Optimization-Based Network Analysis with Applications in Clustering and Data Mining

In this research we develop theoretical foundations and efficient solution methods for two classes of cluster-detection problems from optimization point of view. In particular, the s-club model and the biclique model are considered due to various application areas. An analytical review of the optimization problems is followed by theoretical results and algorithmic solution methods developed in this research. The maximum s-club problem has applications in graph-based data mining and robust network design where high reachability is often considered a critical property. Massive size of real-life instances makes it necessary to devise a scalable solution method for practical purposes. Moreover, lack of heredity property in s-clubs imposes challenges in the design of optimization algorithms. Motivated by these properties, a sufficient condition for checking maximality, by inclusion, of a given s-club is proposed. The sufficient condition can be employed in the design of optimization algorithms to reduce the computational effort. A variable neighborhood search algorithm is proposed for the maximum s-club problem to facilitate the solution of large instances with reasonable computational effort. In addition, a hybrid exact algorithm has been developed for the problem. Inspired by wide usability of bipartite graphs in modeling and data mining, we consider three classes of the maximum biclique problem. Specifically, the maximum edge biclique, the maximum vertex biclique and the maximum balanced biclique problems are considered. Asymptotic lower and upper bounds on the size of these structures in uniform random graphs are developed. These bounds are insightful in understanding the evolution and growth rate of bicliques in large-scale graphs. To overcome the computational difficulty of solving large instances, a scale-reduction technique for the maximum vertex and maximum edge biclique problems, in general graphs, is proposed. The procedure shrinks the underlying network, by confirming and removing edges that cannot be in the optimal solution, thus enabling the exact solution methods to solve large-scale sparse instances to optimality. Also, a combinatorial branch-and-bound algorithm is developed that best suits to solve dense instances where scale-reduction method might be less effective. Proposed algorithms are flexible and, with small modifications, can solve the weighted versions of the problems

On the Rectangles Induced by Points

\newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\rad}{r} \newcommand{\Mh}[1]{#1} \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\polylog}{\mathrm{polylog}} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}[2]{\left\| {#1} - {#2} \right\|} \newcommand{\ptq}{q} \newcommand{\pts}{s} A set $P$ of $n$ points in the plane, induces a set of Delaunay-type axis-parallel rectangles $\mathcal{R}$, potentially of quadratic size, where an axis-parallel rectangle is in $\mathcal{R}$, if it has two points of $P$ as corners, and no other point of $P$ in it. We study various algorithmic problems related to this set of rectangles, including how to compute it, in near linear time, and handle various algorithmic tasks on it, such as computing its union and depth. The set of rectangles $\mathcal{R}$ induces the rectangle influence graph $G = (P,\mathcal{R})$, which we also study. Potentially our most interesting result is showing that this graph can be described as the union of $O(n)$ bicliques, where the total weight of the bicliques is $O(n \log^2 n)$. Here, the weight of a bicliques is the cardinality of its vertices

A new polyhedral approach to combinatorial designs

We consider combinatorial t-design problems as discrete optimization problems. Our motivation is that only a few studies have been done on the use of exact optimization techniques in designs, and that classical methods in design theory have still left many open existence questions. Roughly defined, t-designs are pairs of discrete sets that are related following some strict properties of size, balance, and replication. These highly structured relationships provide optimal solutions to a variety of problems in computer science like error-correcting codes, secure communications, network interconnection, design of hardware; and are applicable to other areas like statistics, scheduling, games, among others. We give a new approach to combinatorial t-designs that is useful in constructing t-designs by polyhedral methods. The first contribution of our work is a new result of equivalence of t-design problems with a graph theory problem. This equivalence leads to a novel integer programming formulation for t-designs, which we call GDP. We analyze the polyhedral properties of GDP and conclude, among other results, the associated polyhedron dimension. We generate new classes of valid inequalities to aim at approximating this integer program by a linear program that has the same optimal solution. Some new classes of valid inequalities are generated as Chv´atal-Gomory cuts, other classes are generated by graph complements and combinatorial arguments, and others are generated by the use of incidence substructures in a t-design. In particular, we found a class of valid inequalities that we call stable-set class that represents an alternative graph equivalence for the problem of finding a t-design. We analyze and give results on the strength of these new classes of valid inequalities. We propose a separation problem and give its integer programming formulation as a maximum (or minimum) edge-weight biclique subgraph problem. We implement a pure cutting-plane algorithm using one of the stronger classes of valid inequalities derived. Several instances of t-designs were solved efficiently by this algorithm at the root node of the search tree. Also, we implement a branch-and-cut algorithm and solve several instances of 2-designs trying different base formulations. Computational results are included

Sum-of-squares representations for copositive matrices and independent sets in graphs

A polynomial optimization problem asks for minimizing a polynomial function (cost) given a set of constraints (rules) represented by polynomial inequalities and equations. Many hard problems in combinatorial optimization and applications in operations research can be naturally encoded as polynomial optimization problems. A common approach for addressing such computationally hard problems is by considering variations of the original problem that give an approximate solution, and that can be solved efficiently. One such approach for attacking hard combinatorial problems and, more generally, polynomial optimization problems, is given by the so-called sum-of-squares approximations. This thesis focuses on studying whether these approximations find the optimal solution of the original problem.We investigate this question in two main settings: 1) Copositive programs and 2) parameters dealing with independent sets in graphs. Among our main new results, we characterize the matrix sizes for which sum-of-squares approximations are able to capture all copositive matrices. In addition, we show finite convergence of the sums-of-squares approximations for maximum independent sets in graphs based on their continuous copositive reformulations. We also study sum-of-squares approximations for parameters asking for maximum balanced independent sets in bipartite graphs. In particular, we find connections with the Lovász theta number and we design eigenvalue bounds for several related parameters when the graphs satisfy some symmetry properties.<br/

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM

On Reconfiguration Problems: Structure and Tractability

Given an n-vertex graph G and two vertices s and t in G, determining whether there exists a path and computing the length of the shortest path between s and t are two of the most fundamental graph problems. In the classical battle of P versus NP or ``easy'' versus ``hard'', both of these problems are on the easy side. That is, they can be solved in poly(n) time, where poly is any polynomial function. But what if our input consisted of a 2^n-vertex graph? Of course, we can no longer assume G to be part of the input, as reading the input alone requires more than poly(n) time. Instead, we are given an oracle encoded using poly(n) bits and that can, in constant or poly(n) time, answer queries of the form ``is u a vertex in G'' or ``is there an edge between u and v?''. Given such an oracle and two vertices of the 2^n-vertex graph, can we still determine if there is a path or compute the length of the shortest path between s and t in poly(n) time? A slightly different, but equally insightful, formulation of the question above is as follows. Given a set S of n objects, consider the graph R(S) which contains one vertex for each set in the power set of S, 2^S, and two vertices are adjacent in R(S) whenever the size of their symmetric difference is equal to one. Clearly, this graph contains 2^n vertices and can be easily encoded in poly(n) bits using the oracle described above. It is not hard to see that there exists a path between any two vertices of R(S). Moreover, computing the length of a shortest path can be accomplished in constant time; it is equal to the size of the symmetric difference of the two underlying sets. If the vertex set of R(S) were instead restricted to a subset of 2^S, both of our problems can become NP-complete or even PSPACE-complete. Therefore, another interesting question is whether we can determine what types of ``restriction'' on the vertex set of R(S) induce such variations in the complexity of the two problems. These two seemingly artificial questions are in fact quite natural and appear in many practical and theoretical problems. In particular, these are exactly the types of questions asked under the reconfiguration framework, the main subject of this thesis. Under the reconfiguration framework, instead of finding a feasible solution to some instance I of a search problem Q, we are interested in structural and algorithmic questions related to the solution space of Q. Naturally, given some adjacency relation A defined over feasible solutions of Q, size of the symmetric difference being one such relation, the solution space can be represented using a graph R_Q(I). R_Q(I) contains one vertex for each feasible solution of Q on instance I and two vertices share an edge whenever their corresponding solutions are adjacent under A. An edge in R_Q(I) corresponds to a reconfiguration step, a walk in R_Q(I) is a sequence of such steps, a reconfiguration sequence, and R_Q(I) is a reconfiguration graph. Studying problems related to reconfiguration graphs has received considerable attention in recent literature, the most popular problem being to determine whether there exists a reconfiguration sequence between two given feasible solutions; for most NP-complete problems, this problem has been shown to be PSPACE-complete. The purpose of our work is to embark on a systematic investigation of the tractability and structural properties of such problems under both classical and parameterized complexity assumptions. Parameterized complexity is another framework which has become an essential tool for researchers in computational complexity during the last two decades or so and one of its main goals is to provide a better explanation of why some hard problems (in a classical sense) can be in fact much easier than others. Hence, we are interested in what separates the tractable instances from the intractable ones and the fixed-parameter tractable instances from the fixed-parameter intractable ones. It is clear from the generic definition of reconfiguration problems that several factors affect their complexity status. Our work aims at providing a finer classification of the complexity of reconfiguration problems with respect to some of these factors, including the definition of the adjacency relation A, structural properties of the input instance I, structural properties of the reconfiguration graph, and the length of a reconfiguration sequence. As most of these factors can be numerically quantified, we believe that the investigation of reconfiguration problems under both parameterized and classical complexity assumptions will help us further understand the boundaries between tractability and intractability. We consider reconfiguration problems related to Satisfiability, Coloring, Dominating Set, Vertex Cover, Independent Set, Feedback Vertex Set, and Odd Cycle Transversal, and provide lower bounds, polynomial-time algorithms, and fixed-parameter tractable algorithms. In doing so, we answer some of the questions left open in recent work and push the known boundaries between tractable and intractable even closer. As a byproduct of our initiating work on parameterized reconfiguration problems, we present a generic adaptation of parameterized complexity techniques which we believe can be used as a starting point for studying almost any such problem