Matching (graph theory)

Summary

In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a network flow problem. Definitions Given a graph G = (V, E), a matching M in G is a set of pairwise non-adjacent edges, none of which are loops; that is, no two edges share common vertices. A vertex is matched (or saturated) if it is an endpoint of one of the edges in the matching. Otherwise the vertex is unmatched (or unsaturated). A maximal matching is a matching M of a graph G that is not a subset of any other matching. A matching M of a graph G is maximal if every edge in G has a non-empty intersection with at least one edge in M. The following figure shows examples of maximal matchings (red) in three graphs. : A maximum matching (also kn

Official source

https://en.wikipedia.org/wiki/Matching_(graph_theory)

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Submodular functions are a widely studied topic in theoretical computer science. They have found several applications both theoretical and practical in the fields of economics, combinatorial optimization and machine learning. More recently, there have also been numerous works that study combinatorial problems with submodular objective functions. This is motivated by their natural diminishing returns property which is useful in real-world applications. The thesis at hand is concerned with the study of streaming and matching problems with submodular functions.Firstly, motivated by developing robust algorithms, we propose a new adversarial injections model, in which the input is ordered randomly, but an adversary may inject misleading elements at arbitrary positions. We study the maximum matching problem and cardinality constrained monotone submodular maximization. We show that even under this seemingly powerful adversary, it is possible to break the barrier of 1/2 for both these problems in the streaming setting. Our main result is a novel streaming algorithm that computes a 0.55-approximation for cardinality constrained monotone submodular maximization.In the second part of the thesis, we study the problem of matroid intersection in the semi-streaming setting. Our main result is a (2 + e)-approximate semi-streaming algorithm for weighted matroid inter- section improving upon the previous best guarantee of 4 + e. While our algorithm is based on the local ratio technique, its analysis differs from the related problem of weighted maximum matching and uses the concept of matroid kernels. We are also able to generalize our results to work for submodular functions by adapting ideas from a recent result by Levin and Wajc (SODA'21) on submodular maximization subject to matching constraints.Finally, we study the submodular Santa Claus problem in the restricted assignment case. The submodular Santa Claus problem was introduced in a seminal work by Goemans, Harvey, Iwata, and Mirrokni (SODA'09) as an application of their structural result. In the mentioned problem n unsplittable resources have to be assigned to m players, each with a monotone submodular utility function fi. The goal is to maximize mini fi(Si) where S1, . . . , Sm is a partition of the resources. The result by Goemans et al. implies a polynomial time O(n1/2+e)-approximation algorithm. In the restricted assignment case, each player is given a set of desired resources Gi and the individual valuation functions are defined as fi(S)= f(SnGi). OurmainresultisaO(loglog(n))-approximation algorithm for the problem. Our proof is inspired by the approach of Bansal and Srividenko (STOC'06) to the Santa Claus problem. Com- pared to the more basic linear setting, the introduction of submodularity requires a much more involved analysis and several new ideas.

EPFL2022

Graph orientations with edge-connection and parity constraints

Parity (matching theory) and connectivity (network flows) are two main branches of combinatorial optimization. In an attempt to understand better their interrelation, we study a problem where both parity and connectivity requirements are imposed. The main result is a characterization of undirected graphs G = (V,E) having a k-edge-connected T-odd orientation for every subset with |E| + |T| even. (T-odd orientation: the in-degree of v is odd precisely if v is in T.) As a corollary, we obtain that every (2k)-edge- connected graph with |V| + |E| even has a (k-1)-edge- connected orientation in which the in-degree of every node is odd. Along the way, a structural characterization will be given for digraphs with a root-node s having k edge- disjoint paths from s to every node and k-1 edge-disjoint paths from every node to s.

2001

Induced Matchings In Graphs Of Degree At Most 4

Viet Hang Nguyen

For a graph G, let nu(s)(G) be the strong matching number of G. We prove the sharp bound nu(s)(G) >= n(G)/9 for every graph G of maximum degree at most 4 and without isolated vertices that does not contain a certain blown-up 5-cycle as a component. This result implies a strengthening of a consequence, namely, nu(s)(G) >= m(G)/18 for such graphs and nu(s)(G) >= m(G)/20 for a graph of maximum degree 4, of the well-known conjecture of Erdos and Nesetril, which says that the strong chromatic index chi(s)'(G) of a graph G is at most 5/4 Delta(G)(2), since nu(s)(G) >= m(G)/chi(s)'(G) and n(G) >= 2m(G)/Delta(G). This bound is tight and the proof implies a polynomial time algorithm to find an induced matching of this size.

Siam Publications2016

EPFL2022