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URN: urn:nbn:de:0030-drops-48337
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Vertex Exponential Algorithms for Connected f-Factors

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Abstract

Given a graph G and a function f:V(G) -> [V(G)], an f-factor is a subgraph H of G such that deg_H(v)=f(v) for every vertex v in V(G); we say that H is a connected f-factor if, in addition, the subgraph H is connected. Tutte (1954) showed that one can check whether a given graph has a specified f-factor in polynomial time. However, detecting a connected f-factor is NP-complete, even when f is a constant function - a foremost example is the problem of checking whether a graph has a Hamiltonian cycle; here f is a function which maps every vertex to 2. The current best algorithm for this latter problem is due to Björklund (FOCS 2010), and runs in randomized O^*(1.657^n) time (the O^*() notation hides polynomial factors). This was the first superpolynomial improvement, in nearly fifty years, over the previous best algorithm of Bellman, Held and Karp (1962) which checks for a Hamiltonian cycle in deterministic O(2^n*n^2) time. In this paper we present the first vertex-exponential algorithms for the more general problem of finding a connected f-factor. Our first result is a randomized algorithm which, given a graph G on n vertices and a function f:V(G) -> [n], checks whether G has a connected f-factor in O^*(2^n) time. We then extend our result to the case when f is a mapping from V(G) to {0,1} and the degree of every vertex v in the subgraph H is required to be f(v)(mod 2). This generalizes the problem of checking whether a graph has an Eulerian subgraph; this is a connected subgraph whose degrees are all even (f(v) equiv 0). Furthermore, we show that the min-cost editing and edge-weighted versions of these problems can be solved in randomized O^*(2^n) time as long as the costs/weights are bounded polynomially in n.

BibTeX - Entry

@InProceedings{philip_et_al:LIPIcs:2014:4833,
  author =	{Geevarghese Philip and M. S. Ramanujan},
  title =	{{Vertex Exponential Algorithms for Connected f-Factors}},
  booktitle =	{34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)},
  pages =	{61--71},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-77-4},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{29},
  editor =	{Venkatesh Raman and S. P. Suresh},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2014/4833},
  URN =		{urn:nbn:de:0030-drops-48337},
  doi =		{10.4230/LIPIcs.FSTTCS.2014.61},
  annote =	{Keywords: Exact Exponential Time Algorithms, f-Factors}
}

Keywords: Exact Exponential Time Algorithms, f-Factors
Seminar: 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)
Issue date: 2014
Date of publication: 2014


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