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157
Biconnectivity Approximations and Graph Carvings
, 1994
"... A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified) ? Unfortunately, the problem is known to be ..."
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Cited by 97 (5 self)
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A spanning tree in a graph is the smallest connected spanning subgraph. Given a graph, how does one find the smallest (i.e., least number of edges) 2connected spanning subgraph (connectivity refers to both edge and vertex connectivity, if not specified) ? Unfortunately, the problem is known to be NP hard. We consider the problem of finding a better approximation to the smallest 2connected subgraph, by an efficient algorithm. For 2edge connectivity our algorithm guarantees a solution that is no more than 3 2 times the optimal. For 2vertex connectivity our algorithm guarantees a solution that is no more than 5 3 times the optimal. The previous best approximation factor is 2 for each of these problems. The new algorithms (and their analyses) depend upon a structure called a carving of a graph, which is of independent interest. We show that approximating the optimal solution to within an additive constant is NP hard as well. We also consider the case where the graph has edge weigh...
Restricted colorings of graphs
 in Surveys in Combinatorics 1993, London Math. Soc. Lecture Notes Series 187
, 1993
"... The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of attention. Here we describe the techniques applied in the study of this subject, which combine combinatorial, al ..."
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Cited by 97 (17 self)
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The problem of properly coloring the vertices (or edges) of a graph using for each vertex (or edge) a color from a prescribed list of permissible colors, received a considerable amount of attention. Here we describe the techniques applied in the study of this subject, which combine combinatorial, algebraic and probabilistic methods, and discuss several intriguing conjectures and open problems. This is mainly a survey of recent and less recent results in the area, but it contains several new results as well.
Unsolved Problems in . . .
, 2009
"... We have sorted the problems into sections: A. Taking and Breaking B. Pushing and Placing Pieces C. Playing with Pencil and Paper D. Disturbing and Destroying ... They have been given new numbers. The numbers in parentheses are the old numbers used in each of the lists of unsolved problems given on ..."
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Cited by 76 (0 self)
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We have sorted the problems into sections: A. Taking and Breaking B. Pushing and Placing Pieces C. Playing with Pencil and Paper D. Disturbing and Destroying ... They have been given new numbers. The numbers in parentheses are the old numbers used in each of the lists of unsolved problems given on pp. 183–189 of AMS Proc. Sympos. Appl. Math. 43 (1991), called PSAM 43 below; on pp. 475–491 of Games of No Chance, hereafter referred to as GONC; and on pp. 457–473 of More Games of No Chance (MGONC). Missing numbers are of problems which have been solved, or for which we have nothing new to add. References [year] may be found in Fraenkel’s Bibliography at the end of this volume. References [#] are at the end of this article. A useful reference for the rules and an introduction to many of the specific games mentioned below is
Computational Experience with Approximation Algorithms for the Set Covering Problem
, 1994
"... The Set Covering problem (SCP) is a well known combinatorial optimization problem, which is NPhard. We conducted a comparative study of eight different approximation algorithms for the SCP, including several greedy variants, fractional relaxations, randomized algorithms and a neural network algorit ..."
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Cited by 50 (2 self)
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The Set Covering problem (SCP) is a well known combinatorial optimization problem, which is NPhard. We conducted a comparative study of eight different approximation algorithms for the SCP, including several greedy variants, fractional relaxations, randomized algorithms and a neural network algorithm. The algorithms were tested on a set of randomgenerated problems with up to 500 rows and 5000 columns, and on two sets of problems originating in combinatorial questions with up to 28160 rows and 11264 columns. On the random problems and on one set of combinatorial problems, the best algorithm among those we tested was the neural network algorithm, with greedy variants very close in second and third place. On the other set of combinatorial problems, the best algorithm was a greedy variant and the neural network performed quite poorly. The other algorithms we tested were always inferior to the ones mentioned above. Theoretical Division and CNLS, MS B213 Los Alamos National Lab, Los Ala...
Random kSAT: two moments suffice to cross a sharp threshold
 CoRR
, 2006
"... Abstract. Many NPcomplete constraint satisfaction problems appear to undergo a “phase transition” from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that abo ..."
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Cited by 41 (4 self)
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Abstract. Many NPcomplete constraint satisfaction problems appear to undergo a “phase transition” from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that the threshold for both random hypergraph 2colorability (Property B) and random NotAllEqual kSAT is 2 k−1 ln 2 − O(1). As a corollary, we establish that the threshold for random kSAT is of order Θ(2 k), resolving a longstanding open problem.
Hardness of approximate hypergraph coloring
 SICOMP: SIAM Journal on Computing
"... We introduce the notion of covering complexity of a verifier for probabilistically checkable proofs (PCP). Such a verifier is given an input, a claimed theorem, and an oracle, representing a purported proof of the theorem. The verifier is also given a random string and decides whether to accept the ..."
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Cited by 39 (4 self)
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We introduce the notion of covering complexity of a verifier for probabilistically checkable proofs (PCP). Such a verifier is given an input, a claimed theorem, and an oracle, representing a purported proof of the theorem. The verifier is also given a random string and decides whether to accept the proof or not, based on the given random string. We define the covering complexity of such a verifier, on a given input, to be the minimum number of proofs needed to “satisfy ” the verifier on every random string, i.e., on every random string, at least one of the given proofs must be accepted by the verifier. The covering complexity of PCP verifiers offers a promising route to getting stronger inapproximability results for some minimization problems, and in particular, (hyper)graph coloring problems. We present a PCP verifier for NP statements that queries only four bits and yet has a covering complexity of one for true statements and a superconstant covering complexity for statements not in the language. Moreover, the acceptance predicate of this verifier is a simple NotallEqual check on the four bits it reads. This enables us to prove that for any constant c, it is NPhard to color a 2colorable 4uniform hypergraph using just c colors, and also yields a superconstant inapproximability result under a stronger hardness
Crossing patterns of semialgebraic sets
 J. Combin. Theory Ser. A
, 2005
"... We prove that, for every family F of n semialgebraic sets in R d of constant description complexity, there exist a positive constant ε that depends on the maximum complexity of the elements of F, and two subfamilies F1, F2 ⊆ F with at least εn elements each, such that either every element of F1 int ..."
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Cited by 31 (10 self)
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We prove that, for every family F of n semialgebraic sets in R d of constant description complexity, there exist a positive constant ε that depends on the maximum complexity of the elements of F, and two subfamilies F1, F2 ⊆ F with at least εn elements each, such that either every element of F1 intersects all elements of F2 or no element of F1 intersects any element of F2. This implies the existence of another constant δ such that F has a subset F ′ ⊆ F with n δ elements, so that either every pair of elements of F ′ intersect each other or the elements of F ′ are pairwise disjoint. The same results hold when the intersection relation is replaced by any other semialgebraic relation. We apply these results to settle several problems in discrete geometry and in Ramsey theory. 1
Problems and Results on Combinatorial Number Theory
 J. N. SRIVASTAVA ET AL., EDS., A SURVEY OF COMBINATORIAL THEORY OC NORTHHOLLAND PUBLISHING COMPANY, 1973
, 1973
"... I will discuss in this paper number theoretic problems which are of combinatorial nature. I certainly do not claim to cover the field completely and the paper will be biased heavily towards problems considered by me and my collaborators. Combinatorial methods have often been used successfully in num ..."
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Cited by 25 (1 self)
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I will discuss in this paper number theoretic problems which are of combinatorial nature. I certainly do not claim to cover the field completely and the paper will be biased heavily towards problems considered by me and my collaborators. Combinatorial methods have often been used successfully in number theory (e.g. sieve methods), but here we will try to restrict ourselves to problems which themselves have a combinatorial flavor. I have written several papers in recent years on such problems and in order to avoid making this paper too long, wherever possible, will discuss either problems not mentioned in the earlier papers or problems where some progress has been made since these papers were written. Before starting the discussion of our problems I give a few of the principal papers where similar problems were discussed and where further literature can be found.
The ErdösSzekeres theorem: upper bounds and related results
 in Combinatorial and Computational Geometry
, 2004
"... Let ES(n) denote the least integer such that among any ES(n) points in general position in the plane there are always n in convex position. In 1935, P. Erdös and G. Szekeres showed that ES(n) exists and ES(n) ≤ + 1. About 62 years later, the upper bound has been slightly improved by Chung and ..."
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Cited by 21 (1 self)
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Let ES(n) denote the least integer such that among any ES(n) points in general position in the plane there are always n in convex position. In 1935, P. Erdös and G. Szekeres showed that ES(n) exists and ES(n) &le; + 1. About 62 years later, the upper bound has been slightly improved by Chung and Graham, a few months later it was further improved by Kleitman and Pachter, and another few months later it was further improved by the present authors. Here we review the original proof of Erdös and Szekeres, the improvements, and finally we combine the methods of the first and third improvements to obtain yet another tiny improvement. We also briefly review some...