Abstract
Given a graph G and a function f:V(G) > [V(G)], an ffactor 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 ffactor if, in addition, the subgraph H is connected. Tutte (1954) showed that one can check whether a given graph has a specified ffactor in polynomial time. However, detecting a connected ffactor is NPcomplete, 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 vertexexponential algorithms for the more general problem of finding a connected ffactor. 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 ffactor 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 mincost editing and edgeweighted 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 fFactors}},
booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)},
pages = {6171},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897774},
ISSN = {18688969},
year = {2014},
volume = {29},
editor = {Venkatesh Raman and S. P. Suresh},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2014/4833},
URN = {urn:nbn:de:0030drops48337},
doi = {10.4230/LIPIcs.FSTTCS.2014.61},
annote = {Keywords: Exact Exponential Time Algorithms, fFactors}
}
Keywords: 

Exact Exponential Time Algorithms, fFactors 
Collection: 

34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014) 
Issue Date: 

2014 
Date of publication: 

12.12.2014 