This paper considers combinatorial auctions among such submodular buyers. The valuations of such buyers are placed within a hierarchy of valuations that exhibit no complementarities, a hierarchy that includes also OR and XOR combinations of singleton valuations, and valuations satisfying the gross substitutes property. Those last valuations are shown to form a zero-measure subset of the submodular valuations that have positive measure. While we show that the allocation problem among submodular valuations is NP-hard, we present an efficient greedy 2-approximation algorithm for this case and generalize it to the case of limited complementarities. No such approximation algorithm exists in a setting allowing for arbitrary complementarities. Some results about strategic aspects of combinatorial auctions among players with decreasing marginal utilities are also presented.

This paper describes a method for generating test sequences for checking the conformance of a protocol implementation to its specification. A Rural Chinese Postman Tour is used to determine a minimum-cost tour of the transition graph of a finite-state machine. When used in combination with Unique Input/Output Sequences [18], the technique yields an efficient method for computing a test sequence for protocol conformance testing.

The grammar problem, a generalization of the single-source shortest-path problem introduced by Knuth, is to compute the minimum-cost derivation of a terminal string from each non-terminal of a given context-free grammar, with the cost of a derivation being suitably defined. This problem also subsumes the problem of finding optimal hyperpaths in directed hypergraphs (under varying optimization criteria) that has received attention recently. In this paper we present an incremental algorithm for a version of the grammar problem. As a special case of this algorithm we obtain an efficient incremental algorithm for the single-source shortest-path problem with positive edge lengths. The aspect of our work that distinguishes it from other work on the dynamic shortest-path problem is its ability to handle "multiple heterogeneous modifications": between updates, the input graph is allowed to be restructured by an arbitrary mixture of edge insertions, edge deletions, and edge-length changes.

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This paper describes the mixtures-of-trees model, a probabilistic model for discrete multidimensional domains. Mixtures-of-trees generalize the probabilistic trees of Chow and Liu [6] in a different and complementary direction to that of Bayesian networks. We present efficient algorithms for learning mixtures-of-trees models in maximum likelihood and Bayesian frameworks. We also discuss additional efficiencies that can be obtained when data are “sparse, ” and we present data structures and algorithms that exploit such sparseness. Experimental results demonstrate the performance of the model for both density estimation and classification. We also discuss the sense in which tree-based classifiers perform an implicit form of feature selection, and demonstrate a resulting insensitivity to irrelevant attributes.

Efficient implementations of Dijkstra’s shortest path algorithm are investigated. A new data structure, called the radix heap, is proposed for use in this algorithm. On a network with n vertices, m edges, and nonnegative integer arc costs bounded by C, a one-level form of radix heap gives a time bound for Dijkstra’s algorithm of O(m + n log C). A two-level form of radix heap gives a bound of O(m + n log C/log log C). A combination of a radix heap and a previously known data structure called a Fibonacci heap gives a bound of O(m + nm). The best previously known bounds are O(m + n log n) using Fibonacci heaps alone and O(m log log C) using the priority queue structure of Van Emde Boas et al. [17].

Abstract. This paper describes new algorithms for approximately solving the concurrent multicommodity flow problem with uniform capacities. These algorithms are much faster than algorithms discovered previously. Besides being an important problem in its own right, the uniform-capacity concurrent flow problem has many interesting applications. Leighton and Rao used uniform-capacity concurrent flow to find an approximately "sparsest cut " in a graph and thereby approximately solve a wide variety of graph problems, including minimum feedback arc set, minimum cut linear arrangement, and minimum area layout. However, their method appeared to be impractical as it required solving a large linear program. This paper shows that their method might be practical by giving an O(m log m) expectedtime randomized algorithm for their concurrent flow problem on an m-edge graph. Raghavan and Thompson used uniform-capacity concurrent flow to solve approximately a channel width minimization problem in very large scale integration. An O (k 3/2 (m + n log n)) expected-time randomized algorithm and an O (k min {n, k} (m + n log n) log k) deterministic algorithm is given for this problem when the channel width is f2 (log n), where k denotes the number of wires to be routed in an n-node, m-edge network. Key words, multicommodity flow, approximation, concurrent flow, graph separators, VLSI routing AMS subject classification. 68Q25, 90C08, 90C27 1. Introduction. The

We study the problem of finding a shortest path between two vertices in a directed graph. This is an important problem with many applications, including that of computing driving directions. We allow preprocessing the graph using a linear amount of extra space to store auxiliary information, and using this information to answer shortest path queries quickly. Our approach uses A ∗ search in combination with a new graph-theoretic lower-bounding technique based on landmarks and the triangle inequality. We also develop new bidirectional variants of A ∗ search and investigate several variants of the new algorithms to find those that are most efficient in practice. Our algorithms compute optimal shortest paths and work on any directed graph. We give experimental results showing that the most efficient of our new algorithms outperforms previous algorithms, in particular A ∗ search with Euclidean bounds, by a wide margin on road networks. We also experiment with several synthetic graph families.

. Ambivalent data structures are presented for several problems on undirected graphs. These data structures are used in finding the k smallest spanning trees of a weighted undirected graph in O(m log #(m, n) + min{k 3/2 ,km 1/2 }) time, where m is the number of edges and n the number of vertices in the graph. The techniques are extended to find the k smallest spanning trees in an embedded planar graph in O(n + k(log n) 3 ) time. Ambivalent data structures are also used to dynamically maintain 2-edge-connectivity information. Edges and vertices can be inserted or deleted in O(m 1/2 ) time, and a query as to whether two vertices are in the same 2-edge-connected component can be answered in O(log n) time, where m and n are understood to be the current number of edges and vertices, respectively. Key words. analysis of algorithms, data structures, embedded planar graph, fully persistent data structures, k smallest spanning trees, minimum spanning tree, on-line updating, topology tr...

Let G = (V