Bounds on Maximum Matchings in 1-Planar Graphs

In this thesis, we study lower bounds on maximum matchings in 1-planar graphs. We expand upon the tools used for proofs of matching bounds in other classes of graphs as well as some original ideas in order to find these bounds. The first novel results we provide are lower bounds on maximum matchings in 1-planar graphs as a function of their minimum degree. We show that for sufficiently large n, 1-planar graphs with minimum degree 3 have a maximum matching of size at least (n+12)/7, 1-planar 7 graphs with minimum degree 4 have a maximum matching of size at least (n+4)/3, and 1-planar 3 graphs with minimum degree 5 have a maximum matching of size at least (2n+3)/5. All of these 5 bounds are tight. We also give examples of 1-planar graphs with small maximum matching and minimum degree 6 and 7. We conjecture that the 1-planar graph of minimum degree 6 presented has the smallest maximum matching over all 1-planar graphs of minimum degree 6, but it is unclear if the method used for the cases of minimum degree 3, 4, and 5 would work for minimum degree 6. We also study lower bounds in the class of maximal 1-plane graphs, and 3-connected maximal 1-plane graphs. We find that 3-connected, maximal 1-plane graphs have a maximum matching of size at least (n+4)/3, and that maximal 1-plane graphs have a maximum matching 3 of size at least (n+6)/4. Again, we present examples of such a graph to show this bound is tight. 4 We also show that every simple 3-connected maximum 1-planar graph has a matching of size at least (2n+6)/5, and provide some evidence that this is tight

The Cost of Perfection for Matchings in Graphs

Perfect matchings and maximum weight matchings are two fundamental combinatorial structures. We consider the ratio between the maximum weight of a perfect matching and the maximum weight of a general matching. Motivated by the computer graphics application in triangle meshes, where we seek to convert a triangulation into a quadrangulation by merging pairs of adjacent triangles, we focus mainly on bridgeless cubic graphs. First, we characterize graphs that attain the extreme ratios. Second, we present a lower bound for all bridgeless cubic graphs. Third, we present upper bounds for subclasses of bridgeless cubic graphs, most of which are shown to be tight. Additionally, we present tight bounds for the class of regular bipartite graphs

Computing Optimal Morse Matchings

Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NP-hard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results

Approximating Semi-Matchings in Streaming and in Two-Party Communication

We study the communication complexity and streaming complexity of approximating unweighted semi-matchings. A semi-matching in a bipartite graph G = (A, B, E), with n = |A|, is a subset of edges S that matches all A vertices to B vertices with the goal usually being to do this as fairly as possible. While the term 'semi-matching' was coined in 2003 by Harvey et al. [WADS 2003], the problem had already previously been studied in the scheduling literature under different names. We present a deterministic one-pass streaming algorithm that for any 0 <= \epsilon <= 1 uses space O(n^{1+\epsilon}) and computes an O(n^{(1-\epsilon)/2})-approximation to the semi-matching problem. Furthermore, with O(log n) passes it is possible to compute an O(log n)-approximation with space O(n). In the one-way two-party communication setting, we show that for every \epsilon > 0, deterministic communication protocols for computing an O(n^{1/((1+\epsilon)c + 1)})-approximation require a message of size more than cn bits. We present two deterministic protocols communicating n and 2n edges that compute an O(sqrt(n)) and an O(n^{1/3})-approximation respectively. Finally, we improve on results of Harvey et al. [Journal of Algorithms 2006] and prove new links between semi-matchings and matchings. While it was known that an optimal semi-matching contains a maximum matching, we show that there is a hierarchical decomposition of an optimal semi-matching into maximum matchings. A similar result holds for semi-matchings that do not admit length-two degree-minimizing paths.Comment: This is the long version including all proves of the ICALP 2013 pape