By Erik D. Demaine, Nicole Immorlica (auth.), Sanjeev Arora, Klaus Jansen, José D. P. Rolim, Amit Sahai (eds.)

This publication constitutes the joint refereed lawsuits of the sixth foreign Workshop on Approximation Algorithms for Optimization difficulties, APPROX 2003 and of the seventh overseas Workshop on Randomization and Approximation ideas in laptop technological know-how, RANDOM 2003, held in Princeton, new york, united states in August 2003.

The 33 revised complete papers provided have been conscientiously reviewed and chosen from seventy four submissions. one of the matters addressed are layout and research of randomized and approximation algorithms, on-line algorithms, complexity concept, combinatorial buildings, error-correcting codes, pseudorandomness, derandomization, community algorithms, random walks, Markov chains, probabilistic facts structures, computational studying, randomness in cryptography, and numerous applications.

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**Extra info for Approximation, Randomization, and Combinatorial Optimization.. Algorithms and Techniques: 6th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2003 and 7th International Workshop on Randomization and Appro**

**Sample text**

The problem is to determine if there is a partition of U into k parts U1 , . . , Uk such that for every i = 1, . . , k, we have u∈Ui su ≤ B. This was shown to be NP-hard in [9]. Theorem 1. The min-max R-centered star cover problem is NP-complete. Proof. Given an instance Π = U, {su }u , k, B of BIN-PACK, we transform it to an instance of R-centered star cover as follows. We create a complete bipartite graph G(Π) with a vertex set R ∪ U , where R is a set of k new nodes R = {r1 , . . , rk }. For every ri and every u ∈ U , the weight of an edge e = (ri , u) is set to w(e) = su .

We begin by showing the NP-completeness of R-centered star cover, and then extend the result to the other three problems. We show the NP-completeness of R-centered star cover by reducing BINPACK to it. An instance of BIN-PACK consists of (i) a set U of elements, where the size of an element u ∈ U is su , (ii) k bins, and (iii) a positive bin capacity B. The problem is to determine if there is a partition of U into k parts U1 , . . , Uk such that for every i = 1, . . , k, we have u∈Ui su ≤ B. This was shown to be NP-hard in [9].

The algorithms in [1] do not seem to extend to rooted versions. These problems fall in the general class of “vehicle routing” problems (see [16] for a recent survey). e. starting and ending point). The objective is to minimize the total length of tours. The k-traveling salesperson problem was ﬁrst approximated to a constant by Frederickson, Hecht and Kim [8] (see also [11]). Recently, Fakcharoenphol, Harrelson and Rao [7] provided a constant-factor approximation algorithm for the k-traveling repairman problem, where the objective is to minimize the average waiting time of the customers.