By Yuval Rabani (auth.), Klaus Jansen, Stefano Leonardi, Vijay Vazirani (eds.)

This ebook constitutes the refereed lawsuits of the fifth foreign Workshop on Approximation Algorithms for Combinatorial Optimization difficulties, APPROX 2002, held in Rome, Italy in September 2002.
The 20 revised complete papers offered have been rigorously reviewed and chosen from fifty four submissions. one of the subject matters addressed are layout and research of approximation algorithms, inapproximability effects, on-line difficulties, randomization strategies, average-case research, approximation periods, scheduling difficulties, routing and move difficulties, coloring and partitioning, cuts and connectivity, packing and protecting, geometric difficulties, community layout, and functions to video game concept and different fields.

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Example text

Set cover is one of the oldest and most well-studied NP-hard problems. Johnson [8] and Lov´ asz [9] proposed a simple greedy algorithm which they showed provides a H(n)-approximation, where H(n) is the n-th harmonic number, and n = |U |. Chv´ atal [3] extended their results to the weighted case. 2. , currently uncovered) elements. In this paper, we refer to this algorithm as the greedy algorithm for set cover, to distinguish it from other priority algorithms that one can define. It is interesting to note that certain primal-dual algorithms can be seen as priority algorithms, or, more precisely, certain priority algorithms have an alternative primal-dual statement.

Let P1 , P2 , . . , Pni be any disjoint partition of P in sets of size si . We claim that for any m with 1 ≤ m ≤ ni there exists at least one set in Ri that covers Pm . Define the set S as the set that contains exactly sh − sh+1 elements of Ch , for every h with 1 ≤ h < i and also contains Pm . Since the Ch ’s with h < i and Pm are all disjoint, the set S has size i−1 (sh − sh+1 ) + si = (s1 − si ) + si = s1 = d h=1 hence S belongs in R1 . It remains to show that S is in Ri . To this end, note that for all j < i, j |S ∩ j Ch | = h=1 (sh − sh+1 ) = s1 − sj+1 , h=1 therefore S is in Rj+1 , for all j < i (and in particular, for j = i − 1).

It is worth mentioning that in this context every priority set cover algorithm belongs in the class of Greedy algorithms4 . 3 Orderings and the Role of the Adversary To show a lower bound on the approximation ratio we must evaluate the performance of every priority algorithm for an appropriately constructed nemesis input set. The construction of such a nemesis set can be seen as a game between an adversary and the algorithm and is conceptually similar to the construction of an adversarial input in competitive analysis.

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