Some results on triangle partitions

We show that there exist efficient algorithms for the triangle packing problem in colored permutation graphs, complete multipartite graphs, distance-hereditary graphs, k-modular permutation graphs and complements of k-partite graphs (when k is fixed). We show that there is an efficient algorithm for C_4-packing on bipartite permutation graphs and we show that C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite graphs that have a triangle partition

Algorithmic aspects of fixed-size coalition formation

We study algorithmic aspects of models in which a set of agents is to be organised into coalitions of a fixed size. Such models can be viewed as a type of hedonic game, coalition formation game, or multidimensional matching problem. We mostly consider models in which coalitions have size three and are formalisms of Three-Dimensional Roommates (3DR). Models of 3DR are characterised by a combination of the system by which agents have preferences over coalitions, and the solution concept (e.g. stability). Since the computational problems associated with 3DR are typically hard, we consider approximate solutions and restricted cases, with the aim of characterising the boundary between tractable and intractable variants. Part of our contribution relates to two new models of 3DR, which involve two existing systems of preferences called [B- and W-preferences]. In each model, we consider the existence of matchings that are stable. We show that the related decision problems are NP-complete and devise approximation algorithms for corresponding optimisation problems. In a model of 3DR with additively separable preferences, we investigate stable matchings and envy-free matchings, for three successively weaker definitions of envy-freeness. We consider restrictions on the agents’ preferences including symmetric, binary, and ternary valuations. We identify dichotomies based on these restrictions and provide a comprehensive complexity classification. Interestingly, we identify a general trend that, for successively weaker solution concepts, existence and polynomial-time solvability hold under successively weaker preference restrictions. We also consider a variant of 3DR known as Three-Dimensional Stable Matching with Cyclic preferences (3-DSM-CYC), which has been of independent interest. It was recently shown that finding a stable matching is NP-hard, so we consider a related optimisation problem and present an approximation algorithm based on serial dictatorship. We also consider a situation in which the preferences of some agents are sufficiently similar to some master list, and show that the approximation ratio of this algorithm can be improved in relation to a specific similarity measure. Finally, we consider a problem in graph theory that generalises the notion of assigning agents to coalitions of a fixed size. Rather than organising a set of agents, the problem is to find a maximum-cardinality set of r-cliques in an undirected graph subject to that set being either vertex disjoint or edge disjoint, for a fixed integer r ≥ 3. This general problem is known as the Kr- packing problem. Here we study the restriction of this problem in which the graph has a fixed maximum degree ∆. It is known for r = 3 that the vertex-disjoint (edge-disjoint) variant is solvable in linear time if ∆ = 3 (∆ = 4) but APX-hard if ∆ ≥ 4 (∆ ≥ 5). We generalise these results to an arbitrary but fixed r ≥ 3, and provide a full complexity classification for both the vertex- and edge-disjoint variants in graphs of maximum degree ∆, for all r ≥ 3

Generalized Set and Graph Packing Problems

Many complex systems that exist in nature and society can be expressed in terms of networks (e.g., social networks, communication networks, biological networks, Web graph, among others). Usually a node represents an entity while an edge represents an interaction between two entities. A community arises in a network when two or more entities have common interests, e.g., related proteins, industrial sectors, groups of people, documents of a collection. There exist applications that model a community as a fixed graph H [98, 10, 119, 2, 142, 136]. Additionally, it is not expected that an entity of the network belongs to only one community; that is, communities tend to share their members. The community discovering or community detection problem consists on finding all communities in a given network. This problem has been extensively studied from a practical perspective [61, 137, 122, 116]. However, we believe that this problem also brings many interesting theoretical questions. Thus in this thesis, we will address this problem using a more rigorous approach. To that end, we first introduce graph problems that we consider capture well the community discovering problem. These graph problems generalize the classical H-Packing problem [88] in two different ways. In the H-Packing with t-Overlap problem, the goal is to find in a given graph G (the network) at least k subgraphs (the communities) isomorphic to a member of a family of graphs H (the community models) such that each pair of subgraphs overlaps in at most t vertices (the shared members). On the other hand, in the H-Packing with t-Membership problem instead of limiting the pairwise overlap, each vertex of G is contained in at most t subgraphs of the solution. For both problems each member of H has at most r vertices and m edges. An instance of the H-Packing with t-Overlap and t-Membership problems corresponds to an instance of the H-Packing problem for t = 0 and t = 1, respectively. We also restrict the overlap between the edges of the subgraphs in the solution instead of the vertices (called H-Packing with t-Edge Overlap and t-Edge Membership problems). Given the closeness of the r-Set Packing problem [87] to the H-Packing problem, we also consider overlap in the problem of packing disjoint sets of size at most r. As usual for set packing problems, given a collection S drawn from a universe U, we seek a sub-collection S'⊆S consisting of at least k sets subject to certain disjointness restrictions. In the r-Set Packing with t-Membership, each element of U belongs to at most t sets of S' while in the r-Set Packing with t-Overlap each pair of sets in S' overlaps in at most t elements. For both problems, each set of S has at most r elements. We refer to all the problems introduced in this thesis simply as packing problems with overlap. Also, we group as the family of t-Overlap problems: H-Packing with t-Overlap, H-Packing with t-Edge Overlap, and r-Set Packing with t-Overlap. While we call the family of t-Membership problems: H-Packing with t-Membership, H-Packing with t-Edge Membership, and r-Set Packing with t-Membership. The classical H-Packing and r-Set Packing problems are NP-complete [87, 88]. We will show in this thesis that allowing overlap in a packing does not make the problems "easier". More precisely, we show that the H-Packing with t-Membership and the r-Set Packing with t-Membership are NP-complete when H = {H'} and H' is an arbitrary connected graph with at least three vertices and r≥3, respectively. Parameterized complexity, introduced by Downey and Fellows [44], is an exciting and interesting approach to deal with NP-complete problems. The underlying idea of this approach is to isolate some aspects or parts of the input (known as the parameters) to investigate whether these parameters make the problem tractable or intractable. The main goal of this thesis is to study the parameterized complexity of our packing problems with overlap. We set up as a parameter k the size of the solution (number of communities), and we consider as fixed-constants r, m and t. We show that our problems admit polynomial kernels via two types of techniques: polynomial parametric transformations (PPTs) [16] and classical reduction algorithms [43]. PPTs are mainly used to show lower bounds and as far as we know they have not been used as extensively to obtain kernel results as classical kernelization techniques [96, 42]. Thus, we believe that employing PPTs is a promising approach to obtain kernel reductions for other problems as well. On the other hand, with non-trivial generalizations of kernelization algorithms for the classical H-Packing problem [114], we are able to improve our kernel sizes obtained via PPTs. These improved kernel sizes are equivalent to the kernel sizes for the disjoint version when t = 0 and t = 1 for the t-Overlap and t-Membership problems, respectively. We also obtain fixed-parameter algorithms for our packing problems with overlap (other than running brute force on the kernel). Our algorithms combine a search tree and a greedy localization technique and generalize a fixed-parameter algorithm for the problem of packing disjoint triangles [54]. In addition, we obtain faster FPT-algorithms by transforming our overlapping problems into an instance of the disjoint version of our problems. Finally, we introduce the Π-Packing with α()-Overlap problem to allow for more complex overlap constraints than the ones considered by the t-Overlap and t-Membership problems and also to include more general communities definitions. This problem seeks at least k induced subgraphs in a graph G subject to: each subgraph has at most r vertices and obeys a property Π (a community definition) and for any pair of subgraphs Hi,Hj, with i≠j, we have that α(Hi,Hj) = 0 holds (an overlap constraint). We show that the Π-Packing with α()-Overlap problem is fixed-parameter tractable provided that Π is computable in polynomial time in n and α() obeys some natural conditions. Motivated by practical applications we give several examples of α() functions which meet those conditions

Engineering Graph Clustering Algorithms

Networks in the sense of objects that are related to each other are ubiquitous. In many areas, groups of objects that are particularly densely connected, so called clusters, are semantically interesting. In this thesis, we investigate two different approaches to partition the vertices of a network into clusters. The first quantifies the goodness of a clustering according to the sparsity of the cuts induced by the clusters, whereas the second is based on the recently proposed measure surprise