Packing triangles in low degree graphs and indifference graphs

We consider the problems of finding the maximum number of vertex-disjoint triangles (VTP) and edge-disjoint triangles (ETP) in a simple graph. Both problems are NP-hard. The algorithm with the best approximation guarantee known so far for these problems has ratio $3/2 + ɛ$, a result that follows from a more general algorithm for set packing obtained by Hurkens and Schrijver in 1989. We present improvements on the approximation ratio for restricted cases of VTP and ETP that are known to be APX-hard: we give an approximation algorithm for VTP on graphs with maximum degree 4 with ratio slightly less than 1.2, and for ETP on graphs with maximum degree 5 with ratio 4/3. We also present an exact linear-time algorithm for VTP on the class of indifference graphs

Maximum weight cycle packing in directed graphs, with application to kidney exchange programs

Centralized matching programs have been established in several countries to organize kidney exchanges between incompatible patient-donor pairs. At the heart of these programs are algorithms to solve kidney exchange problems, which can be modelled as cycle packing problems in a directed graph, involving cycles of length 2, 3, or even longer. Usually, the goal is to maximize the number of transplants, but sometimes the total benefit is maximized by considering the differences between suitable kidneys. These problems correspond to computing cycle packings of maximum size or maximum weight in directed graphs. Here we prove the APX-completeness of the problem of finding a maximum size exchange involving only 2-cycles and 3-cycles. We also present an approximation algorithm and an exact algorithm for the problem of finding a maximum weight exchange involving cycles of bounded length. The exact algorithm has been used to provide optimal solutions to real kidney exchange problems arising from the National Matching Scheme for Paired Donation run by NHS Blood and Transplant, and we describe practical experience based on this collaboration

Sufficient conditions for the existence of specified subgraphs in graphs

A classical problem in combinatorics is, given graphs G and H, to determine if H is a subgraph of G. It is usually computationally complex to determine if H is a subgraph of G. Therefore, we often prove conditions that are sufficient to guarantee that a graph G contains H as a subgraph. In Chapter 2, we consider a theorem of Dirac and Erdős from 1963 that considers when a graph contains many disjoint cycles. Generalizing the seminal result of Corrádi and Hajnal, they prove that if a graph G contains many more vertices of degree at least 2k than vertices of degree at most 2k-2, then G contains k vertex-disjoint cycles. We strengthen their result, proving that if G contains 3k more vertices of high degree than vertices of low degree, then G contains k disjoint cycles and that this bound is sharp. Moreover, when G has many vertices, G is planar, or G contains few triangles, this value can be improved to 2k. The value 2k is the best possible, as shown by examples of Dirac and Erdős. In Chapter 3, we rephrase the problem of subgraphs in the language of graph packing. Two graphs G and G' pack if G is a subgraph of the complement of G' or, equivalently, if G' is a subgraph of the complement of G. Graph packing is a restatement of the subgraph problem that does not require one graph to be specified as the underlying graph and the other as the subgraph. Theorems of Sauer and Spencer and, independently, Bollobás and Eldridge prove that if G and G' together have few edges or if the maximum degree of G and the maximum degree of G' are small, then G and G' pack. We explore two results that combine bounds on the maximum degrees and number of edges in G and G'. Recently, Alon and Yuster proved that if G and G' are graphs on n vertices such that G has a bounded number of edges and G' has bounded degree, then G and G' pack. We characterize the pairs of graphs for which their theorem is sharp. In particular, we show that for sufficiently large n, if the vertex of maximum degree in G can be appropriately placed, then G can contain more edges and still pack with G'. We also consider a conjecture of Żak that states if the sum of the number of edges in G, the number of edges in G', and the degree of the largest vertex in G or G' is bounded above by 3n - 7, then G and G' pack. We prove that, up to an additive constant, this conjecture is correct. Using the notion of list packing, we prove that there is a constant C such that if the same sum is bounded above by 3n - C, then G and G' pack. This improves a theorem of Żak from 2014. Finally, we consider a generalization of finding a matching in a graph. The stable marriage problem was introduced by Gale and Shapley in 1962 and the generalization to multiple dimensions was first mentioned by Knuth in 1976. We consider a generalization of the Stable Marriage problem with s-dimensions and purely cyclic preferences (cyclic s-DSM). In 2004, Boros et al. showed that if there are at most s agents of each gender, then every instance of cyclic s-DSM admits a stable matching. In 2006, Eriksson et al. showed this is also true when s = 3 and there are 4 agents of each gender. We extend their result, proving that when there are s+1 agents of each gender, each instance of s-DSM admits a stable matching. We also provide a minimal example of an instance of s-DSM which admits no strongly stable matching

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view

Matchings and Covers in Hypergraphs

In this thesis, we study three variations of matching and covering problems in hypergraphs. The first is motivated by an old conjecture of Ryser which says that if \mcH is an $r$-uniform, $r$-partite hypergraph which does not have a matching of size at least $\nu +1$, then \mcH has a vertex cover of size at most $(r-1)\nu$. In particular, we examine the extremal hypergraphs for the $r=3$ case of Ryser's conjecture. In 2014, Haxell, Narins, and Szab{\'{o}} characterized these $3$-uniform, tripartite hypergraphs. Their work relies heavily on topological arguments and seems difficult to generalize. We reprove their characterization and significantly reduce the topological dependencies. Our proof starts by using topology to show that every $3$-uniform, tripartite hypergraph has two matchings which interact with each other in a very restricted way. However, the remainder of the proof uses only elementary methods to show how the extremal hypergraphs are built around these two matchings. Our second motivational pillar is Tuza's conjecture from 1984. For graphs $G$ and $H$, let $\nu_{H}(G)$ denote the size of a maximum collection of pairwise edge-disjoint copies of $H$ in $G$ and let $\tau_{H}(G)$ denote the minimum size of a set of edges which meets every copy of $H$ in $G$. The conjecture is relevant to the case where $H=K_{3}$ and says that $\tau_{\triangledown}(G) \leq 2\nu_{\triangledown}(G)$ for every graph $G$. In 1998, Haxell and Kohayakawa proved that if $G$ is a tripartite graph, then $\tau_{\triangledown}(G) \leq 1.956\nu_{\triangledown}(G)$. We use similar techniques plus a topological result to show that $\tau_{\triangledown}(G) \leq 1.87\nu_{\triangledown}(G)$ for all tripartite graphs $G$. We also examine a special subclass of tripartite graphs and use a simple network flow argument to prove that $\tau_{\triangledown}(H) = \nu_{\triangledown}(H)$ for all such graphs $H$. We then look at the problem of packing and covering edge-disjoint $K_{4}$'s. Yuster proved that if a graph $G$ does not have a fractional packing of $K_{4}$'s of size bigger than $\nu_{\boxtimes}^{*}(G)$, then $\tau_{\boxtimes}(G) \leq 4\nu_{\boxtimes}^{*}(G)$. We give a complementary result to Yuster's for $K_{4}$'s: We show that every graph $G$ has a fractional cover of $K_{4}$'s of size at most $\frac{9}{2}\nu_{\boxtimes}(G)$. We also provide upper bounds on $\tau_{\boxtimes}$ for several classes of graphs. Our final topic is a discussion of fractional stable matchings. Tan proved that every graph has a $\frac{1}{2}$-integral stable matching. We consider hypergraphs. There is a natural notion of fractional stable matching for hypergraphs, and we may ask whether an analogous result exists for this setting. We show this is not the case: Using a construction of Chung, F{\"{u}}redi, Garey, and Graham, we prove that, for all n \in \mbN, there is a $3$-uniform hypergraph with preferences with a fractional stable matching that is unique and has denominators of size at least $n$

Approximation Algorithms for Effective Team Formation

This dissertation investigates the problem of creating multiple disjoint teams of maximum efficacy from a fixed set of workers. We identify three parameters which directly correlate to the team effectiveness — team expertise, team cohesion and team size — and propose efficient algorithms for optimizing each in various settings. We show that under standard assumptions the problems we explore are not optimally solvable in polynomial time, and thus we focus on developing efficient algorithms with guaranteed worst case approximation bounds. First, we investigate maximizing team expertise in a setting where each worker has different expertise for each job and each job may be completed only by teams of certain sizes. Second, we consider the problem of maximizing team cohesion when the set of workers form a social network with known pairwise compatibility. Third, we explore the problem from a game theoretic perspective in which multiple teams compete on a fixed number of workers and the true needs of each team are pri- vate. We present allocation algorithms that both incentivize teams to state their needs accurately and allocate workers effectively. Finally, we experimentally measure the correlation between team cohesiveness, team expertise and team efficacy on a social network graph of computer science research co-authorship