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 span-ning 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 connected-components 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

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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 well-known (see for example [O]): Theorem 1.1 Let B be a non-empty 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).

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...

. Coxeter matroids, introduced by I.M. Gelfand and V. Serganova, are combinatorial structures associated with any nite Coxeter group and its parabolic subgroup; they include ordinary matroids as a special case. A basic result in the subject is a geometric characterization of Coxeter matroid, rst stated by Gelfand and Serganova. This paper presents a self-contained, simple proof of a more general version of this geometric characterization. 1991 Mathematics Subject Classication. 05E99 05B35 20F55. Key words and phrases. matroid, Coxeter group, Coxeter matroid, Bruhat order . We would like to thank A. Borovik for bringing to the attention of the two groups of coauthors Serganova-Zelevinsky and Vince, who had previously worked independently, their common work on the geometric characterization of Coxeter matroids. The original work by Serganova and Zelevinsky greatly beneted from the discussions with I.M. Gelfand, N. Bagotskaya, V. Levit and I. Losev, while Vince thanks A. Boro...

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 NP-Complete, 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...

. The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W; P ) consisting of a nite irreducible Coxeter group W and parabolic subgroup P is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are a special case, the case W = An (isomorphic to the symmetric group Symn+1 ) and P a maximal parabolic subgroup. The main result of this paper is that for Coxeter matroids, just as for ordinary matroids, the greedy algorithm provides a solution to a naturally associated combinatorial optimization problem. Indeed, in many important cases, Coxeter matroids are characterized by this property. This result generalizes the classical Rado-Edmonds and Gale theorems. A corollary of our theorem is that, for Coxeter matroids L, the greedy algorithm solves the L-assignment problem. Let W be a nite group acting as linear transformations on a Euclidean space E, and let f ; (w) = hw; i for ; 2 E; w 2 W: The ...

. This paper is concerned with a new distance in undirected graphs with weighted edges, which gives new insights into the structure of all minimum spanning trees of a graph. This distance is a generalized one, in the sense that it takes values in a certain Heyting semigroup. More precisely, it associates with each pair of distinct vertices in a connected component of a graph the set of all paths joining them in the minimum spanning trees of that component. A partial order and an addition of these sets of paths are defined. We show how general algorithms for path algebra problems can be used to compute the generalized distance. Some theoretical problems concerning this distance are formulated. The main application of our generalized distance is related to recent clustering procedures. Given a connected graph with weighted edges and certain vertices labeled as centers, we define a centered forest to be a spanning forest with exactly one center in each tree component. A partition of the v...

This paper continues the works [1, 2] and uses, with some modification, their terminology and notation. Throughout the paper W is a Coxeter group (possibly infinite) and P a finite standard parabolic subgroup of W . We identify the Coxeter group W with its Coxeter complex and refer to elements of W as chambers, to cosets with respect to a parabolic subgroup as residues, etc. We shall use the calligraphic letter as a notation for the Coxeter complex of W and the symbol for the set of left cosets of the parabolic subgroup P . We shall use the Bruhat ordering on in its geometric interpretation, as defined in [2, Theorem 5.7]. The w-Bruhat ordering on is denoted by the same symbol as the w-Bruhat ordering on . Notation , < > has obvious meaning

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 α-approximate 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. 1