Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which finds such a partition A, B, C in O(n) time.

The width of a Resolution proof is de ned to be the maximal number of literals in any clause of the proof. In this paper we relate proof width to proof length (=size), in both general Resolution, and its tree-like variant. The following consequences of these relations reveal width as a crucial "resource" of Resolution proofs. In one direction, the relations allow us to give simple, unified proofs for almost all known exponential lower bounds on size of resolution proofs, as well as several interesting new ones. They all follow from width lower bounds, and we show how these follow from natural expansion property of clauses of the input tautology. In the other direction, the width-size relations naturally suggest a simple dynamic programming procedure for automated theorem proving - one which simply searches for small width proofs. This relation guarantees that the running time (and thus the size of the produced proof) is at most quasi-polynomial in the smallest tree-like proof. This algorithm is never much worse than any of the recursive automated provers (such as DLL) used in practice. In contrast, we present a family of tautologies on which it is exponentially faster.

Abstract. The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Lapla-cian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph. Some new results and generalizations are added. † This article appeared in “Graph Theory, Combinatorics, and Applications”, Vol. 2,

We present the best known separation between tree-like and general resolution, improving on the recent exp(n ) separation of [BEGJ98].

Dedicated to the memory of Misha Alekhnovich Abstract: This paper gives two distinct proofs of an exponential separation between regular resolution and unrestricted resolution. The previous best known separation between these systems was quasi-polynomial. ACM Classification: F.2.2, F.2.3 AMS Classification: 03F20, 68Q17 Key words and phrases: resolution, proof complexity, lower bounds

We analyze size and space complexity of Res(k), a family of propositional proof systems introduced by Kraj'icek in [21] which extends Resolution by allowing disjunctions of conjunctions of up to k 1 literals. We show that the treelike Res(k) proof systems form a strict hierarchy with respect to proof size and also with respect to space. Moreover

We investigate tradeoffs of various important complexity measures such as size, space and width. We show examples of CNF formulas that have optimal proofs with respect to any one of these parameters, but optimizing one parameter must cost an increase in the other. These results, the first of their kind, have implications on the efficiency (or rather, inefficiency) of some commonly used SAT solving heuristics. Our proof

Abstract. Most CSP algorithms are based on refinements and extensions of backtracking, and employ one of two simple “branching schemes”: 2-way branching or d-way branching, for domain size d. The schemes are not equivalent, but little is known about their relative power. Here we compare them in terms of how efficiently they can refute an unsatisfiable instance with optimal branching choices, by studying two variants of the resolution proof system, denoted C-RES and NG-RES, which model the reasoning of CSP algorithms. The tree-like restrictions, tree-C-RES and tree-NG-RES, exactly capture the power of backtracking with 2-way branching and d-way branching, respectively. We give a family instances which require exponential sized search trees for backtracking with d-way branching, but have size O(d 2 n) search trees for backtracking with 2way branching. We also give a natural branching strategy with which backtracking with 2-way branching finds refutations of these instances in time O(d 2 n 2). The unrestricted variants of C-RES and NG-RES can simulate the reasoning of algorithms which incorporate learning and k-consistency enforcement. We show exponential separations between C-RES and NG-RES, as well as between the tree-like and unrestricted versions of each system. All separations given are nearly optimal. 1

Most state-of-the-art satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of the largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space is still quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether on the contrary these measures are completely unrelated in the sense that short proofs can be arbitrarily complex with respect to space. In this paper, we present some evidence that the true answer should be that the latter case holds and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of Θ ( √ n) on the space needed for so-called pebbling contradictions over pyramid graphs of size n. This yields the first polynomial lower bound on space that is not a consequence of a corresponding lower bound on width, as well as an improvement of the weak separation of space and width in (Nordström 2006) from logarithmic to polynomial. Also, continuing the line of research initiated by (Ben-Sasson 2002) into trade-offs between different proof complexity measures, we present a simplified proof of the recent length-space trade-off result in (Hertel and Pitassi 2007), and show how our ideas can be used to prove a couple of other exponential trade-offs in resolution.

In standard implementations of the Grobner basis algorithm, the original polynomials are homogenized so that each term in a given polynomial has the same degree. In this paper, we study the effect of homogenization on the runtime of the Grobner basis algorithm for instances of satisfiability, and also on the degree of Polynomial Calculus (PC) and Nullstellensatz refutations of unsatisfiable formulas. We show that the Nullstellensatz degree is equivalent to the homogenized PC degree. Using this relationship, we prove nearly linear separations between Nullstellensatz and PC degree, for 3CNF formulas. Research partially supported by NSF grant CCR-9457782 and a scholarship from the Arizona Chapter of the ARCS Foundation. + Research supported by NSF CCR-9734911, Sloan Research Fellowship BR-3311, and by a cooperative research grant INT9600919 /ME-103 from NSF and the M SMT (Czech Republic), and USA-Israel-BSF Grant 97-00188 # Research supported by NSF grant CCR-9457782 and US-I...