Introduction to algorithms
Network flows: theory, algorithms, and applications
Network flows: theory, algorithms, and applications
Fast approximation algorithms for multicommodity flow problems
Selected papers of the 23rd annual ACM symposium on Theory of computing
Efficiency of a Good But Not Linear Set Union Algorithm
Journal of the ACM (JACM)
On the impact of combinatorial structure on congestion games
Journal of the ACM (JACM)
Pure Nash equilibria in player-specific and weighted congestion games
Theoretical Computer Science
Pure nash equilibria in player-specific and weighted congestion games
WINE'06 Proceedings of the Second international conference on Internet and Network Economics
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Given a weighted graph G = (V, E), a positive integer k, and a penalty function wp, we want to find k spanning trees on G, not necessarily disjoint, of minimum total weight, such that the weight of each edge is subject to a penalty given by wp if it belongs to more than one tree. The objective function to be minimized is Σe∈EWe(ie), where ie is the number of times edge e appears in the solution and We(ie) = iewp(e, ie) is the aggregate cost of using edge e ie times. For the case when We is weakly convex, which should have wide application in congestion problems, we present a polynomial time algorithm; the algorithm's complexity is quadratic in k. We also present two heuristics with complexity linear in k. In an experimental study we show that these heuristics are much faster than the exact algorithm also in practice. These experiments present a diverse combination of input families (four), varying k (up to 1000), and penalty functions (two). In most inputs tested the solutions given by the heuristics were within 1% of optimal or much better, especially for large k. The worst quality observed was 3.2% of optimal.