LP duality in infinite hypergraphs

AbstractIn any graph there exist a fractional cover and a fractional matching satisfying the complementary slackness conditions of linear programming. The proof uses a Gallai-Edmonds decomposition result for infinite graphs. We consider also the same problem for infinite hypergraphs, in particular in the case that the edges of the hypergraph are intervals on the real line. We prove an extension of a theorem of Gallai to the infinite case

Distributed Approximations of f-Matchings and b-Matchings in Graphs of Sub-Logarithmic Expansion

We give a distributed algorithm which given ? > 0 finds a (1-?)-factor approximation of a maximum f-matching in graphs G = (V,E) of sub-logarithmic expansion. Using a similar approach we also give a distributed approximation of a maximum b-matching in the same class of graphs provided the function b: V ? ?^+ is L-Lipschitz for some constant L. Both algorithms run in O(log^* n) rounds in the LOCAL model, which is optimal

On the Factorization of Graphs with Exactly One Vertex of Infinite Degree

AbstractWe give a necessary and sufficient condition for the existence of a 1-factor in graphs with exactly one vertex of infinite degree

On the Number of 1-Factors of Locally Finite Graphs

AbstractEvery infinite locally finite graph with exactly one 1-factor is at most 2-connected is shown. More generally a lower bound for the number of 1-factors in locally finite n-connected graphs is given

A Randomized Polynomial Kernelization for Vertex Cover with a Smaller Parameter

In the Vertex Cover problem we are given a graph $G=(V,E)$ and an integer $k$ and have to determine whether there is a set $X\subseteq V$ of size at most $k$ such that each edge in $E$ has at least one endpoint in $X$. The problem can be easily solved in time $O^*(2^k)$, making it fixed-parameter tractable (FPT) with respect to $k$. While the fastest known algorithm takes only time $O^*(1.2738^k)$, much stronger improvements have been obtained by studying parameters that are smaller than $k$. Apart from treewidth-related results, the arguably best algorithm for Vertex Cover runs in time $O^*(2.3146^p)$, where $p=k-LP(G)$ is only the excess of the solution size $k$ over the best fractional vertex cover (Lokshtanov et al.\ TALG 2014). Since $p\leq k$ but $k$ cannot be bounded in terms of $p$ alone, this strictly increases the range of tractable instances. Recently, Garg and Philip (SODA 2016) greatly contributed to understanding the parameterized complexity of the Vertex Cover problem. They prove that $2LP(G)-MM(G)$ is a lower bound for the vertex cover size of $G$, where $MM(G)$ is the size of a largest matching of $G$, and proceed to study parameter $\ell=k-(2LP(G)-MM(G))$. They give an algorithm of running time $O^*(3^\ell)$, proving that Vertex Cover is FPT in $\ell$. It can be easily observed that $\ell\leq p$ whereas $p$ cannot be bounded in terms of $\ell$ alone. We complement the work of Garg and Philip by proving that Vertex Cover admits a randomized polynomial kernelization in terms of $\ell$, i.e., an efficient preprocessing to size polynomial in $\ell$. This improves over parameter $p=k-LP(G)$ for which this was previously known (Kratsch and Wahlstr\"om FOCS 2012).Comment: Full version of ESA 2016 pape