Max-weight integral multicommodity flow in spiders and high-capacity trees

Abstract. We consider the max-weight integer multicommodity flow problem in trees. In this problem we are given an edge-capacitated tree and weighted pairs of terminals, and the objective is to find a max-weight integral flow between terminal pairs subject to the capacities. This problem was shown to be APX-hard by Garg, Vazirani and Yannakakis [Algorithmica, 1997], and a 4-approximation was given by Chekuri, Mydlarz and Shepherd [ACM Trans. Alg., 2007]. Some special cases are known to be solvable in polynomial time, including when the graph is a star (via b-matching) or a path. First, when every edge has capacity at least µ ≥ 2, we use iterated relaxation to obtain an improved approximation ratio of min{3, 1+4/µ+6/(µ 2 −µ)}. We show this ratio bounds the integrality gap of the natural LP relaxation. A complementary hardness result yields a 1+Θ(1/µ) threshold of approximability (if P ̸ = NP). Second, we extend the range of instances for which exact solutions can be found efficiently. When the tree is a spider (i.e. if only one vertex has degree greater than 2) we give a polynomial-time algorithm to find an optimal solution, as well as a polyhedral description of the integer hull of all In the max-weight integral multicommodity flow problem (WMCF), we are given an undirected supply graph G = (V,E), terminal pairs (s1,t1),...,(sk,tk) where si,ti ∈ V, non-negative weight

Approximability of Sparse Integer Programs

The main focus of this paper is a pair of new approximation algorithms for certain integer programs. First, for covering integer programs {min cx: Ax >= b, 0 <= x <= d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A, b, c, d are nonnegative.) For any k >= 2 and eps>0, if P != NP this ratio cannot be improved to k-1-eps, and under the unique games conjecture this ratio cannot be improved to k-eps. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs {max cx: Ax <= b, 0 <= x <= d} where A has at most k nonzeroes per column, we give a (2k^2+2)-approximation algorithm. Our approach builds on the iterated LP relaxation framework. In addition, we obtain improved approximations for the second problem when k=2, and for both problems when every A_{ij} is small compared to b_i. Finally, we demonstrate a 17/16-inapproximability for covering integer programs with at most two nonzeroes per column.Comment: Version submitted to Algorithmica special issue on ESA 2009. Previous conference version: http://dx.doi.org/10.1007/978-3-642-04128-0_