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264
Improved algorithms for optimal winner determination in combinatorial auctions and generalizations
, 2000
"... Combinatorial auctions can be used to reach efficient resource and task allocations in multiagent systems where the items are complementary. Determining the winners is NPcomplete and inapproximable, but it was recently shown that optimal search algorithms do very well on average. This paper present ..."
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Cited by 519 (53 self)
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Combinatorial auctions can be used to reach efficient resource and task allocations in multiagent systems where the items are complementary. Determining the winners is NPcomplete and inapproximable, but it was recently shown that optimal search algorithms do very well on average. This paper presents a more sophisticated search algorithm for optimal (and anytime) winner determination, including structural improvements that reduce search tree size, faster data structures, and optimizations at search nodes based on driving toward, identifying and solving tractable special cases. We also uncover a more general tractable special case, and design algorithms for solving it as well as for solving known tractable special cases substantially faster. We generalize combinatorial auctions to multiple units of each item, to reserve prices on singletons as well as combinations, and to combinatorial exchanges  all allowing for substitutability. Finally, we present algorithms for determining the winners in these generalizations.
A general approximation technique for constrained forest problems
 in Proceedings of the 3rd Annual ACMSIAM Symposium on Discrete Algorithms
, 1992
"... Abstract. We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimizatio ..."
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Cited by 355 (21 self)
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Abstract. We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles, or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimumcost spanning tree, minimumweight perfect matching, traveling salesman, and Steiner tree problems. Our technique produces approximation algorithms that run in O(n log n) time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2approximation algorithm for the minimumweight perfect matching problem under the triangle inequality. Our running time of O(n log n) time compares favorably with the best strongly polynomial exact algorithms running in O(n 3) time for dense graphs. A similar result is obtained for the 2matching problem and its variants. We also derive the first approximation algorithms for many NPcomplete problems, including the nonfixed pointtopoint connection problem, the exact path partitioning problem, and complex locationdesign problems. Moreover, for the prizecollecting traveling salesman or Steiner tree problems, we obtain 2approximation algorithms, therefore improving the previously bestknown performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413420].
Primaldual approximation algorithms for metric facility location and kmedian problems
 Journal of the ACM
, 1999
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Computationally Manageable Combinatorial Auctions
, 1998
"... There is interest in designing simultaneous auctions for situations in which the value of assets to a bidder depends upon which other assets he or she wins. In such cases, bidders may well wish to submit bids for combinations of assets. When this is allowed, the problem of determining the revenue ma ..."
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Cited by 314 (1 self)
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There is interest in designing simultaneous auctions for situations in which the value of assets to a bidder depends upon which other assets he or she wins. In such cases, bidders may well wish to submit bids for combinations of assets. When this is allowed, the problem of determining the revenue maximizing set of nonconflicting bids can be a difficult one. We analyze this problem, identifying several different structures of combinatorial bids for which computational tractability is constructively demonstrated and some structures for which computational tractability 1 Introduction Some auctions sell many assets simultaneously. Often these assets, like U.S. treasury bills, are interchangeable. However, sometimes the assets and the bids for them are distinct. This happens frequently, as in the U.S. Department of the Interior's simultaneous sales of offshore oil leases, in some private farm land auctions, and in the Federal Communications Commission's recent multibillion dollar sales...
Cones of matrices and setfunctions and 01 optimization
 SIAM JOURNAL ON OPTIMIZATION
, 1991
"... It has been recognized recently that to represent a polyhedron as the projection of a higher dimensional, but simpler, polyhedron, is a powerful tool in polyhedral combinatorics. We develop a general method to construct higherdimensional polyhedra (or, in some cases, convex sets) whose projection a ..."
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Cited by 268 (7 self)
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It has been recognized recently that to represent a polyhedron as the projection of a higher dimensional, but simpler, polyhedron, is a powerful tool in polyhedral combinatorics. We develop a general method to construct higherdimensional polyhedra (or, in some cases, convex sets) whose projection approximates the convex hull of 01 valued solutions of a system of linear inequalities. An important feature of these approximations is that one can optimize any linear objective function over them in polynomial time. In the special case of the vertex packing polytope, we obtain a sequence of systems of inequalities, such that already the first system includes clique, odd hole, odd antihole, wheel, and orthogonality constraints. In particular, for perfect (and many other) graphs, this first system gives the vertex packing polytope. For various classes of graphs, including tperfect graphs, it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets. We also discuss an extension of the method, which establishes a connection with certain submodular functions and the Möbius function of a lattice.
Learning structured prediction models: a large margin approach
, 2004
"... We consider large margin estimation in a broad range of prediction models where inference involves solving combinatorial optimization problems, for example, weighted graphcuts or matchings. Our goal is to learn parameters such that inference using the model reproduces correct answers on the training ..."
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Cited by 164 (7 self)
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We consider large margin estimation in a broad range of prediction models where inference involves solving combinatorial optimization problems, for example, weighted graphcuts or matchings. Our goal is to learn parameters such that inference using the model reproduces correct answers on the training data. Our method relies on the expressive power of convex optimization problems to compactly capture inference or solution optimality in structured prediction models. Directly embedding this structure within the learning formulation produces concise convex problems for efficient estimation of very complex and diverse models. We describe experimental results on a matching task, disulfide connectivity prediction, showing significant improvements over stateoftheart methods. 1.
Edmonds polytopes and a hierarchy of combinatorial problems
, 2006
"... Let S be a set of linear inequalities that determine a bounded polyhedron P. The closure of S is the smallest set of inequalities that contains S and is closed under two operations: (i) taking linear combinations of inequalities, (ii) replacing an inequality Σaj xj ≤ a0, where a1,a2,...,an are integ ..."
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Cited by 146 (0 self)
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Let S be a set of linear inequalities that determine a bounded polyhedron P. The closure of S is the smallest set of inequalities that contains S and is closed under two operations: (i) taking linear combinations of inequalities, (ii) replacing an inequality Σaj xj ≤ a0, where a1,a2,...,an are integers, by the inequality Σaj xj ≤ a with a ≥[a0]. Obviously, if integers x1,x2,...,xn satisfy all the inequalities in S, then they satisfy also all inequalities in the closure of S. Conversely, let Σcj xj ≤ c0 hold for all choices of integers x1,x2,...,xn, that satisfy all the inequalities in S. Then we prove that Σcj xj ≤ c0 belongs to the closure of S. To each integer linear programming problem, we assign a nonnegative integer, called its rank. (The rank is the minimum number of iterations of the operation (ii) that are required in order to eliminate the integrality constraint.) We prove that there is no upper bound on the rank of problems arising from the search for largest independent sets in graphs.