Results 1  10
of
28
A Separator Theorem for Planar Graphs f
, 1977
"... Let G be any nvertex 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 su ..."
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Cited by 397 (1 self)
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Let G be any nvertex 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.
Short Proofs are Narrow  Resolution made Simple
 Journal of the ACM
, 2000
"... 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 treelike variant. The following consequences of these relations reveal width as a crucial "reso ..."
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Cited by 181 (15 self)
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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 treelike 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 widthsize 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 quasipolynomial in the smallest treelike 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.
The Laplacian spectrum of graphs
 Graph Theory, Combinatorics, and Applications
, 1991
"... 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 Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, m ..."
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Cited by 151 (1 self)
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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 Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidthtype parameters of a graph. Some new results and generalizations are added. † This article appeared in “Graph Theory, Combinatorics, and Applications”, Vol. 2,
NearOptimal Separation of Treelike and General Resolution
 Electronic Colloquium in Computation Complexity
, 2000
"... We present the best known separation between treelike and general resolution, improving on the recent exp(n ) separation of [BEGJ98]. ..."
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Cited by 47 (3 self)
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We present the best known separation between treelike and general resolution, improving on the recent exp(n ) separation of [BEGJ98].
Eigenvalues, geometric expanders, sorting in rounds and
 Ramsey Theory, Cornbinatorica
, 1986
"... Expanding graphs are relevant to theoretical computer science in several ways. Here we show that the points versus hyperplanes incidence graphs of finite geometries form highly (nonlinear) expanding graphs with essentially the smallest possible number of edges. The expansion properties of the graphs ..."
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Cited by 47 (12 self)
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Expanding graphs are relevant to theoretical computer science in several ways. Here we show that the points versus hyperplanes incidence graphs of finite geometries form highly (nonlinear) expanding graphs with essentially the smallest possible number of edges. The expansion properties of the graphs are proved using the eigenvalues of their adjacency matrices. These graphs enable us to improve previous results on a parallel sorting problem that arises in structural modeling, by describing an explicit algorithm to sort n elements in k time units using O(n ~k) parallel processors, where, e.g., cq=7/4, ~q8/5, 0q=26/17 and ~q=22/15. Our approach also yields several applications to Ramsey Theory and other extremal problems
An exponential separation between regular and general resolution
 Theory of Computing
"... 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 quasipolynomial. ACM Classification: F.2.2, F.2.3 AMS Classif ..."
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Cited by 36 (5 self)
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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 quasipolynomial. ACM Classification: F.2.2, F.2.3 AMS Classification: 03F20, 68Q17 Key words and phrases: resolution, proof complexity, lower bounds
On the Complexity of Resolution with Bounded Conjunctions
 IN 29TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING
, 2004
"... 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 ..."
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Cited by 28 (4 self)
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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
Size Space tradeoffs for Resolution
, 2002
"... 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 ..."
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Cited by 22 (4 self)
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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
2way vs dway branching for CSP
 In Proceedings of CP’05
, 2005
"... Abstract. Most CSP algorithms are based on refinements and extensions of backtracking, and employ one of two simple “branching schemes”: 2way branching or dway branching, for domain size d. The schemes are not equivalent, but little is known about their relative power. Here we compare them in term ..."
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Cited by 16 (0 self)
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Abstract. Most CSP algorithms are based on refinements and extensions of backtracking, and employ one of two simple “branching schemes”: 2way branching or dway 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 CRES and NGRES, which model the reasoning of CSP algorithms. The treelike restrictions, treeCRES and treeNGRES, exactly capture the power of backtracking with 2way branching and dway branching, respectively. We give a family instances which require exponential sized search trees for backtracking with dway 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 2way branching finds refutations of these instances in time O(d 2 n 2). The unrestricted variants of CRES and NGRES can simulate the reasoning of algorithms which incorporate learning and kconsistency enforcement. We show exponential separations between CRES and NGRES, as well as between the treelike and unrestricted versions of each system. All separations given are nearly optimal. 1
Homogenization and the Polynomial Calculus
 Computational Complexity
, 1999
"... 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 ..."
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Cited by 10 (4 self)
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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 CCR9457782 and a scholarship from the Arizona Chapter of the ARCS Foundation. + Research supported by NSF CCR9734911, Sloan Research Fellowship BR3311, and by a cooperative research grant INT9600919 /ME103 from NSF and the M SMT (Czech Republic), and USAIsraelBSF Grant 9700188 # Research supported by NSF grant CCR9457782 and USI...