbounded width in [Alekhnovich et al. '02], where the space lower bound is smaller than the initial width of the clauses in the formulas.  Thus, in particular, it has been consistent with current knowledge that polynomial calculus could be able to refute any k-CNF formula in constant space.

    In this paper, we prove several new results on space in polynomial calculus (PC), and in the extended proof system polynomial calculus resolution (PCR) studied in [Alekhnovich et al. '02]:

    (1) We prove an Omega(n) space lower bound in PC for the canonical 3-CNF version of the pigeonhole principle formulas PHP^m_n with m pigeons and n holes, and show that this is tight.

    (2) For PCR, we prove an Omega(n) space lower bound for a bitwise encoding of the functional pigeonhole principle.  These formulas have width O(log n), and hence this is an exponential improvement over [Alekhnovich et al. '02] measured in the width of the formulas.

    (3) We then present another encoding of the pigeonhole principle that has constant width, and prove an Omega(n) space lower bound in PCR for these formulas as well.

    (4) Finally, we prove that any k-CNF formula can be refuted in PC in simultaneous exponential size and linear space (which holds for resolution and thus for PCR, but was not obviously the case for PC). We also characterize a natural class of CNF formulas for which the space complexity in resolution and PCR does not change when the formula is transformed into 3-CNF in the canonical way, something that we believe can be useful when proving PCR space lower bounds for other well-studied formula families in proof complexity.

  • 12.
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    A rank lower bound for cutting planes proofs of Ramsey Theorem2012In: Electronic Colloquium on Computational Complexity (ECCC), ISSN 1433-8092, no 124Article in journal (Other academic)
    Abstract [en]

    Ramsey Theorem is a cornerstone of combinatorics and logic. In its simplest formulation it says that there is a function r such that any simple graph with r(k, s) vertices contains either a clique of size k or an independent set of size s. We study the complexity of proving upper bounds for the number r(k, k). In particular we focus on the propositional proof system cutting planes; we prove that the upper bound “r(k, k) 4k” requires cutting planes proof of high rank. In order to do that we show a protection lemma which could be of independent interest.

  • 13.
    Lauria, Massimo
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    A rank lower bound for cutting planes proofs of Ramsey's theorem2013In: Theory and Applications of Satisfiability Testing – SAT 2013: 16th International Conference, Helsinki, Finland, July 8-12, 2013. Proceedings, Springer, 2013, p. 351-364Conference paper (Refereed)
    Abstract [en]

    Ramsey's Theorem is a cornerstone of combinatorics and logic. In its simplest formulation it says that there is a function r such that any simple graph with r(k,s) vertices contains either a clique of size k or an independent set of size s. We study the complexity of proving upper bounds for the number r(k,k). In particular we focus on the propositional proof system cutting planes; we prove that the upper bound "r(k,k) ≤ 4k" requires cutting planes proof of high rank. In order to do that we show a protection lemma which could be of independent interest.

  • 14.
    Lauria, Massimo
    et al.
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Nordström, Jakob
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Tight size-degree bounds for sums-of-squares proofs2015In: Leibniz International Proceedings in Informatics, LIPIcs, Dagstuhl Publishing , 2015, Vol. 33, p. 448-466Conference paper (Refereed)
    Abstract [en]

    We exhibit families of 4-CNF formulas over n variables that have sums-of-squares (SOS) proofs of unsatisfiability of degree (a.k.a. rank) d but require SOS proofs of size nΩ(d) for values of d = d(n) from constant all the way up to nδ for some universal constant δ. This shows that the nO(d) running time obtained by using the Lasserre semidefinite programming relaxations to find degree-d SOS proofs is optimal up to constant factors in the exponent. We establish this result by combining NP-reductions expressible as low-degree SOS derivations with the idea of relativizing CNF formulas in [Krajícek’04] and [Dantchev and Riis’03], and then applying a restriction argument as in [Atserias, Müller, and Oliva’13] and [Atserias, Lauria, and Nordström’14]. This yields a generic method of amplifying SOS degree lower bounds to size lower bounds, and also generalizes the approach in [ALN14] to obtain size lower bounds for the proof systems resolution, polynomial calculus, and Sherali-Adams from lower bounds on width, degree, and rank, respectively.

  • 15.
    Lauria, Massimo
    et al.
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Pudlak, Pavel
    Rodl, Vojtch
    Thapen, Neil
    The complexity of proving that a graph is Ramsey2017In: Combinatorica, ISSN 0209-9683, E-ISSN 1439-6912, Vol. 37, no 2, p. 253-268Article in journal (Refereed)
    Abstract [en]

    We say that a graph with n vertices is c-Ramsey if it does not contain either a clique or an independent set of size c log n. We define a CNF formula which expresses this property for a graph G. We show a superpolynomial lower bound on the length of resolution proofs that G is c-Ramsey, for every graph G. Our proof makes use of the fact that every c-Ramsey graph must contain a large subgraph with some properties typical for random graphs.

  • 16.
    Lauria, Massimo
    et al.
    KTH, School of Computer Science and Communication (CSC), Theoretical Computer Science, TCS.
    Pudlák, P.
    Rödl, V.
    Thapen, N.
    The complexity of proving that a graph is Ramsey2013In: Automata, Languages, and Programming: 40th International Colloquium, ICALP 2013, Riga, Latvia, July 8-12, 2013, Proceedings, Part I, Springer, 2013, no PART 1, p. 684-695Conference paper (Refereed)
    Abstract [en]

    We say that a graph with n vertices is c-Ramsey if it does not contain either a clique or an independent set of size c log n. We define a CNF formula which expresses this property for a graph G. We show a superpolynomial lower bound on the length of resolution proofs that G is c-Ramsey, for every graph G. Our proof makes use of the fact that every Ramsey graph must contain a large subgraph with some of the statistical properties of the random graph.

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