## Abstract

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.

Original language | English |
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Pages (from-to) | 29-42 |

Number of pages | 14 |

Journal | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Volume | 5426 LNCS |

DOIs | |

State | Published - 2009 |

Externally published | Yes |

Event | 6th International Workshop on Approximation and Online Algorithms, WAOA 2008 - Karlsruhe, Germany Duration: 18 Sep 2008 → 19 Sep 2008 |

### Bibliographical note

Funding Information:This work has been partially supported by European project IST FET AEOLUS, PACA region of France, Ministerio de Educación y Ciencia of Spain, European Regional Development Fund under project TEC2005-03575, Catalan Research Council under project 2005SGR00256 and COST action 293 GRAAL, and has been done in the context of the crc Corso with France Telecom.

## Keywords

- Approximation Algorithms
- Apx
- Degree-Constrained Subgraphs
- Excluded Minor
- Hardness of Approximation
- PTAS