A Tutorial on Clique Problems in Communications and Signal Processing

Since its first use by Euler on the problem of the seven bridges of K\"onigsberg, graph theory has shown excellent abilities in solving and unveiling the properties of multiple discrete optimization problems. The study of the structure of some integer programs reveals equivalence with graph theory problems making a large body of the literature readily available for solving and characterizing the complexity of these problems. This tutorial presents a framework for utilizing a particular graph theory problem, known as the clique problem, for solving communications and signal processing problems. In particular, the paper aims to illustrate the structural properties of integer programs that can be formulated as clique problems through multiple examples in communications and signal processing. To that end, the first part of the tutorial provides various optimal and heuristic solutions for the maximum clique, maximum weight clique, and $k$-clique problems. The tutorial, further, illustrates the use of the clique formulation through numerous contemporary examples in communications and signal processing, mainly in maximum access for non-orthogonal multiple access networks, throughput maximization using index and instantly decodable network coding, collision-free radio frequency identification networks, and resource allocation in cloud-radio access networks. Finally, the tutorial sheds light on the recent advances of such applications, and provides technical insights on ways of dealing with mixed discrete-continuous optimization problems

Low-Diameter Clusters in Network Analysis

In this dissertation, we introduce several novel tools for cluster-based analysis of complex systems and design solution approaches to solve the corresponding optimization problems. Cluster-based analysis is a subfield of network analysis which utilizes a graph representation of a system to yield meaningful insight into the system structure and functions. Clusters with low diameter are commonly used to characterize cohesive groups in applications for which easy reachability between group members is of high importance. Low-diameter clusters can be mathematically formalized using a clique and an s-club (with relatively small values of s), two concepts from graph theory. A clique is a subset of vertices adjacent to each other and an s-club is a subset of vertices inducing a subgraph with a diameter of at most s. A clique is actually a special case of an s-club with s = 1, hence, having the shortest possible diameter. Two topics of this dissertation focus on graphs prone to uncertainty and disruptions, and introduce several extensions of low-diameter models. First, we introduce a robust clique model in graphs where edges may fail with a certain probability and robustness is enforced using appropriate risk measures. With regard to its ability to capture underlying system uncertainties, finding the largest robust clique is a better alternative to the problem of finding the largest clique. Moreover, it is also a hard combinatorial optimization problem, requiring some effective solution techniques. To this aim, we design several heuristic approaches for detection of large robust cliques and compare their performance. Next, we consider graphs for which uncertainty is not explicitly defined, studying connectivity properties of 2-clubs. We notice that a 2-club can be very vulnerable to disruptions, so we enhance it by reinforcing additional requirements on connectivity and introduce a biconnected 2-club concept. Additionally, we look at the weak 2-club counterpart which we call a fragile 2-club (defined as a 2-club that is not biconnected). The size of the largest biconnected 2-club in a graph can help measure overall system reachability and connectivity, whereas the largest fragile 2-club can identify vulnerable parts of the graph. We show that the problem of finding the largest fragile 2-club is polynomially solvable whereas the problem of finding the largest biconnected 2-club is NP-hard. Furthermore, for the former, we design a polynomial time algorithm and for the latter - combinatorial branch-and-bound and branch-and-cut algorithms. Lastly, we once again consider the s-club concept but shift our focus from finding the largest s-club in a graph to the problem of partitioning the graph into the smallest number of non-overlapping s-clubs. This problem cannot only be applied to derive communities in the graph, but also to reduce the size of the graph and derive its hierarchical structure. The problem of finding the minimum s-club partitioning is a hard combinatorial optimization problem with proven complexity results and is also very hard to solve in practice. We design a branch-and-bound combinatorial optimization algorithm and test it on the problem of minimum 2-club partitioning

A min-flow algorithm for Minimal Critical Set detection in Resource Constrained Project Scheduling

AbstractWe propose a min-flow algorithm for detecting Minimal Critical Sets (MCS) in Resource Constrained Project Scheduling Problems (RCPSP). The MCS detection is a fundamental step in the Precedence Constraint Posting method (PCP), one of the most successful approaches for the RCPSP. The proposed approach is considerably simpler compared to existing flow based MCS detection procedures and has better scalability compared to enumeration- and envelope-based ones, while still providing good quality Critical Sets. The method is suitable for problem variants with generalized precedence relations or uncertain/variable durations

Where Graph Topology Matters: The Robust Subgraph Problem

Robustness is a critical measure of the resilience of large networked systems, such as transportation and communication networks. Most prior works focus on the global robustness of a given graph at large, e.g., by measuring its overall vulnerability to external attacks or random failures. In this paper, we turn attention to local robustness and pose a novel problem in the lines of subgraph mining: given a large graph, how can we find its most robust local subgraph (RLS)? We define a robust subgraph as a subset of nodes with high communicability among them, and formulate the RLS-PROBLEM of finding a subgraph of given size with maximum robustness in the host graph. Our formulation is related to the recently proposed general framework for the densest subgraph problem, however differs from it substantially in that besides the number of edges in the subgraph, robustness also concerns with the placement of edges, i.e., the subgraph topology. We show that the RLS-PROBLEM is NP-hard and propose two heuristic algorithms based on top-down and bottom-up search strategies. Further, we present modifications of our algorithms to handle three practical variants of the RLS-PROBLEM. Experiments on synthetic and real-world graphs demonstrate that we find subgraphs with larger robustness than the densest subgraphs even at lower densities, suggesting that the existing approaches are not suitable for the new problem setting.Comment: 13 pages, 10 Figures, 3 Tables, to appear at SDM 2015 (9 pages only

Link communities reveal multiscale complexity in networks

Networks have become a key approach to understanding systems of interacting objects, unifying the study of diverse phenomena including biological organisms and human society. One crucial step when studying the structure and dynamics of networks is to identify communities: groups of related nodes that correspond to functional subunits such as protein complexes or social spheres. Communities in networks often overlap such that nodes simultaneously belong to several groups. Meanwhile, many networks are known to possess hierarchical organization, where communities are recursively grouped into a hierarchical structure. However, the fact that many real networks have communities with pervasive overlap, where each and every node belongs to more than one group, has the consequence that a global hierarchy of nodes cannot capture the relationships between overlapping groups. Here we reinvent communities as groups of links rather than nodes and show that this unorthodox approach successfully reconciles the antagonistic organizing principles of overlapping communities and hierarchy. In contrast to the existing literature, which has entirely focused on grouping nodes, link communities naturally incorporate overlap while revealing hierarchical organization. We find relevant link communities in many networks, including major biological networks such as protein-protein interaction and metabolic networks, and show that a large social network contains hierarchically organized community structures spanning inner-city to regional scales while maintaining pervasive overlap. Our results imply that link communities are fundamental building blocks that reveal overlap and hierarchical organization in networks to be two aspects of the same phenomenon.Comment: Main text and supplementary informatio