Results 1  10
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30
A Survey of Combinatorial Gray Codes
 SIAM Review
, 1996
"... The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing nbit binary numbers so that ..."
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Cited by 81 (2 self)
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The term combinatorial Gray code was introduced in 1980 to refer to any method for generating combinatorial objects so that successive objects differ in some prespecified, small way. This notion generalizes the classical binary reflected Gray code scheme for listing nbit binary numbers so that successive numbers differ in exactly one bit position, as well as work in the 1960's and 70's on minimal change listings for other combinatorial families, including permutations and combinations. The area of combinatorial Gray codes was popularized by Herbert Wilf in his invited address at the SIAM Discrete Mathematics Conference in 1988 and his subsequent SIAM monograph in which he posed some open problems and variations on the theme. This resulted in much recent activity in the area and most of the problems posed by Wilf are now solved. In this paper, we survey the area of combinatorial Gray codes, describe recent results, variations, and trends, and highlight some open problems. ...
The Complexity of Counting in Sparse, Regular, and Planar Graphs
 SIAM Journal on Computing
, 1997
"... We show that a number of graphtheoretic counting problems remain NPhard, indeed #Pcomplete, in very restricted classes of graphs. In particular, it is shown that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to ..."
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Cited by 71 (0 self)
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We show that a number of graphtheoretic counting problems remain NPhard, indeed #Pcomplete, in very restricted classes of graphs. In particular, it is shown that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree. To achieve these results, a new interpolationbased reduction technique which preserves properties such as constant degree is introduced. In addition, the problem of approximately counting minimum cardinality vertex covers is shown to remain NPhard even when restricted to graphs of maximal degree 3. Previously, restrictedcase complexity results for counting problems were elusive; we believe our techniques may help obtain similar results for many other counting problems. 1 Introduction Ever since the introduction of NPcompleteness in the early 1970's, the primary focus of complexity theory has been on decision ...
Computing The Volume Of Convex Bodies: A Case Where Randomness Provably Helps
, 1991
"... We discuss the problem of computing the volume of a convex body K in IR n . We review worstcase results which show that it is hard to deterministically approximate volnK and randomised approximation algorithms which show that with randomisation one can approximate very nicely. We then provide som ..."
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Cited by 62 (6 self)
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We discuss the problem of computing the volume of a convex body K in IR n . We review worstcase results which show that it is hard to deterministically approximate volnK and randomised approximation algorithms which show that with randomisation one can approximate very nicely. We then provide some applications of this latter result. Supported by NATO grant RG0088/89 y Supported by NSF grant CCR8900112 and NATO grant RG0088/89 1 Introduction The mathematical study of areas and volumes is as old as civilization itself, and has been conducted for both intellectual and practical reasons. As far back as 2000 B.C., the Egyptians 1 had methods for approximating the areas of fields (for taxation purposes) and the volumes of granaries. The exact study of areas and volumes began with Euclid 2 and was carried to a high art form by Archimedes 3 . The modern study of this subject began with the great astronomer Johann Kepler's treatise 4 Nova stereometria doliorum vinariorum, wh...
The Complexity of Counting Graph Homomorphisms
 In 11th ACM/SIAM Symposium on Discrete Algorithms
, 1999
"... The problem of counting graph homomorphisms is considered. We show that the counting problem corresponding to a given graph is #Pcomplete unless every connected component of the graph is an isolated vertex without a loop, a complete graph with all loops present, or a complete unlooped bipartite gra ..."
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Cited by 15 (4 self)
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The problem of counting graph homomorphisms is considered. We show that the counting problem corresponding to a given graph is #Pcomplete unless every connected component of the graph is an isolated vertex without a loop, a complete graph with all loops present, or a complete unlooped bipartite graph. 1 Introduction Many combinatorial counting problems on graphs can be restated as the problem of counting the number of homomorphisms to a particular graph H. The vertices of H correspond to colours, and the edges show which colours may be adjacent. The graph H may contain loops. Specifically, let C be a set of k colours, where k is a constant. Let H = (C; EH ) be a graph with vertex set C. Given a graph G = (V; E) with vertex set V , a map X : V 7! C is called a Hcolouring if fX(v); X(w)g 2 EH for all fv; wg 2 E: In other words, X is a homomorphism from G to H. Let\Omega H (G) denote the set of all Hcolourings of G. Two wellknown combinatorial counting problems which can be c...
From a zoo to a zoology: Towards a general theory of graph polynomials
 Theory of Computing Systems
, 2007
"... Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We in ..."
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Cited by 13 (4 self)
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Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce two classes of (hyper)graph polynomials definable in second order logic, and outline a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities. 1
Some Algorithmic Problems in Polytope Theory
 IN ALGEBRA, GEOMETRY, AND SOFTWARE SYSTEMS
, 2003
"... Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact... ..."
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Cited by 12 (1 self)
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Convex polytopes, i.e.. the intersections of finitely many closed affine halfspaces in R^d, are important objects in various areas of mathematics and other disciplines. In particular, the compact...
On Enumerating All Minimal Solutions of Feedback Problems
"... We present an alg orithm thatg enerates all (inclusionwise) minimal feedback vertex sets of a directedg raph G =(V,E). The feedback vertex sets of G areg enerated with a polynomial delay of O # V  2 (V  + E) # . We further show that the underlying technique can be tailored tog enerate all ..."
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Cited by 12 (0 self)
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We present an alg orithm thatg enerates all (inclusionwise) minimal feedback vertex sets of a directedg raph G =(V,E). The feedback vertex sets of G areg enerated with a polynomial delay of O # V  2 (V  + E) # . We further show that the underlying technique can be tailored tog enerate all minimal solutions for the undirected case and the directed feedback arc set problem, both with a polynomial delay of O # V E (V  + E) # . Finally we prove that computing the number of minimal feedback arc sets is #Phard.
The Complexity of Planar Counting Problems
, 1998
"... . We prove the #Phardness of the counting problems associated with various satisfiability, graph, and combinatorial problems, when restricted to planar instances. These problems include 3Sat, 13Sat, 1Ex3Sat, Minimum Vertex Cover, Minimum Dominating Set, Minimum Feedback Ve ..."
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Cited by 10 (0 self)
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.<F3.862e+05> We prove the #Phardness of the counting problems associated with various satisfiability, graph, and combinatorial problems, when restricted to planar instances. These problems include<F3.771e+05> 3Sat, 13Sat, 1Ex3Sat, Minimum Vertex Cover, Minimum Dominating Set, Minimum Feedback Vertex Set, X3C, Partition Into Triangles, and Clique Cover.<F3.862e+05> We also prove the<F4.039e+05><F3.862e+05> NPcompleteness of the<F3.771e+05> Ambiguous Satisfiability<F3.862e+05> problems [J. B. Saxe,<F3.783e+05> Two Papers on Graph Embedding<F3.862e+05> Problems, Tech. Report CMUCS80102, Dept. of Computer Science, Carnegie Mellon Univ., Pittsburgh, PA, 1980] and the<F4.039e+05> D<F2.539e+05> P<F3.862e+05> completeness (with respect to random polynomial reducibility) of the unique satisfiability problems [L. G. Valiant and V. V. Vazirani,<F3.783e+05> NP is as easy as detecting unique<F3.862e+05> solutions, in Proc. 17th ACM Symp. on Theory of Computing, 1985, pp. 458463] associ...
Signable Posets and Partitionable Simplicial Complexes
, 1996
"... The notion of a partitionable simplicial complex is extended to that of a signable partially ordered set. It is shown in a unified way that face lattices of shellable polytopal complexes, polyhedral cone fans, and oriented matroid polytopes, are all signable. Each of these classes, which are belie ..."
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Cited by 9 (5 self)
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The notion of a partitionable simplicial complex is extended to that of a signable partially ordered set. It is shown in a unified way that face lattices of shellable polytopal complexes, polyhedral cone fans, and oriented matroid polytopes, are all signable. Each of these classes, which are believed to be mutually incomparable, strictly contains the class of convex polytopes. A general sufficient condition, termed total signability, for a simplicial complex to satisfy McMullen's Upper Bound Theorem on the numbers of faces, is provided. The simplicial members of each of the three classes above are concluded to be partitionable and to satisfy the upper bound theorem. The computational complexity of face enumeration and of deciding partitionability is discussed. It is shown that under a suitable presentation, the face numbers of a signable simplicial complex can be efficiently computed. In particular, the face numbers of simplicial fans can be computed in polynomial time, extending the analogous statement for convex polytopes.