Vector Bin Packing with Multiple-Choice

We consider a variant of bin packing called multiple-choice vector bin packing. In this problem we are given a set of items, where each item can be selected in one of several $D$-dimensional incarnations. We are also given $T$ bin types, each with its own cost and $D$-dimensional size. Our goal is to pack the items in a set of bins of minimum overall cost. The problem is motivated by scheduling in networks with guaranteed quality of service (QoS), but due to its general formulation it has many other applications as well. We present an approximation algorithm that is guaranteed to produce a solution whose cost is about $\ln D$ times the optimum. For the running time to be polynomial we require $D=O(1)$ and $T=O(\log n)$. This extends previous results for vector bin packing, in which each item has a single incarnation and there is only one bin type. To obtain our result we also present a PTAS for the multiple-choice version of multidimensional knapsack, where we are given only one bin and the goal is to pack a maximum weight set of (incarnations of) items in that bin

A NOTE ON DUAL APPROXIMATION ALGORITHMS FOR CLASS CONSTRAINED BIN PACKING PROBLEMS

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)In this paper we present a dual approximation scheme for the class constrained shelf bin packing problem. In this problem, we are given bins of capacity 1, and n items of Q different classes, each item e with class c(e) and size s(e). The problem is to pack the items into bins, such that two items of different classes packed in a same bin must be in different shelves. Items in a same shelf are packed consecutively. Moreover, items in consecutive shelves must be separated by shelf divisors of size d. In a shelf bin packing problem, we have to obtain a shelf packing such that the total size of items and shelf divisors in any bin is at most 1. A dual approximation scheme must obtain a shelf packing of all items into N bins, such that, the total size of all items and shelf divisors packed in any bin is at most 1 + epsilon for a given epsilon > 0 and N is the number of bins used in an optimum shelf bin packing problem. Shelf divisors are used to avoid contact between items of different classes and can hold a set of items until a maximum given weight. We also present a dual approximation scheme for the class constrained bin packing problem. In this problem, there is no use of shelf divisors, but each bin uses at most C different classes.432239248Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Faepex [31608]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)FAPESP [2008/01490-3]Faepex [31608]CNPq [478470/06-1, 472504/07-0, 306624/07-9

Approximation algorithms for distributed and selfish agents

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (p. 157-165).Many real-world systems involve distributed and selfish agents who optimize their own objective function. In these systems, we need to design efficient mechanisms so that system-wide objective is optimized despite agents acting in their own self interest. In this thesis, we develop approximation algorithms and decentralized mechanisms for various combinatorial optimization problems in such systems. First, we investigate the distributed caching and a general set of assignment problems. We develop an almost tight LP-based ... approximation algorithm and a local search ... approximation algorithm for these problems. We also design efficient decentralized mechanisms for these problems and study the convergence of the corresponding games. In the following chapters, we study the speed of convergence to high quality solutions on (random) best-response paths of players. First, we study the average social value on best response paths in basic-utility, market sharing, and cut games. Then, we introduce the sink equilibrium as a new equilibrium concept. We argue that, unlike Nash equilibria, the selfish behavior of players converges to sink equilibria and all strategic games have a sink equilibrium. To illustrate the use of this new concept, we study the social value of sink equilibria in weighted selfish routing (or weighted congestion) games and valid-utility (or submodular-utility) games. In these games, we bound the average social value on random best-response paths for sink equilibria.. Finally, we study cross-monotonic cost sharings and group-strategyproof mechanisms.(cont.) We study the limitations imposed by the cross-monotonicity property on cost-sharing schemes for several combinatorial optimization games including set cover and metric facility location. We develop a novel technique based on the probabilistic method for proving upper bounds on the budget-balance factor of cross-monotonic cost sharing schemes, deriving tight or nearly-tight bounds for these games. At the end, we extend some of these results to group-strategyproof mechanisms.by Vahab S. Mirrokni.Ph.D

The Class Constrained Bin Packing Problem With Applications To Video-on-demand

In this paper we present approximation results for the class constrained bin packing problem that has applications to Video-on-Demand Systems. In this problem we are given bins of capacity B with C compartments, and n items of Q different classes. The problem is to pack the items into the minimum number of bins, where each bin contains items of at most C different classes and has total items size at most B. We present several approximation algorithms for off-line and online versions of the problem. © Springer-Verlag Berlin Heidelberg 2006.4112 LNCS439448Coffman Jr., E.G., Garey, M.R., Johnson, D.S., Approximation algorithms for bin packing: A survey (1997) Approximation Algorithms for NP-hard Problems, pp. 46-93. , D. Hochbaum, editor, chapter 2. PWSDawande, M., Kalagnanam, J., Sethuraman, J., Variable sized bin packing with color constraints (1998) Technical Report, , IBM, T.J. Watson Research Center, NYDawande, M., Kalagnanam, J., Sethuraman, J., Variable sized bin packing with color constraints (2001) Electronic Notes in Dicrete Mathematics, 7. , Proceedings of Graco 2001De La Vega, W.F., Lueker, G.S., Bin packing can be solved within 1 + ε in linear time (1981) Combinatorica, 1 (4), pp. 349-355Golubchik, L., Khanna, S., Khuller, S., Thurimella, R., Zhu, A., Approximation algorithms for data placement on parallel disks (2000) Proceedings of SODA, pp. 223-232Johnson, D.S., Demers, A., Ullman, J.D., Garey, M.R., Graham, R.L., Worst-case performance bounds for simple one-dimensional packing algorithms (1974) SIAM Journal on Computing, 3, pp. 299-325Kashyap, S.R., Khuller, S., Algorithms for non-uniform size data placement on parallel disks (2003) Lecture Notes in Computer Science, 2914, pp. 265-276. , Proceedings of FSTTCSShachnai, H., Tamir, T., On two class-constrained versions of the multiple knapsack problem (2001) Algorithmica, 29, pp. 442-467Shachnai, H., Tamir, T., Polynomial time approximation schemes for class-constrained packing problems (2001) Journal of Scheduling, 4 (6), pp. 313-338Shachnai, H., Tamir, T., Approximation schemes for generalized 2-dimensional vector packing with application to data placement (2003) Lecture Notes in Computer Science, 2764, pp. 165-177. , Proceedings of 6th RANDOM-APPROXShachnai, H., Tamir, T., Tight bounds for online class-constrained packing (2004) Theoretical Computer Science, 321 (1), pp. 103-12