GraphCombEx: A Software Tool for Exploration of Combinatorial Optimisation Properties of Large Graphs

We present a prototype of a software tool for exploration of multiple combinatorial optimisation problems in large real-world and synthetic complex networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial Explorer), provides a unified framework for scalable computation and presentation of high-quality suboptimal solutions and bounds for a number of widely studied combinatorial optimisation problems. Efficient representation and applicability to large-scale graphs and complex networks are particularly considered in its design. The problems currently supported include maximum clique, graph colouring, maximum independent set, minimum vertex clique covering, minimum dominating set, as well as the longest simple cycle problem. Suboptimal solutions and intervals for optimal objective values are estimated using scalable heuristics. The tool is designed with extensibility in mind, with the view of further problems and both new fast and high-performance heuristics to be added in the future. GraphCombEx has already been successfully used as a support tool in a number of recent research studies using combinatorial optimisation to analyse complex networks, indicating its promise as a research software tool

Lazy Parameter Tuning and Control:Choosing All Parameters Randomly from a Power-Law Distribution

Most evolutionary algorithms have multiple parameters and their values drastically affect the performance. Due to the often complicated interplay of the parameters, setting these values right for a particular problem (parameter tuning) is a challenging task. This task becomes even more complicated when the optimal parameter values change significantly during the run of the algorithm since then a dynamic parameter choice (parameter control) is necessary. In this work, we propose a lazy but effective solution, namely choosing all parameter values (where this makes sense) in each iteration randomly from a suitably scaled power-law distribution. To demonstrate the effectiveness of this approach, we perform runtime analyses of the $(1+(\lambda,\lambda))$ genetic algorithm with all three parameters chosen in this manner. We show that this algorithm on the one hand can imitate simple hill-climbers like the $(1+1)$ EA, giving the same asymptotic runtime on problems like OneMax, LeadingOnes, or Minimum Spanning Tree. On the other hand, this algorithm is also very efficient on jump functions, where the best static parameters are very different from those necessary to optimize simple problems. We prove a performance guarantee that is comparable, sometimes even better, than the best performance known for static parameters. We complement our theoretical results with a rigorous empirical study confirming what the asymptotic runtime results suggest.Comment: Extended version of the paper accepted to GECCO 2021, including all the proofs omitted in the conference versio

The Optimum Communication Spanning Tree Problem : properties, models and algorithms

For a given cost matrix and a given communication requirement matrix, the OCSTP is defined as finding a spanning tree that minimizes the operational cost of the network. OCST can be used to design of more efficient communication and transportation networks, but appear also, as a subproblem, in hub location and sequence alignment problems. This thesis studies several mixed integer linear optimization formulations of the OCSTP and proposes a new one. Then, an efficient Branch & Cut algorithm derived from the Benders decomposition of one of such formulations is used to successfully solve medium-sized instances of the OCSTP. Additionally, two new combinatorial lower bounds, two new heuristic algorithms and a new family of spanning tree neighborhoods based on the Dandelion Code are presented and tested.Postprint (published version

Topology Network Optimization of Facility Planning and Design Problems

The research attempts to provide a graphical theory-based approach to solve the facility layout problem. Which has generally been approached using Quadratic Assignment Problem (QAP) in the past, an algebraic method. It is a very complex problem and is part of the NP-Hard optimization class, because of the nonlinear quadratic objective function and (0,1) binary variables. The research is divided into three phases which together provide an optimal facility layout, block plan solution consisting of MHS (material handling solution) projected onto the block plan. In phase one, we solve for the position of departments in a facility based on flow and utility factor (weight for location). The position of all the departments is identified on the vertices of MPG (maximal planar graph), which maximizes the possibility of flow. We use named MPG produced in literature, throughout the research. The grouping of the department is achieved through GMAFLAD, a QSP (quadratic set packing) based optimizer. In Phase 2, the dual for the MPG’s is solved consisting of department location as per phase 1, to generate Voronoi graphs. These graphs are then, expanded by an ingenious parameter optimization formulation to achieve area fitting for individual cases. Optimization modeling software, Lingo17.0 is used for solving the parameter optimization for generating coordinates of the block plan. The plotting of coordinates for the block plan graphics is done via Autodesk inventor 2019. In phase 3, the solution for MHS is achieved using an RSMT (Rectilinear Steiner minimal tree) graph approach. The Voronoi seed coordinates produced through phase 2 results are computed by GeoSteiner package to generated the RSMT graph for projection onto the block plan (Also, done by Inventor 2019). The graphical method employed in this research, itself has complex and NP-hard problem segments in it, which have been relaxed to polynomial time complexity by fragmenting into groups and solving them in sections. Solving for MPG & RSMT are a class of NP-Hard problem, which have been restricted to N=32 here. Finally, to validate the research and its methodology a real-life case study of a shipyard building for the data set of PDVSA, Venezuela is performed and verified

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