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Generating Linear Extensions Fast
"... One of the most important sets associated with a poset P is its set of linear extensions, E(P) . "ExtensionFast.html" 87 lines, 2635 characters One of the most important sets associated with a poset P is its set of linear extensions, E(P) . In this paper, we present an algorithm to generat ..."
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Cited by 40 (6 self)
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One of the most important sets associated with a poset P is its set of linear extensions, E(P) . "ExtensionFast.html" 87 lines, 2635 characters One of the most important sets associated with a poset P is its set of linear extensions, E(P) . In this paper, we present an algorithm to generate all of the linear extensions of a poset in constant amortized time; that is, in time O(e(P)) , where e ( P ) =  E(P) . The fastest previously known algorithm for generating the linear extensions of a poset runs in time O(n e(P)) , where n is the number of elements of the poset. Our algorithm is the first constant amortized time algorithm for generating a ``naturally defined'' class of combinatorial objects for which the corresponding counting problem is #Pcomplete. Furthermore, we show that linear extensions can be generated in constant amortized time where each extension differs from its predecessor by one or two adjacent transpositions. The algorithm is practical and can be modified to efficiently count linear extensions, and to compute P(x < y) , for all pairs x,y , in time O( n^2 + e ( P )).
Algorithms for Square Roots of Graphs
 SIAM Journal on Discrete Mathematics
, 1991
"... The nth power (n 1) of a graph G = (V; E), written G n , is defined to be the graph having V as its vertex set with two vertices u; v adjacent in G n if and only if there exists a path of length at most n between them. Similarly, graph H has an nth root G if G n = H . For the case of n = 2, ..."
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Cited by 38 (0 self)
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The nth power (n 1) of a graph G = (V; E), written G n , is defined to be the graph having V as its vertex set with two vertices u; v adjacent in G n if and only if there exists a path of length at most n between them. Similarly, graph H has an nth root G if G n = H . For the case of n = 2, we say that G 2 is the square of G and G is the square root of G 2 . Here we give a linear time algorithm for finding the tree square roots of a given graph and a linear time algorithm for finding the square roots of planar graphs. We also give a polynomial time algorithm for finding the square roots of subdivision graphs, which is equivalent to the problem of the inversion of total graphs. Further, we give a linear time algorithm for finding a Hamiltonian cycle in a cubic graph, and we prove the NPcompleteness of finding the maximum cliques in powers of graphs and the chordality of powers of trees. Keywords: Square graphs, power graphs, tree square, planar square graphs. 1 Introduct...
Computing Roots of Graphs is Hard
 DISCRETE APPLIED MATHEMATICS
, 1994
"... The square of an undirected graph G is the graph G² on the same vertex set such that there is an edge between two vertices in G² if and only if they are at distance at most 2 in G. The k'th power of a graph is defined analogously. It has been conjectured that the problem of computing any squ ..."
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Cited by 26 (1 self)
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The square of an undirected graph G is the graph G² on the same vertex set such that there is an edge between two vertices in G² if and only if they are at distance at most 2 in G. The k'th power of a graph is defined analogously. It has been conjectured that the problem of computing any square root of a square graph, or even that of deciding whether a graph is a square, is NPhard. We settle this conjecture in the affirmative.
Approximating DisjointPath Problems Using Packing Integer Programs
, 1998
"... In a packing integer program, we are given a matrix A and column vectors b; c with nonnegative entries. We seek a vector x of nonnegative integers, which maximizes c^T x; subject to Ax ≤ b: The edge and vertexdisjoint path problems together with their unsplittable ow generalization are NPha ..."
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Cited by 16 (2 self)
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In a packing integer program, we are given a matrix A and column vectors b; c with nonnegative entries. We seek a vector x of nonnegative integers, which maximizes c^T x; subject to Ax &le; b: The edge and vertexdisjoint path problems together with their unsplittable ow generalization are NPhard problems with a multitude of applications in areas such as routing, scheduling and bin packing. These two categories of problems are known to be conceptually related, but this connection has largely been ignored in terms of approximation algorithms. We explore the topic of approximating disjointpath problems using polynomialsize packing integer programs. Motivated by the...
Infinite Hamilton cycles in squares of locally finite graphs, preprint 2007
"... We prove Diestel’s conjecture that the square G2 of a 2connected locally finite graph G has a Hamilton circle, a homeomorphic copy of the unit circle S1 in the Freudenthal compactification of G2. 1 ..."
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Cited by 12 (5 self)
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We prove Diestel’s conjecture that the square G2 of a 2connected locally finite graph G has a Hamilton circle, a homeomorphic copy of the unit circle S1 in the Freudenthal compactification of G2. 1
Performance Guarantees for the TSP with a Parameterized Triangle Inequality
 IN PROC. 6TH INT. WORKSHOP ON ALGORITHMS AND DATA STRUCTURES (WADS), VOLUME 1663 OF LECTURE NOTES IN COMPUT. SCI
, 2000
"... We consider the approximability of the tsp problem in graphs that satisfy a relaxed form of triangle inequality. More precisely, we assume that for some parameter 1, the distances satisfy the inequality dist(x; y) \Delta (dist(x; z) + dist(z; y)) for every triple of vertices x, y, and z. We o ..."
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Cited by 9 (0 self)
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We consider the approximability of the tsp problem in graphs that satisfy a relaxed form of triangle inequality. More precisely, we assume that for some parameter 1, the distances satisfy the inequality dist(x; y) \Delta (dist(x; z) + dist(z; y)) for every triple of vertices x, y, and z. We obtain a 4 approximation and also show that for some ffl ? 0 it is nphard to obtain a (1 + ffl ) approximation. Our upper bound improves upon the earlier known ratio of (3 =2 + =2) [1] for all values of ? 7/3.
Toughness in graphs a survey
 Graphs and Combinatorics 22 (1): 135, 2006. SERGEY
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Cycles in Networks
, 1993
"... We study the presence of cycles and long paths in graphs that have been proposed as interconnection networks for parallel architectures. The study surveys and complements known results. 1 Introduction This paper is devoted to studying embeddings of the simplest possible guest graphs, the path P ..."
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Cited by 6 (0 self)
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We study the presence of cycles and long paths in graphs that have been proposed as interconnection networks for parallel architectures. The study surveys and complements known results. 1 Introduction This paper is devoted to studying embeddings of the simplest possible guest graphs, the path PN and the cycle CN , in graphs that have been proposed as interconnection networks for parallel architectures. In addition to their intrinsic interest, in terms of the development of algorithms on parallel architectures, these two guest graphs are important because of the fact that many structurally richer graphs can be constructed from paths and cycles by various product constructions. A few of the results we present are original; several appear in the literature and are duly cited; many belong to the folklore of the field. Indeed this paper is motivated by a desire to find a single repository for this important, yet scattered material. Before proceeding further, we define formally the ...
Hamiltonian paths in Cayley graphs
, 2008
"... The classical question raised by Lovász asks whether every Cayley graph is Hamiltonian. We present a short survey of various results in that direction and make some additional observations. In particular, we prove that every finite group G has a generating set of size at most log 2 G, such that th ..."
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Cited by 5 (0 self)
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The classical question raised by Lovász asks whether every Cayley graph is Hamiltonian. We present a short survey of various results in that direction and make some additional observations. In particular, we prove that every finite group G has a generating set of size at most log 2 G, such that the corresponding Cayley graph contains a Hamiltonian cycle. We also present an explicit construction of 3regular Hamiltonian expanders.
Exact and Approximation Algorithms for Network Flow and DisjointPath Problems
, 1998
"... Network flow problems form a core area of Combinatorial Optimization. Their significance arises both from their very large number of applications and their theoretical importance. This thesis focuses on efficient exact algorithms for network flow problems in P and on approximation algorithms for NP ..."
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Cited by 5 (3 self)
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Network flow problems form a core area of Combinatorial Optimization. Their significance arises both from their very large number of applications and their theoretical importance. This thesis focuses on efficient exact algorithms for network flow problems in P and on approximation algorithms for NP hard variants such as disjoint paths and unsplittable flow. Given an nvertex