Showing papers by "Alexander A. Razborov published in 2011"

On the Caccetta-Haggkvist conjecture with forbidden subgraphs

Abstract: The Caccetta-Haggkvist conjecture made in 1978 asserts that every orgraph on n vertices without oriented cycles of length <= l must contain a vertex of outdegree at most (n-1)/l. It has a rather elaborate set of (conjectured) extremal configurations. In this paper we consider the case l=3 that received quite a significant attention in the literature. We identify three orgraphs on four vertices each that are missing as an induced subgraph in all known extremal examples and prove the Caccetta-Haggkvist conjecture for orgraphs missing as induced subgraphs any of these orgraphs, along with cycles of length 3. Using a standard trick, we can also lift the restriction of being induced, although this makes graphs in our list slightly more complicated.

Non-three-colorable common graphs exist

Abstract: A graph H is called common if the total number of copies of H in every graph and its complement asymptotically minimizes for random graphs. A former conjecture of Burr and Rosta, extending a conjecture of Erdos asserted that every graph is common. Thomason disproved both conjectures by showing that the complete graph of order four is not common. It is now known that in fact the common graphs are very rare. Answering a question of Sidorenko and of Jagger, Stovicek and Thomason from 1996 we show that the 5-wheel is common. This provides the first example of a common graph that is not three-colorable.

On the Fon-Der-Flaass interpretation of extremal examples for Turán’s (3, 4)-problem

Abstract: Fon-Der-Flaass (1988) presented a general construction that converts an arbitrary \(\vec C_4 \)-free oriented graph Γ into a Turan (3, 4)-graph. He observed that all Turan-Brown-Kostochka examples result from his construction, and proved the lower bound \(\tfrac{4} {9} \) (1 − o(1)) on the edge density of any Turan (3, 4)-graph obtainable in this way. In this paper we establish the optimal bound \(\tfrac{3} {7} \) (1 − o(1)) on the edge density of any Turan (3, 4)-graph resulting from the Fon-Der-Flaass construction under any of the following assumptions on the undirected graph G underlying the oriented graph Γ: (i) G is complete multipartite; (ii) the edge density of G is not less than \(\tfrac{2} {3} - \varepsilon \) for some absolute constant e > 0. We are also able to improve Fon-Der-Flaass’s bound to \(\tfrac{7} {{16}} \) (1 − o(1)) without any extra assumptions on Γ.

Satisfiability, Branch-Width and Tseitin tautologies

Abstract: For a CNF τ, let w b (τ) be the branch-width of its underlying hypergraph, that is the smallest w for which the clauses of τ can be arranged in the form of leaves of a rooted binary tree in such a way that for every vertex its descendants and non-descendants have at most w variables in common. In this paper we design an algorithm for solving SAT in time $${n^{O(1)}2^{O(w_b(\tau))}}$$. This in particular implies a polynomial algorithm for testing satisfiability on instances with branch-width O(log n). Our algorithm is a modification of the width based automated theorem prover (WBATP) which is a popular (at least on the theoretical level) heuristic for finding resolution refutations of unsatisfiable CNFs, and we call it Branch-Width Based Automated Theorem Prover (BWBATP). As opposed to WBATP, our algorithm always produces regular refutations. Perhaps more importantly, its running time is bounded in terms of a clean combinatorial characteristic that can be efficiently approximated, and that the algorithm also produces, within the same time, a satisfying assignment if τ happens to be satisfiable. In the second part of the paper we investigate the behavior of BWBATP on the well-studied class of Tseitin tautologies. We argue that in this case BWBATP is better than WBATP. Namely, we show that its running time on any Tseitin tautology τ is $${|\tau|^{O(1)} \cdot 2^{O(w(\tau\vdash\emptyset))}}$$, as opposed to the obvious bound $${n^{O(w(\tau\vdash\emptyset))}}$$ provided by WBATP. This in particular implies that resolution is automatizable on those Tseitin tautologies for which we know the relation $${w(\tau\vdash\emptyset)\leq O(\log S(\tau))}$$. We identify one such subclass and prove partial results toward establishing this relation for larger classes of graphs.

Parameterized bounded-depth Frege is not optimal

Abstract: A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [9]. There the authors concentrate 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 the present paper significantly improves upon this by showing optimal lower bounds for a parameterized version of boundeddepth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size nΩ(k) in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [9]. 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 treelike Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNF's.

On minimal unsatisfiability and time-space trade-offs for k-DNF resolution

Abstract: A well-known theorem by Tarsi states that a minimally unsatisfiable CNF formula with m clauses can have at most m-1 variables, and this bound is exact. In the context of proving lower bounds on proof space in k-DNF resolution, [Ben-Sasson and Nordstrom 2009] extended the concept of minimal unsatisfiability to sets of k-DNF formulas and proved that a minimally unsatisfiable k-DNF set with m formulas can have at most (mk)k+1 variables. This result is far from tight, however, since they could only present explicit constructions of minimally unsatisfiable sets with Ω(mk2) variables. In the current paper, we revisit this combinatorial problem and significantly improve the lower bound to (Ω(m))k, which almost matches the upper bound above. Furthermore, using similar ideas we show that the analysis of the technique in [Ben-Sasson and Nordstrom 2009] for proving time-space separations and trade-offs for k-DNF resolution is almost tight. This means that although it is possible, or even plausible, that stronger results than in [Ben-Sasson and Nordstrom 2009] should hold, a fundamentally different approach would be needed to obtain such results.