Discrete Mathematics
Approximating the minimum degree spanning tree to within one from the optimal degree
SODA '92 Proceedings of the third annual ACM-SIAM symposium on Discrete algorithms
The traveling salesman problem with distances one and two
Mathematics of Operations Research
Approximating the minimum-degree Steiner tree to within one of optimal
SODA selected papers from the third annual ACM-SIAM symposium on Discrete algorithms
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Succinct Representation of Balanced Parentheses and Static Trees
SIAM Journal on Computing
The Approximation of Maximum Subgraph Problems
ICALP '93 Proceedings of the 20th International Colloquium on Automata, Languages and Programming
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Finding a Path of Superlogarithmic Length
SIAM Journal on Computing
Ruling Out PTAS for Graph Min-Bisection, Densest Subgraph and Bipartite Clique
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Discrete Applied Mathematics
Hardness and approximation of traffic grooming
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Parameterized complexity of the smallest degree-constrained subgraph problem
IWPEC'08 Proceedings of the 3rd international conference on Parameterized and exact computation
On approximating the d-girth of a graph
SOFSEM'11 Proceedings of the 37th international conference on Current trends in theory and practice of computer science
Parameterized complexity of finding small degree-constrained subgraphs
Journal of Discrete Algorithms
Tight complexity bounds for FPT subgraph problems parameterized by clique-width
IPEC'11 Proceedings of the 6th international conference on Parameterized and Exact Computation
Parameterized Domination in Circle Graphs
Theory of Computing Systems
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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.