Further Approximations for Demand Matching: Matroid Constraints and MinorClosed Graphs
Abstract
We pursue a study of the Generalized Demand Matching problem, a common generalization of the $b$Matching and Knapsack problems. Here, we are given a graph with vertex capacities, edge profits, and asymmetric demands on the edges. The goal is to find a maximumprofit subset of edges so the demands of chosen edges do not violate vertex capacities. This problem is APXhard and constantfactor approximations are known. Our results fall into two categories. First, using iterated relaxation and various filtering strategies, we show with an efficient rounding algorithm if an additional matroid structure $\mathcal M$ is given and we further only allow sets $F \subseteq E$ that are independent in $\mathcal M$, the natural LP relaxation has an integrality gap of at most $\frac{25}{3} \approx 8.333$. This can be improved in various special cases, for example we improve over the 15approximation for the previouslystudied Coupled Placement problem [Korupolu et al. 2014] by giving a $7$approximation. Using similar techniques, we show the problem of computing a minimumcost base in $\mathcal M$ satisfying vertex capacities admits a $(1,3)$bicriteria approximation. This improves over the previous $(1,4)$approximation in the special case that $\mathcal M$ is the graphic matroid over the given graph [Fukanaga and Nagamochi, 2009]. Second, we show Demand Matching admits a polynomialtime approximation scheme in graphs that exclude a fixed minor. If all demands are polynomiallybounded integers, this is somewhat easy using dynamic programming in boundedtreewidth graphs. Our main technical contribution is a sparsification lemma allowing us to scale the demands to be used in a more intricate dynamic programming algorithm, followed by randomized rounding to filter our scaleddemand solution to a feasible solution.
 Publication:

arXiv eprints
 Pub Date:
 May 2017
 arXiv:
 arXiv:1705.10396
 Bibcode:
 2017arXiv170510396A
 Keywords:

 Computer Science  Data Structures and Algorithms