Let G be a graph with weights on the edges, S a subset of vertices, and k an integer. The problem of computing a minimum-weight subset of edges that meets all the cuts of cardinality less-than-or-equal-to k that separ...
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Let G be a graph with weights on the edges, S a subset of vertices, and k an integer. The problem of computing a minimum-weight subset of edges that meets all the cuts of cardinality less-than-or-equal-to k that separate pairs of vertices in S is considered. This problem is motivated by issues in network survivability. Assuming Absolute value of S = 2, it is shown that although this problem is NP-hard, it can be solved in linear time for each fixed value of k. Furthermore, if Absolute value of S > 2, the problem is NP-hard even for small values of k but can be solved in linear time for each fixed k and Absolute value of S.
We consider the problem of determining whether a directed graph contains a pair of vertices connected by two distinct simple paths. A straightforward implementation using n depth-first searches requires O(nm) time on ...
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We consider the problem of determining whether a directed graph contains a pair of vertices connected by two distinct simple paths. A straightforward implementation using n depth-first searches requires O(nm) time on an n-vertex, m-arc digraph;we obtain an O(n2)-time algorithm by using contraction wherever possible.
Assume that each edge of a graph G=(V,E) is given a weight, which is an element of some group G. The weight of a path P is defined as the product of the weights of the edges along P. The group path problem is to find ...
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Assume that each edge of a graph G=(V,E) is given a weight, which is an element of some group G. The weight of a path P is defined as the product of the weights of the edges along P. The group path problem is to find a chordless path of a given weight between two given vertices. It generalizes the parity path problem considered by Hsu. We show that the recognition problem associated with the group path problem is NP-complete in general, and present an 0(Absolute value of G . Absolute value of E + Absolute value of V) time algorithm for the group path problem on a chordal graph.
The authors determine the algorithmic complexity of domination and variants on cocomparability graphs, a class of perfect graphs containing both the interval and the permutation graphs. Minimum dominating total domina...
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The authors determine the algorithmic complexity of domination and variants on cocomparability graphs, a class of perfect graphs containing both the interval and the permutation graphs. Minimum dominating total dominating, connected dominating, and independent dominating sets can be constructed in polynomial time. On the other hand. DOMINATING CLIQUE and MINIMUM DOMiNATING CLIQUE remain NP-complete on cocomparability graphs.
It is shown that the pathwidth of a cograph equals its treewidth, and a linear time algorithm to determine the pathwidth of a cograph and build a corresponding path-decomposition is given.
It is shown that the pathwidth of a cograph equals its treewidth, and a linear time algorithm to determine the pathwidth of a cograph and build a corresponding path-decomposition is given.
This paper presents efficient algorithms for finding the largest tree S such that there are vertex disjoint subtrees S1, ... , S(k) (k > 1) of T each of which is isomorphic to S and every leaf of T is a leaf of som...
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This paper presents efficient algorithms for finding the largest tree S such that there are vertex disjoint subtrees S1, ... , S(k) (k > 1) of T each of which is isomorphic to S and every leaf of T is a leaf of some S(i). The algorithms are useful for learning a macro table.
An optimal algorithm for decomposing a special type of the Hasse diagram into a minimum set of disjoint paths is described. It is useful for testing the consistency of functional dependencies.
An optimal algorithm for decomposing a special type of the Hasse diagram into a minimum set of disjoint paths is described. It is useful for testing the consistency of functional dependencies.
This article introduces an approach for characterizing the classes of empirical distributions that satisfy certain positive dependence notions. Mathematically, this can be expressed as studying certain subsets of the ...
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The graph realization problem is that of computing the relative locations of a set of vertices placed in Euclidean space, relying only upon some set of inter-vertex distance measurements. This paper is concerned with ...
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The graph realization problem is that of computing the relative locations of a set of vertices placed in Euclidean space, relying only upon some set of inter-vertex distance measurements. This paper is concerned with the closely related problem of determining whether or not a graph has a unique realization. Both these problems are NP-hard, but the proofs rely upon special combinations of edge lengths. If one assumes the vertex locations are unrelated, then the uniqueness question can be approached from a purely graph theoretic angle that ignores edge lengths. This paper identifies three necessary graph theoretic conditions for a graph to have a unique realization in any dimension. Efficient sequential and NC algorithms are presented for each condition, although these algorithms have very different flavors in different dimensions.
This paper determines upper bounds on the expected time complexity for a variety of parallel algorithms for undirected and directed random graph problems. For connectivity, biconnectivity, transitive closure, minimum ...
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This paper determines upper bounds on the expected time complexity for a variety of parallel algorithms for undirected and directed random graph problems. For connectivity, biconnectivity, transitive closure, minimum spanning trees, and all pairs minimum cost paths, we prove the expected time to be O(log log n) for the CRCW PRAM (this parallel RAM machine allows resolution of write conflicts) and O(log *** log n) for the CREW PRAM (which allows simultaneous reads but not simultaneous writes). We also show that the problem of graph isomorphism has expected parallel time O(log log n) for the CRCW PRAM and O(log n) for the CREW PRAM. Most of these results follow because of upper bounds on the mean depth of a graph, derived in this paper, for more general graphs than was known before. For undirected connectivity especially, we present a new probabilistic algorithm which runs on a randomized input and has an expected running time of O(log log n) on the CRCW PRAM, with O(n) expected number of processors only. Our results also improve known upper bounds on the expected space required for sequential graph algorithms. For example, we show that the problems of finding connected components, transitive closure, minimum spanning trees, and minimum cost paths have expected sequential space O(log *** log n) on a deterministic Turing Machine. We use a simulation of the CRCW PRAM to get these expected sequential space bounds.
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