Abstract
We pursue a study of the Generalized Demand Matching problem, a common generalization of the bMatching 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 the vertex capacities. This problem is APXhard and constantfactor approximations are already known.
Our main results fall into two categories. First, using iterated relaxation and various filtering strategies, we show with an efficient rounding algorithm that if an additional matroid structure M is given and we further only allow sets that are independent in M, the natural LP relaxation has an integrality gap of at most 25/3. This can be further improved
in various special cases, for example we improve over the 15approximation for the previously studied Coupled Placement problem [Korupolu et al. 2014] by giving a 7approximation.
Using similar techniques, we show the problem of computing a minimumcost base in M satisfying vertex capacities admits a (1,3)bicriteria approximation: the cost is at most the optimum and the capacities are violated by a factor of at most 3. This improves over the previous (1,4)approximation in the special case that 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 that allows us to scale the demands of some items to be used in a more intricate dynamic programming algorithm, followed by some randomized rounding to filter our scaleddemand solution to one whose original demands satisfy all constraints.
BibTeX  Entry
@InProceedings{ahmadian_et_al:LIPIcs:2017:7460,
author = {Sara Ahmadian and Zachary Friggstad},
title = {{Further Approximations for Demand Matching: Matroid Constraints and MinorClosed Graphs}},
booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
pages = {55:155:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770415},
ISSN = {18688969},
year = {2017},
volume = {80},
editor = {Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7460},
URN = {urn:nbn:de:0030drops74600},
doi = {10.4230/LIPIcs.ICALP.2017.55},
annote = {Keywords: Approximation Algorithms, ColumnRestricted Packing, Demand Matching, Matroids, Planar Graphs}
}
Keywords: 

Approximation Algorithms, ColumnRestricted Packing, Demand Matching, Matroids, Planar Graphs 
Collection: 

44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) 
Issue Date: 

2017 
Date of publication: 

07.07.2017 