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 $\omega : E \rightarrow \mathbb R^+$ 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 ∑  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 $(\min \{m/ \log n,\ nd/(2 \log n)\})$-approximation algorithm for unweighted graphs, and a $(\min\{n/2,\ m/d\})$-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 $\mathcal{O}(n/\log n)$-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.