On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem

We study the parameterized complexity of the Bounded-Degree Vertex Deletion problem (BDD), where the aim is to find a maximum induced subgraph whose maximum degree is below a given degree bound. Our focus lies on parameters that measure the structural properties of the input instance. We first show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. We thereby resolve the main open question stated in Betzler, Bredereck, Niedermeier and Uhlmann (2012) concerning the complexity of BDD parameterized by the feedback vertex set number. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to the decompositional parameter treecut width and a novel problem-specific parameter called the core fracture number

On the Parameterized Complexity of Maximum Degree Contraction Problem

In the Maximum Degree Contraction problem, input is a graph G on n vertices, and integers k, d, and the objective is to check whether G can be transformed into a graph of maximum degree at most d, using at most k edge contractions. A simple brute-force algorithm that checks all possible sets of edges for a solution runs in time n^O(k). As our first result, we prove that this algorithm is asymptotically optimal, upto constants in the exponents, under Exponential Time Hypothesis (ETH). Belmonte, Golovach, van't Hof, and Paulusma studied the problem in the realm of Parameterized Complexity and proved, among other things, that it admits an FPT algorithm running in time (d + k)^(2k) ⋅ n^O(1) = 2^O(k log (k+d)) ⋅ n^????(1), and remains NP-hard for every constant d ≥ 2 (Acta Informatica (2014)). We present a different FPT algorithm that runs in time 2^O(dk) ⋅ n^O(1). In particular, our algorithm runs in time 2^O(k) ⋅ n^O(1), for every fixed d. In the same article, the authors asked whether the problem admits a polynomial kernel, when parameterized by k + d. We answer this question in the negative and prove that it does not admit a polynomial compression unless NP ⊆ coNP/poly

Greedy Randomized Adaptive Search Procedure for the Maximum Co-k-plex Problem

The focus of this thesis is a degree based relaxation of independent sets in graphs called co-k-plexes and the related combinatorial optimization problem of finding a maximum cardinality co-k-plex in G. This thesis develops a metaheuristic approach for solving the maximum co-k-plex problem which is known to be NP-hard. The approach is further extended for finding a maximum weighted co-k-plex in G where vertices of G are associated with specific weights. As the maximum co-k-plex problem in G is equivalent to the maximum k-plex problem in complement graph of G, many applications of this problem can be found in clustering and data mining social networks, biological networks, internet graphs and stock market graphs among others. In this thesis, a Greedy Randomized Adaptive Search Procedure (GRASP) is developed to solve the maximum co-k-plex and maximum weighted co-k-plex problems. Computational experiments are performed to study the effectiveness of the proposed metaheuristic on benchmark instances. Finally, the performance of the developed GRASP algorithms for both versions was confirmed by comparing the running time and solution quality with results obtained by an exact algorithm.Industrial Engineering & Managemen

On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem

We study the parameterized complexity of the Bounded-Degree Vertex Deletion problem (BDD), where the aim is to find a maximum induced subgraph whose maximum degree is below a given degree bound. Our focus lies on parameters that measure the structural properties of the input instance. We first show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. We thereby resolve an open question stated in Betzler, Bredereck, Niedermeier and Uhlmann (2012) concerning the complexity of BDD parameterized by the feedback vertex set number. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to the decompositional parameter treecut width and a novel problem-specific parameter called the core fracture number

Characterizing and Detecting Cohesive Subgroups with Applications to Social and Brain Networks

Many complex systems involve entities that interact with each other through various relationships (e.g., people in social systems, neurons in the brain). These entities and interactions are commonly represented using graphs due to several advantages. This dissertation focuses on developing theory and algorithms for novel methods in graph theory and optimization, and their applications to social and brain networks. Specifically, the major contributions of this dissertation are three fold. First, this dissertation aims not only to develop a new clique relaxation model based on a structural metric, clustering coefficient, but also to introduce a novel graph clustering algorithm using this model. Clique relaxations are used in classical models of cohesive subgroups in social network analysis. Clustering coefficient was introduced more recently as a structural feature characterizing small-world networks. Leveraging the similarities between the concepts of cohesive subgroups and small-world networks (i.e., graphs that are highly clustered with small path lengths). The first part of this dissertation introduces a new clique relaxation, α-cluster, defined by enforcing a lower bound α on the clustering coefficient in the corresponding induced subgraph. Two different definitions of the clustering coefficient are considered, namely, the local and global clustering coefficient. Certain structural properties of α-clusters are analyzed, and mathematical optimization models for determining the largest size α-clusters in a network are developed and applied to several real-life social network instances. In addition, a network clustering algorithm based on local α-cluster is introduced and successfully evaluated. Second, this dissertation explores a novel mathematical model called the maximum independent union of cliques problem (max IUC problem), which arises as a special case of α-clusters. It is an interesting problem for which both the maximum clique and maximum independent sets are feasible solutions and individually their corresponding sizes are lower bounds for the size of the IUC solution. After presenting the structural properties as well as the complexity results of different graph types (planar, unit disk graphs and claw-free graphs), an integer programming formulation is developed, followed by a branch-and-bound algorithm and several heuristic methods to approximate the maximum independent union of cliques problem. The developed methods have been empirically evaluated on many benchmark instances. Finally, this dissertation, in collaboration with Texas Institute of Preclinical Studies (TIPS), applies clique relaxation models to explore a new experimental data to understand the effect of concussion on animal brains. Our research involves cohesive and robust clustering analysis of animal brain networks utilizing a unique and novel experimental data. In collaboration with TIPS, we have analyzed multiple pairs of fMRI data about animal brains that are measured before and after a concussion. We utilize network analysis to first identify the similar regions in animal brains, and then compare how these regions as well as graph structural properties change before and after a concussion. To the best of our knowledge, this study is unique in the literature in that it not only explicitly examines the relation between concussion level and the functional unit interaction but also uses very detailed and fine-grained fMRI measurements of brain data

Optimization Methods for Cluster Analysis in Network-based Data Mining

This dissertation focuses on two optimization problems that arise in network-based data mining, concerning identification of basic community structures (clusters) in graphs: the maximum edge weight clique and maximum induced cluster subgraph problems. We propose a continuous quadratic formulation for the maximum edge weight clique problem, and establish the correspondence between its local optima and maximal cliques in the graph. Subsequently, we present a combinatorial branch-and-bound algorithm for this problem that takes advantage of a polynomial-time solvable nonconvex relaxation of the proposed formulation. We also introduce a linear-time-computable analytic upper bound on the clique number of a graph, as well as a new method of upper-bounding the maximum edge weight clique problem, which leads to another exact algorithm for this problem. For the maximum induced cluster subgraph problem, we present the results of a comprehensive polyhedral analysis. We derive several families of facet-defining valid inequalities for the IUC polytope associated with a graph. We also provide a complete description of this polytope for some special classes of graphs. We establish computational complexity of the separation problems for most of the considered families of valid inequalities, and explore the effectiveness of employing the corresponding cutting planes in an integer (linear) programming framework for the maximum induced cluster subgraph problem

The Maximum K-Dependent and F-Dependent Set Problem

this paper we analyze both problems for bipartite graphs, split graphs, cographs, trees and graphs with bounded treewidth. Among others, we show that the decision version of the maximum k-dependent set problem restricted to planar, bipartite graphs is NP-complete for any given k 1: This contrast with the well known fact that the maximum 0-dependent (i.e. independent) set in a bipartite graph can be found in polynomial time by reduction to maximum matching (see [15]) via Konig-Egervary theorem (see [12] and [19]). Next, we give polynomial algorithms for both problems restricted to cographs, trees and graphs with bounded treewidth. On the other hand, we show that the complexity differs for split graphs; we give a polynomial time algorithm for the maximum k-dependent set problem and show the NP-completeness for the maximum f-dependent set problem. Finally, we provide efficient poly-log time PRAM versions of the aforementioned sequential algorithms