This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We present new faster and much simpler algorithms for these problems.

This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We present new faster and much simpler algorithms for these problems.

The minimum-weight set cover problem is widely known to be O(log n)-approximable, with no improvement possible in the general case. We take the approach of exploiting problem structure to achieve better results, by providing a geometry-inspired algorithm whose approximation guarantee depends solely on an instance-specific combinatorial property known as shallow cell complexity (SCC). Roughly speaking, a set cover instance has low SCC if any column-induced submatrix of the corresponding element-set incidence matrix has few distinct rows. By adapting and improving Varadarajan's recent quasi-uniform random sampling method for weighted geometric covering problems, we obtain strong approximation algorithms for a structurally rich class of weighted covering problems with low SCC. We also show how to derandomize our algorithm.Our main result has several immediate consequences. Among them, we settle an open question of Chakrabarty et al. [8] by showing that weighted instances of the capacitated covering problem with underlying network structure have O(1)-approximations. Additionally, our improvements to Varadarajan's sampling framework yield several new results for weighted geometric set cover, hitting set, and dominating set problems. In particular, for weighted covering problems exhibiting linear (or near-linear) union complexity, we obtain approximability results agreeing with those known for the unweighted case. For example, we obtain a constant approximation for the weighted disk cover problem, improving upon the 2 O(log * n) -approximation known prior to our work and matching the O(1)-approximation known for the unweighted variant.

We show that the standard linear programming relaxation for the tree augmentation problem in undirected graphs has an integrality ratio that approaches 3 2 . This refutes a conjecture of Cheriyan, Jordán, and Ravi (ESA 1999) that the integrality ratio is

Abstract. We consider a game-theoretical variant of the Steiner forest problem in which each player j, out of a set of k players, strives to connect his terminal pair (s j , t j ) of vertices in an undirected, edge-weighted graph G. In this paper we show that a natural adaptation of the primaldual Steiner forest algorithm of Agrawal, Klein, and Ravi [SIAM J. Comput., 24 (1995), pp. 445-456] yields a 2-budget balanced and cross-monotonic cost sharing method for this game. We also present a negative result, arguing that no cross-monotonic cost sharing method can achieve a budget balance factor of less than 2 for the Steiner tree game. This shows that our result is tight. Our algorithm gives rise to a new linear programming relaxation for the Steiner forest problem which we term the lifted-cut relaxation. We show that this new relaxation is stronger than the standard undirected cut relaxation for the Steiner forest problem.