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27
Approximating the Minimum Spanning Tree Weight in Sublinear Time
 In Proceedings of the 28th Annual International Colloquium on Automata, Languages and Programming (ICALP
, 2001
"... We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set {1,...,w}, and given a parameter 0 < ε < 1/2, estimates in time O(dwε−2 log dw ε) the weight of the minimum spanning tree of G with a relativ ..."
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Cited by 42 (6 self)
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We present a probabilistic algorithm that, given a connected graph G (represented by adjacency lists) of average degree d, with edge weights in the set {1,...,w}, and given a parameter 0 < ε < 1/2, estimates in time O(dwε−2 log dw ε) the weight of the minimum spanning tree of G with a relative error of at most ε. Note that the running time does not depend on the number of vertices in G. We also prove a nearly matching lower bound of Ω(dwε−2) on the probe and time complexity of any approximation algorithm for MST weight. The essential component of our algorithm is a procedure for estimating in time O(dε−2 log d ε) the number of connected components of an unweighted graph to within an additive error of εn. (This becomes O(ε−2 log 1 ε) for d = O(1).) The time bound is shown to be tight up to within the log d ε factor. Our connectedcomponents algorithm picks O(1/ε2) vertices in the graph and then grows “local spanning trees” whose sizes are specified by a stochastic process. From the local information collected in this way, the algorithm is able to infer, with high confidence, an estimate of the number of connected components. We then show how estimates on the number of components in various subgraphs of G can be used to estimate the weight of its MST. 1
On Exchange Properties for Coxeter Matroids and Oriented Matroids
"... We introduce new basis exchange axioms for matroids and oriented matroids. These new axioms are special cases of exchange properties for a more general class of combinatorial structures, Coxeter matroids. We refer to them as "properties" in the more general setting because they are not all ..."
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Cited by 14 (12 self)
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We introduce new basis exchange axioms for matroids and oriented matroids. These new axioms are special cases of exchange properties for a more general class of combinatorial structures, Coxeter matroids. We refer to them as "properties" in the more general setting because they are not all equivalent, as they are for ordinary matroids, since the Symmetric Exchange Property is strictly stronger than the others. The weaker ones constitute the definition of Coxeter matroids, and we also prove their equivalence to the matroid polytope property of Gelfand and Serganova. 2 The terminology in the present paper follows [BG, BR] (though we prefer to use the name `Coxeter matroids' rather than `WP matroids,' as used in these papers); see also the forthcoming book [BGW1]. The cited publications also contain all the necessary background material. For more detail, refer to books [We],[Wh], [O] and [R] for the systematic exposition of matroid theory and theory of Coxeter complexes. The authors wish to thank A. Kelmans for several helpful suggestions. 1 Exchange properties for matroids Matroids. The following is wellknown (see for example [O]): Theorem 1.1 Let B be a nonempty collection of subsets of E. Then the following are equivalent: (1) For every A, B # B and a # A \ B there exists b # B \ A such that A \ {a} # {b} # B (the Exchange Property). (2) For every A, B # B and a # A \ B there exists b # B \ A such that B \ {b} # {a} # B (the Dual Exchange Property). (3) For every A, B # B and a # A \ B, there exists b # B \ A such that A\{a}#{b} # B and B \{b}#{a} # B (the Symmetric Exchange Property).
An Exact Characterization of Greedy Structures
 SIAM Journal on Discrete Mathematics
, 1993
"... We present exact characterizations of structures on which the greedy algorithm produces optimal solutions. Our characterization, which we call matroid embeddings, complete the partial characterizations of Rado, Gale, and Edmonds (matroids), and of Korte and Lovasz (greedoids). We show that the gre ..."
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Cited by 10 (1 self)
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We present exact characterizations of structures on which the greedy algorithm produces optimal solutions. Our characterization, which we call matroid embeddings, complete the partial characterizations of Rado, Gale, and Edmonds (matroids), and of Korte and Lovasz (greedoids). We show that the greedy algorithm optimizes all linear objective functions if and only if the problem structure (phrased in terms of either accessible set systems or hereditary languages) is a matroid embedding. We also present an exact characterization of the objective functions optimized by the greedy algorithm on matroid embeddings. Finally, we present an exact characterization of the structures on which the greedy algorithm optimizes all bottleneck functions, structures which are less constrained than matroid embeddings. 1 Introduction Obtaining an exact characterization of the class of problems for which the greedy algorithm returns an optimal solution has been an open problem. Rado [9], Gale [3], a...
Approximation Algorithms for MinMax Tree Partition
, 1997
"... We consider the problem of partitioning the node set of a graph into p equal sized subsets. The objective is to minimize the maximum length, over these subsets, of a minimum spanning tree. We show that no polynomial algorithm with bounded Ž 2 error ratio can be given for the problem unless P � NP. W ..."
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Cited by 5 (1 self)
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We consider the problem of partitioning the node set of a graph into p equal sized subsets. The objective is to minimize the maximum length, over these subsets, of a minimum spanning tree. We show that no polynomial algorithm with bounded Ž 2 error ratio can be given for the problem unless P � NP. We present an On. time algorithm for the problem, where n is the number of nodes in the graph. Assuming that the edge lengths satisfy the triangle inequality, its error ratio is at most 2 p � 1. We also present an improved algorithm that obtains as an input a positive Ž Ž p�x. p 2 integer x. It runs in O 2 n. time, and its error ratio is at most Ž2�x� Ž x�p�1.. p.
Approximation Algorithms for Minimum Tree Partition
 Disc. Applied Math
, 1998
"... We consider a problem of locating communication centers. In this problem, it is required to partition the set of n customers into subsets minimizing the length of nets required to connect all the customers to the communication centers. Suppose that communication centers are to be placed in p of the ..."
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Cited by 4 (3 self)
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We consider a problem of locating communication centers. In this problem, it is required to partition the set of n customers into subsets minimizing the length of nets required to connect all the customers to the communication centers. Suppose that communication centers are to be placed in p of the customers locations. The number of customers each center supports is also given. The problem remains to divide a graph into sets of the given sizes, keeping the sum of the spanning trees minimal. The problem is NPComplete, and no polynomial algorithm with bounded error ratio can be given, unless P = NP . We present an approximation algorithm for the problem assuming that the edge lengths satisfy the triangle inequality. It runs in O(p 2 4 p +n 2 ) time (n = jV j) and comes within a factor of 2p \Gamma 1 of optimal. When the sets' sizes are all equal this algorithm runs in O(n 2 ) time. Next an improved algorithm is presented which obtains as an input a positive integer x (x n \Gamm...
Robust Subgraphs for Trees and Paths
, 2004
"... Consider a graph problem which is associated with a parameter, for example, that of finding a longest tour spanning k vertices. The following question is natural: Is there a small subgraph which contains an optimal or near optimal solution for every possible value of the given parameter? Such a subg ..."
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Cited by 2 (2 self)
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Consider a graph problem which is associated with a parameter, for example, that of finding a longest tour spanning k vertices. The following question is natural: Is there a small subgraph which contains an optimal or near optimal solution for every possible value of the given parameter? Such a subgraph is said to be robust. In this paper we consider the problems of finding heavy paths and heavy trees of k edges. In these two cases we prove surprising bounds on the size of a robust subgraph for a variety of approximation ratios. For both problems we show that in every complete weighted graph on n vertices α there exists a subgraph with approximately 1−α2 n edges which contains an αapproximate solution for every k = 1,..., n − 1. In the analysis of the tree problem we also describe a new result regarding balanced decomposition of trees. In addition, we consider variations in which the subgraph itself is restricted to be a path or a tree. For these problems we describe polynomial time algorithms and corresponding proofs of negative results.