Abstract. Some important classical mechanisms considered in Microeconomics and Game Theory require the solution of a difficult optimization problem. This is true of mechanisms for combinatorial auctions, which have in recent years assumed practical importance, and in particular of the gold standard for combinatorial auctions, the Generalized Vickrey Auction (GVA). Traditional analysis of these mechanisms—in particular, their truth revelation properties—assumes that the optimization problems are solved precisely. In reality, these optimization problems can usually be solved only in an approximate fashion. We investigate the impact on such mechanisms of replacing exact solutions by approximate ones. Specifically, we look at a particular greedy optimization method. We show that the GVA payment scheme does not provide for a truth revealing mechanism. We introduce another scheme that does guarantee truthfulness for a restricted class of players. We demonstrate the latter property by identifying natural properties for combinatorial auctions and showing that, for our restricted class of players, they imply that truthful strategies are dominant. Those properties have applicability beyond the specific auction studied.

When attempting to design a truthful mechanism for a computationally hard problem such as combinatorial auctions, one is faced with the problem that most efficiently computable heuristics can not be embedded in any truthful mechanism (e.g. VCG-like payment rules will not ensure truthfulness). We develop a set of techniques that allow constructing efficiently computable truthful mechanisms for combinatorial auctions in the special case where each bidder desires a specific known subset of items and only the valuation is unknown by the mechanism (the single parameter case). For this case we extend the work of Lehmann O’Callaghan, and Shoham, who presented greedy heuristics. We show how to use IF-THEN-ELSE constructs, perform a partial search, and use the LP relaxation. We apply these techniques for several canonical types of combinatorial auctions, obtaining truthful mechanisms with provable approximation ratios. 1

We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems, and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most Δ, the algorithm achieves a performance ratio of 2 - (1 - o(1)) 2 ln ln \Delta ln \Delta for large \Delta, which improves the previously known ratio of 2 \Gamma log \Delta+O(1) \Delta obtained by Halldórsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For k-uniform hypergraphs with n vertices, we achieve a ratio of k \Gamma (1 \Gamma o(1)) k ln ln n ln n for large n, and for k-uniform hypergraphs with maximum degree at most \Delta, the algorithm achieves a ratio of k \Gamma (1 \Gamma o(1)) k(k\Gamma1) ln ln \Delta ln \Delta for large \Delta. These results considerably improve the previous best ratio of k(1\Gammac=\Delta 1 k\Gamma1 ) for bounded degree k-uniform hypergraphs, and k(1 \Gamma c=n k\Gamma1 k ) for general k-uniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of Halldórsson.

Some important classical mechanisms considered in Microeconomics and Game Theory require the solution of a difficult optimization problem. This is true of mechanisms for combinatorial auctions, which have in recent years assumed practical importance, and in particular of the gold standard for combinatorial auctions, the Generalized Vickrey Auction (GVA). Traditional analysis of these mechanisms - in particular, their truth revelation properties - assumes that the optimization problems are solved precisely. In reality, these optimization problems can usually be solved only in an approximate fashion. We investigate the impact on such mechanisms of replacing exact solutions by approximate ones. Specifically, we look at a particular greedy optimization method, which has empirically been shown to perform well. We show that the GVA payment scheme does not provide for a truth revealing mechanism. We introduce another scheme that does guarantee truthfulness for a restricted class...

Finding optimal solutions for multi-unit combinatorial auctions is a hard problem and nding approximations to the optimal solution is also hard. We investigate the use of Branch-and-Bound techniques: they require both a way to bound from above the value of the best allocation and a good criterion to decide which bids are to be tried rst. Dierent methods for eciently bounding from above the value of the best allocation are considered. Theoretical original results characterize the best approximation ratio and the ordering criterion that provides it. We suggest to use this criterion. Keywords Combinatorial Auctions, Branch and Bound 1. MULTI-UNIT COMBINATORIAL AUCTIONS (MUCAS) Auctions have been used from times immemorial, but the renewed modern interest in auctions stems from: their increased use for selling o government property after WWII and later in extensive denationalizations, and the theoretical breakthroughs started by [14]. A very recent surge of interest in aucti...

We present lower bounds on the competitive ratio of randomized algorithms for a wide class of on-line graph optimization problems and we apply such results to on-line virtual circuit and optical routing problems. Lund and Yannakakis [LY93a] give inapproximability results for the problem of finding the largest vertex induced subgraph satisfying any non-trivial, hereditary, property . E.g., independent set, planar, acyclic, bipartite, etc. We consider the on-line version of this family of problems, where some graph G is fixed and some subgraph H is presented on-line, vertex by vertex. The on-line algorithm must choose a subset of the vertices of H , choosing or rejecting a vertex when it is presented, whose vertex induced subgraph satisfies property . Furthermore, we study the on-line version of graph coloring whose off-line version has also been shown to be inapproximable [LY93b], on-line max edge-disjoint paths and on-line path coloring problems. Irrespective of the time complexity, w...

Scheduling dependent jobs on multiple machines is modeled by the graph multicoloring problem. In this paper we consider the problem of minimizing the average completion time of all jobs. This is formalized as the sum multicoloring problem: Given a graph and the number of colors required by each vertex, find a multicoloring which minimizes the sum of the largest colors assigned to the vertices. It reduces to the known sum coloring problem when each vertex requires exactly one color. This paper

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... (PIP) — the problems of finding a maximum-value 0/1 vector x satisfying Ax ≤ b, with A and b nonnegative. Many combinatorial problems belong to this broad class, including the knapsack problem, maximum clique, stable set, matching, hypergraph matching (a.k.a. set packing), bmatching, and others. PIP can also be seen as a “multidimensional” knapsack problem where we wish to pack a maximum-value collection of items with vector-valued sizes. In our stochastic setting, the vector-valued size of each item is known to us apriori only as a probability distribution, and the size of an item is instantiated once we commit to including the item in our solution. Following the framework of [3], we consider both adaptive and non-adaptive policies for solving such problems, adaptive policies having the flexibility of being able to make ...

We show that aggregate constraints (as opposed to pairwise constraints) that often arise when integrating multiple sources of data, can be leveraged to enhance the quality of deduplication. However, despite its appeal, we show that the problem is challenging, both semantically and computationally. We define a restricted search space for deduplication that is intuitive in our context and we solve the problem optimally for the restricted space. Our experiments on real data show that incorporating aggregate constraints significantly enhances the accuracy of deduplication.