{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article7651","name":"Vertex Exponential Algorithms for Connected f-Factors","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 Bj\u00f6rklund (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.\r\n \r\nIn 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.","keywords":["Exact Exponential Time Algorithms","f-Factors"],"author":[{"@type":"Person","name":"Philip, Geevarghese","givenName":"Geevarghese","familyName":"Philip"},{"@type":"Person","name":"Ramanujan, M. S.","givenName":"M. S.","familyName":"Ramanujan"}],"position":7,"pageStart":61,"pageEnd":71,"dateCreated":"2014-12-12","datePublished":"2014-12-12","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":[{"@type":"Person","name":"Philip, Geevarghese","givenName":"Geevarghese","familyName":"Philip"},{"@type":"Person","name":"Ramanujan, M. S.","givenName":"M. S.","familyName":"Ramanujan"}],"copyrightYear":"2014","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.FSTTCS.2014.61","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6232","volumeNumber":29,"name":"34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)","dateCreated":"2014-12-12","datePublished":"2014-12-12","editor":[{"@type":"Person","name":"Raman, Venkatesh","givenName":"Venkatesh","familyName":"Raman"},{"@type":"Person","name":"Suresh, S. P.","givenName":"S. P.","familyName":"Suresh"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article7651","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6232"}}}