Degree-constrained subgraph problems: Hardness and approximation results

A general instance of a Degree-Constrained Subgraph problem consists of an edge-weighted or vertex-weighted graph G and the objective is to find an optimal weighted subgraph, subject to certain degree constraints on the vertices of the subgraph. This paper considers two natural Degree-Constrained Subgraph problems and studies their behavior in terms of approximation algorithms. These problems take as input an undirected graph G=(V,E), with |V|=n and |E|=m. Our results, together with the definition of the two problems, are listed below. The Maximum Degree-Bounded Connected Subgraph problem (MDBCS _d) takes as input a weight function w: E → ℝ⁺ and an integer d≥2, and asks for a subset E∈E such that the subgraph G'= (V,E') is connected, has maximum degree at most d, and Σ_e∈E'ω(e) is maximized. This problem is one of the classical NP-hard problems listed by Garey and Johnson in [Computers and Intractability, W.H. Freeman, 1979], but there were no results in the literature except for d∈=∈2. We prove that MDBCS _d is not in Apx for any d≥2 (this was known only for d=2) and we provide a -approximation algorithm for unweighted graphs, and a -approximation algorithm for weighted graphs. We also prove that when G has a low-degree spanning tree, in terms of d, MDBCS _d can be approximated within a small constant factor in unweighted graphs. The Minimum Subgraph of Minimum Degree _≥d (MSMD _d ) problem requires finding a smallest subgraph of G (in terms of number of vertices) with minimum degree at least d. We prove that MSMD _d is not in Apx for any d≥3 and we provide an -approximation algorithm for the class of graphs excluding a fixed graph as a minor, using dynamic programming techniques and a known structural result on graph minors.