Abstract
In the partial degree bounded edge packing problem (PDBEP), the input is an undirected graph \(G=(V,E)\) with capacity \(c_v\in {\mathbb {N}}\) on each vertex. The objective is to find a feasible subgraph \(G'=(V,E')\) maximizing \(|E'|\), where \(G'\) is said to be feasible if for each \(e=\{u,v\}\in E'\), \(\deg _{G'}(u)\le c_u\) or \(\deg _{G'}(v)\le c_v\). In the weighted version of the problem, additionally each edge \(e\in E\) has a weight w(e) and we want to find a feasible subgraph \(G'=(V,E')\) maximizing \(\sum _{e\in E'} w(e)\). The problem is already NP-hard if \(c_v = 1\) for all \(v\in V\) [Zhang, FAW-AAIM 2012].
In this paper, we introduce a generalization of the PDBEP problem. We let the edges have weights as well as demands, and we present the first constant-factor approximation algorithms for this problem. Our results imply the first constant-factor approximation algorithm for the weighted PDBEP problem, improving the result of Aurora et al. [FAW-AAIM 2013] who presented an \(O(\log n)\)-approximation for the weighted case.
We also present a PTAS for H-minor free graphs, if the demands on the edges are bounded above by a constant, and we show that the problem is APX-hard even for cubic graphs and bounded degree bipartite graphs with \(c_v = 1, \; \forall v\in V\).
M. Jena—The author is supported by a TCS Scholarship.
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Aurora, P., Jena, M., Raman, R. (2016). Constant Factor Approximation for the Weighted Partial Degree Bounded Edge Packing Problem. In: Chan, TH., Li, M., Wang, L. (eds) Combinatorial Optimization and Applications. COCOA 2016. Lecture Notes in Computer Science(), vol 10043. Springer, Cham. https://doi.org/10.1007/978-3-319-48749-6_14
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