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Vertex Exponential Algorithms for Connected f-Factors Philip, Geevarghese Ramanujan, M. S. Exact Exponential Time Algorithms f-Factors 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Geevarghese Philip and M. S. Ramanujan 2014 Is Part Of LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.FSTTCS.2014.61 urn:nbn:de:0030-drops-48337 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.61 eng https://creativecommons.org/licenses/by/3.0/legalcode