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

This report introduces a classification scheme for the global constraints. This classification is based on four basic ingredients from which one can generate almost all existing global constraints and come up with new interesting constraints. Global constraints are defined in a very concise way, in term of graph properties that have to hold, where the graph is a structured network of same elementary constraints. Since this classification is based on the internal structure of the global constraints it is also a strong hint for the pruning algorithms of the global constraints. Keywords Constraint, finite domain, global constraint, classification, resource constraint scheduling, graph partitioning, timetabling. 2 Table of contents Table of contents ....................................................................................................................................................... 2 Table of figures.........................................................................

Abstract—Ant colony optimization is a metaheuristic approach belonging to the class of model-based search algorithms. In this paper, we propose a new framework for implementing ant colony optimization algorithms called the hyper-cube framework for ant colony optimization. In contrast to the usual way of implementing ant colony optimization algorithms, this framework limits the pheromone values to the interval [0,1]. This is obtained by introducing changes in the pheromone value update rule. These changes can in general be applied to any pheromone value update rule used in ant colony optimization. We discuss the benefits coming with this new framework. The benefits are twofold. On the theoretical side, the new framework allows us to prove that in Ant System, the ancestor of all ant colony optimization algorithms, the average quality of the solutions produced increases in expectation over time when applied to unconstrained problems. On the practical side, the new framework automatically handles the scaling of the objective function values. We experimentally show that this leads on average to a more robust behavior of ant colony optimization algorithms. Index Terms—Ant colony optimization (ACO), metaheuristics. I.

Abstract. One of the promises of the service-oriented architecture (SOA) is that complex services can be composed using individual services. Individual services can be selected and integrated either statically or dynamically based on the service functionalities and performance constraints. For many distributed applications, the runtime performance (e.g. end-to-end delay, cost, reliability and availability) of complex services are very important. In our earlier work, we have studied the service selection problem for complex services with only one QoS constraint. This paper extends the service selection problem to multiple QoS constraints. The problem can be modelled in two ways: the combinatorial model and the graph model. The combinatorial model defines the problem as the multi-dimension multi-choice 0-1 knapsack problem (MMKP). The graph model defines the problem as the multi-constraint optimal path (MCOP) problem. We propose algorithms for both models and study their performances by test cases. We also compare the pros & cons between the two models. 1

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. We consider variants of the classic bin packing and multiple knapsack problems, in which sets of items of different classes (colors) need to be placed in bins; the items may have different sizes and values. Each bin has a limited capacity, and a bound on the number of distinct classes of items it can hold. In the class-constrained multiple knapsack (CCMK) problem, our goal is to maximize the total value of packed items, whereas in the class-constrained bin-packing (CCBP), we seek to minimize the number of (identical) bins, needed for packing all the items. We give a polynomial time approximation scheme (PTAS) for CCMK and a dual PTAS for CCBP. We also show that the 0-1 class-constrained knapsack admits a fully polynomial time approximation scheme, even when the number of distinct colors of items depends on the input size. Finally, we introduce the generalized class-constrained packing problem (GCCP), where each item may have more than one color. We show that GCCP is APX...

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We study two variants of the classic knapsack problem, in which we need to place items of different types in multiple knapsacks; each knapsack has a limited capacity, and a bound on the number of different types of items it can hold: in the class-constrained multiple knapsack problem (CMKP) we wish to maximize the total number of packed items; in the fair placement problem (FPP) our goal is to place the same (large) portion from each set. We look for a perfect placement, in which both problems are solved optimally. We first show that the two problems are NP-Hard; we then consider some special cases, where a perfect placement exists and can be found in polynomial time. For other cases, we give approximate solutions. Finally, we give a nearly optimal solution for the CMKP. Our results for the CMKP and the FPP are shown to provide efficient solutions for two fundamental problems arising in multimedia storage sub-systems. Key words. knapsack, packing, approximation algorithms, resource a...

The concept of submodularity plays a vital role in combinatorial optimization. In particular, many important optimization problems can be cast as submodular maximization problems, including maximum coverage, maximum facility location and max cut in directed/undirected graphs. In this paper we present the first known approximation algorithms for the problem of maximizing a non-decreasing submodular set function subject to multiple linear constraints. Given a d-dimensional budget vector ¯ L, for some d ≥ 1, and an oracle for a non-decreasing submodular set function f over a universe U, where each element e ∈ U is associated with a d-dimensional cost vector, we seek a subset of elements S ⊆ U whose total cost is at most ¯ L, such that f(S) is maximized. We develop a framework for maximizing submodular functions subject to d linear constraints that yields a (1 − ε)(1 − e−1)-approximation to the optimum for any ε> 0, where d> 1 is some constant. Our study is motivated by a variant of the classical maximum coverage problem that we call maximum coverage with multiple packing constraints. We use our framework to obtain the same approximation ratio for this problem. To the best of our knowledge, this is the first time the theoretical bound of 1 − e−1 is (almost) matched for both of these problems.

Motivated by a real world application, we study the multiple knapsack problem with assignment restrictions (MKAR): We are given a set of items N = f1; : : : ; ng and a set of knapsacks M = f1; : : : ; mg. Each item j 2 N has a positive real weight w j and each knapsack i 2 M has a positive real capacity c i associated with it. In addition, for each item j 2 N a set A j ` M of knapsacks that can hold item j is specified. In a feasible assignment of items to knapsacks, for each knapsack i 2 M , we need to choose a subset S i of items in N to be assigned to knapsack i, such that (i) Each item is assigned to at most one knapsack (ii) Assignment restrictions are satisfied and (iii) For each knapsack, its capacity constraint is satisfied. We consider two objectives (i) Maximize assigned weight P i2M P j2S i w j and (ii) minimize utilized capacity P i:S i 6=; c i Our results include two 1 3 approximation algorithms and two 1 2 approximation algorithms for the single objective problem of maximizing assigned weight. For the bi-criteria problem which considers both the objectives, we present two algorithms with performance ratios ( 1 3 ; 2) and ( 1 2 ; 3) respectively.