Browsing by Author "Hicks, Illya V"
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Item A Branch and Cut Approach to the Feedback Vertex Set Problem(2018-11-29) Bell, Nick; Hicks, Illya VIn this thesis, I use a branch and cut implementation to solve the feedback vertex set problem and add new facet defining inequalities to the literature. Feedback in a system arises when repeated traversal over some cycle causes unwanted growth or decay in the data. This can cause problems in many systems such as excess noise at a concert or deadlock forming cyclical dependence on processes in a group. To eliminate the feedback, I implement a branch and cut algorithm to make the calculations, selecting inequalities based on results from the literature. I report improvements upon previous branch and cut work in a collection of random cases. Along with this implementation, I show that the clique inequalities, which are known to be valid for the feedback vertex problem, are in fact facet inducing for all maximal clique subgraphs of the original network.Item A Combinatorial Disjunctive Constraint Approach to Optimal Footstep Planning(2023-04-07) Garcia, Raul; Hicks, Illya VWalking robots have many uses in manufacturing, agriculture, medical practice, prosthesis research, emergency relief, military operations, space exploration, service, and entertainment. These include industrial inspection, maintenance, personal assistance, surveillance, reconnaissance, delivery, and human interactions. In many of these scenarios, an autonomous agent must find a path through a cluttered environment to safely arrive at a destination, and such a path should in some sense be optimal. However, its workspace is generally non-convex and needs to be efficiently described in the context of an optimization problem. For this reason, we leverage the independent branching scheme to construct small, ideal formulations for constraints that ensure our robot travels only through obstacle-free polyhedral regions, which traditionally have been formulated with (non-ideal) big-M techniques. As our approach requires a biclique cover for an associated graph, we exploit the structure of this class of graphs to develop a fast subroutine for obtaining biclique covers in polynomial time. While experiments have shown the big-M approach to outperform our method on most of our test instances, there are numerous directions for future work which may improve our resolution time as well as determine classes of scenarios where our formulation may be more favorable.Item A Spectral Decomposition Heuristic for Near Optimal Capture Sets In Consensus Models(2016-08-31) Mikesell, Derek Justin; Hicks, Illya VGiven a network G=(V,E), consider the problem of selecting a subset of nodes, A, of a fixed size, k, such that the sum expected walk length from V to A, or hitting time, is minimized. This study is motivated by modeling communication models as a random walk on a weighted loopless directed graph where the desired information is observed when a random walk reaches the chosen set. The origin of this problem is found in the study of how information or ``consensus" flows through a network, introduced by Borkar et al. in 2010. In general, this problem is NP-hard and as a result problems posed on large networks become quickly infeasible. The objective function of interest F(A) is supermodular and therefore, a greedy technique provides a ( 1 - 1/e) approximation to the optimal solution. While the greedy technique provides a guaranteed approximation it comes at the cost of being O(k n^4). This work develops the Spectral Decomposition heuristic for this problem based on spectral clustering. In general, this reduces the model to O(n^4/k^3). Analysis of this technique is completed on the Stochastic Block Model, and under the appropriate assumptions the method reduces to O(n^2.5). This approach is compared to previous approaches, including the greedy method, centrality measures, and more recent near-optimal subset techniques. The Spectral Decomposition heuristic greatly improves on the run-time of the greedy and near-optimal subset method, while maintaining quality of approximation. While the method cannot compete with the complexity of the centrality measures, frequently it greatly outperforms the approximation. Optimality is observed through two metrics; the value of the objective function and the value of the Perron-Frobenius eigenvalue of the Markov submatrix resulting from the capture set. Multiple examples illustrate the method and larger than prior networks are explored.Item Algorithms to Find the Girth and Cogirth of a Linear Matroid(2014-09-18) Arellano, John David; Hicks, Illya V; Tapia, Richard A; Yin, Wotao; Baraniuk, Richard GIn this thesis, I present algorithms to find the cogirth and girth, the cardinality of the smallest cocircuit and circuit respectively, of a linear matroid. A set covering problem (SCP) formulation of the problems is presented. The solution to the linear matroid cogirth problem provides the degree of redundancy of the corresponding sensor network, and allows for the evaluation of the quality of the network. Hence, addressing the linear matroid cogirth problem can lead to significantly enhancing the design process of sensor networks. The linear matroid girth problem is related to reconstructing a signal in compressive sensing. I provide an introduction to matroids and their relation to the degree of redundancy problem as well as compressive sensing. I also provide an overview of the methods used to address linear matroid cogirth/girth problems, the SCP, and reconstructing a signal in compressive sensing. Computational results are provided to validate a branch-and-cut algorithm that addresses the SCP formulation as well as an algorithm which uses branch decompositions and dynamic programming to find the girth of a linear matroid.Item Bilevel Clique Interdiction and Related Problems(2017-04-18) Becker, Timothy; Hicks, Illya VI introduce a formulation of the bilevel clique interdiction problem. Interdiction, a military term, describes the removal of enemy resources. The single level clique interdiction problem describes the attempt of an attacker to interdict a maximum number of cliques. The bilevel form of the problem introduces a defender who attempts to minimize the number of cliques interdicted by the attacker. An algorithm and formulation for the bilevel clique interdiction problem has not previously been investigated. I start by introducing a formulation and a column-generation algorithm to solve the problem of bilevel interdiction of a minimum clique transversal and move forward to the creation of a delayed row-and-column generation algorithm for bilevel clique interdiction. Next, I introduce a formulation and algorithm to solve the bilevel interdiction of a maximum stable set problem. Bilevel interdiction of a maximum stable set is choosing a maximum stable set, but with a defender who is attempting to minimize the maximum stable set that can be chosen by the interdictor. I introduce a deterministic formulation and a delayed column generation algorithm. Additionally, I introduce a stochastic formulation of the problem. I solve this problem using a cross-decomposition method that involves L-shaped cuts into a master problem as well as new ``clique" cuts for the inner problem. Lastly, I define new classes of valid inequalities and facets for the clique transversal polytope. The valid inequalities come from two graph structures who have a closed form for their vertex cover number, which we use as a specific case for finding a minimum clique transversal. The first class of facets are just the maximal clique constraints of the clique transversal polytope. The next class contains an odd hole with distinct cliques on each edge of the hole. Another similar class contains an odd clique with distinct maximal cliques on the edges of one of its spanning cycles. The fourth class contains a clique with distinct maximal cliques on every edge of the initial clique, while the last class is a prism graph with distinct maximal cliques on every edge of the prism.Item Bounding the Forcing Number of a Graph(2015-04-16) Davila, Randy R; Hicks, Illya V; Tapia, Richard A; Zhang, YinThe forcing number, denoted F(G), is an upper bound for the maximum nullity of all symmetric matrices with a sparsity pattern described by the simple graph G. Simple lower and upper bounds are δ ≤ F(G) where δ is the minimum degree and F (G) ≤ n − 1 where n is the order of the graph. This thesis provides improvements on the minimum degree lower bound in the case that G has girth of at least 5. In particular, it is shown that 2δ − 2 ≤ F (G) for graphs with girth of at least 5; this can be further improved when G has a small cut set. Further, this thesis also conjectures a lower bound on F(G) as a function of the girth, g, and δ.Item Clique Generalizations and Related Problems(2016-04-21) Wood, Cynthia Ivette; Hicks, Illya VA large number of real-world problems can be model as optimization problems in graphs. The clique model was introduced to aid the study of network structure for social interaction. Each vertex represented an actor and the edges represented the relations between them. Nevertheless, the model has been shown to be restrictive for modeling real-world problems, since it leaves out subgraphs that do not have all pos- sible edges. As a consequence, clique generalizations were introduced to overcome the disadvantages of the clique model. In this thesis, I present three computationally dif- ficult combinatorial optimization problems related to clique generalization problems: co-2-plexes and k-cores. A k-core is a subgraph with minimum degree greater than or equal to k. In this work, I discuss the minimal k-core problem and the minimum k-core problem. I present a backtracking algorithm to find all minimal k-cores of a given undirected graph and its applications to the study of associative memory. The proposed method is a modification of the Bron and Kerbosch algorithm for finding all cliques of an undirected graph. In addition, I study the polyhedral structure of the k-core polytope. The minimum k-core problem is modeled as a binary integer program and relaxed as a linear program. Since the relaxation yields to a non-integral solution, cuts must be added in order to improve the solution. I show that edge and cycle transversals of the graph give valid inequalities for the convex hull of k-cores. A set of pairwise non-adjacent vertices defines a stable set. A stable set is the complement of a clique. A co-2-plex is a subgraph with degree less than or equal to one, and it is a stable set relaxation. I introduce a study of the maximum weighted co-2-plex (MWC2P) problem for {claw, bull}-free graphs and present two polynomial time algorithms to solve it. One of the algorithms transforms the original graph to solve an instance of the maximum weighted stable set problem utilizing Minty’s algorithm. The second algorithm is an extension of Minty’s algorithm and solves the problem in the original graph. All the algorithms discussed in this thesis were implemented and tested. Numerical results are provided for each one of them.Item Clique Relaxations & the Minority Districting Problem(2024-04-16) Kroger, Samuel; Hicks, Illya VThis thesis studies the intersection of graph theory and mixed integer programming through three combinatorial optimization problems. We model each problem on graphs and exploit the inherent structure of the problem to propose novel integer programming formulations, create fixing procedures, and add valid cuts to the problem. We demonstrate the theoretical and computational improvement for each problem through propositions and computational experiments. The Anchored k-core Problem is a variant of the maximum k-core problem, which itself is a relaxation of a clique. We propose an integer programming formulation for the anchored k-core problem, define and study the polyhedra of the maximum anchored k-core problem, and provide a computational study comparing our model against leading algorithms in the literature. Next, we cover the Maximum Stable Set Problem, which is closely related to the maximum clique problem. We extend a polynomial time algorithm for solving the maximum stable set problem on chordal graphs into a polynomial time fixing procedure for general graphs. We also discover a small class of graphs for which the maximum stable set problem is polynomial-time solvable using our proposed algorithm. Finally, we cover the Minority Districting Problem. In this problem, we hope to identify states with a legal impetus, imposed by Section 2 of the Voting Rights Act,to form minority-majority districts. Identifying and enacting plans with minority districts is paramount to ensure minority populations in America are given the political representation they are constitutionally entitled to. We use a diameter-based metric to enforce compactness based on s-clubs (another clique relaxation). We propose a new mixed integer programming formulation alongside robust fixing procedures, symmetry-breaking constraints, and a framework for finding the maximum number of minority districts possible for a state with a diameter-bounded compactness measure.Item Efficient Computation of Chromatic and Flow Polynomials(2015-10-13) Brimkov, Boris; Hicks, Illya VThis thesis surveys chromatic and flow polynomials, and presents new efficient methods to compute these polynomials on specific families of graphs. The chromatic and flow polynomials of a graph count the number of ways to color and assign flow to the graph; they also contain other important information such as the graph's chromatic number, Hamiltonicity, and number of acyclic orientations. Unfortunately, these graph polynomials are generally difficult to compute; thus, research in this area often focuses on exploiting the structure of specific families of graphs in order to characterize their chromatic and flow polynomials. In this thesis, I present closed formulas and polynomial-time algorithms for computing the chromatic polynomials of novel generalizations of trees, cliques, and cycles; I also use graph duality to compute the flow polynomials of outerplanar graphs and generalized wheel graphs. The proposed methods are validated by computational results.Item Fort Neighborhoods: A Set Cover Formulation for Power Domination(2018-11-14) Smith, Logan; Hicks, Illya VThis thesis introduces a novel separation algorithm for calculating power domination numbers and minimum power dominating sets in graphs. Additionally, it shows how the existence of solutions of special forms can be exploited by computational methods. Power domination studies arise from a key problem in electrical engineering concerning placements of monitoring devices known as Phasor Measurement Units (PMUs). PMUs are installed in electrical networks for early detection of electrical imbalances, enabling corrective actions to mitigate outages. In graph representations of electrical networks, power dominating sets denote locations at which PMUs can be placed to monitor entire networks. Due to PMU installation costs of up to $200,000, it is desirable to identify minimum power dominating sets and their cardinality, known is the graph’s power domination number. Unlike prior methods, this work exploits crucial yet previously neglected graph structures: the neighborhoods of zero forcing forts. Utilizing these structures, algorithms are devised for calculating minimum power dominating sets and the power domination numbers of graphs. Computational experiments demonstrating an order of magnitude improvement over previous methods are presented.Item Graph Coloring, Zero Forcing, and Related Problems(2017-08-09) Brimkov, Boris; Hicks, Illya VThis thesis investigates several problems related to classical and dynamic coloring of graphs, and enumeration of graph attributes. In the first part of the thesis, I present new efficient methods to compute the chromatic and flow polynomials of specific families of graphs. The chromatic and flow polynomials count the number of ways to color and assign flow to the graph, and encode information relevant to various physical applications. The second part of the thesis focuses on zero forcing - a dynamic graph coloring process whereby at each discrete time step, a colored vertex with a single uncolored neighbor forces that neighbor to become colored. Zero forcing has applications in linear algebra, quantum control, and power network monitoring. A connected forcing set is a connected set of initially colored vertices which forces the entire graph to become colored; the connected forcing number is the cardinality of the smallest connected forcing set. I present a variety of structural results about connected forcing, such as the effects of vertex and edge operations on the connected forcing number, the relations between the connected forcing number and other graph parameters, and the computational complexity of connected forcing. I also give efficient algorithms for computing the connected forcing numbers of different families of graphs, and characterize the graphs with extremal connected forcing numbers. Finally, I investigate several enumeration problems associated with zero forcing, such as the exponential growth of certain families of forcing sets, relations of families of forcing sets to matroids and greedoids, and polynomials which count the number of distinct forcing sets of a given size.Item Monitoring on Graphs: An Exploration into k-Cores, Zero Forcing, and Power Domination(2019-10-16) Mikesell, Derek Justin; Hicks, Illya VThis thesis will cover various computational models for approximating and solving multiple graph monitoring problems. The first problem of interest is the Minimum k-core problem, which asks for the smallest induced subgraph of minimum degree k. It has been shown that this problem is difficult and thus sophisticated techniques are required to obtain good solutions. The minimum k-core problem is modeled as a binary integer program and relaxed as a linear program. Since the relaxation may yield a non-integral solution, a Branch-and-Cut framework is used to find the integral optimal solution. It is shown that the edge and cycle transversals of the graph give valid inequalities for the convex hull of the k-core polytope - which can be further generalized to a family of l-core transversals. Further, a heuristic for the hitting set of the minimal l-cores is given with its associated valid inequality. Additionally, improved valid inequalities are generated using bounds generated via the girth of the graph. Multiple low order heuristics are explored for finding initial bounds for the branching process utilizing the degree distribution of the graph. Finally, numerical results are given comparing the Branch-and-Bound, Branch-and-Cut, and heuristic techniques. The next problem that is studied is the Zero Forcing Problem. Zero forcing is a graph coloring process based on the following color change rule: all vertices of a graph G are initially colored either blue or white; in each time-step, a white vertex turns blue if it is the only white neighbor of some blue vertex. A zero forcing set of G is a set of blue vertices such that all vertices eventually become blue after iteratively applying the color change rule. The zero forcing number Z(G) is the cardinality of a minimum zero forcing set. This thesis proposes novel exact algorithms for computing Z(G) based on formulating the zero forcing problem as a two-stage Boolean satisfiability problem with clause generation. Several heuristics are proposed for zero forcing based on iteratively adding blue vertices which color a large part of the remaining white vertices. These heuristics are used to speed up the exact algorithms, and can also be of independent interest in approximating Z(G). Computational results on various types of graphs show that in many cases, the algorithms presented offer a significant improvement on the state-of-the-art algorithms for zero forcing. Lastly, the Connected Power Domination problem is studied. The study of power domination in graphs arises from the problem of placing a minimum number of measurement devices in an electrical network, while monitoring the entire network. A power dominating set of a graph is a set of vertices from which every vertex in the graph can be observed, following a set of rules for power system monitoring. This thesis studies the problem of finding a minimum power dominating set which is connected; the cardinality of such a set is called the connected power domination number of the graph. It is shown that the connected power domination number of a graph is NP-hard to compute in general, but can be computed in linear time in cactus graphs and block graphs. Various structural results about connected power domination are given. Finally, novel integer programming formulations for power domination, connected power domination, and power propagation time are presented, with their respective computational results showing they perform better than other integer programming models in the literature.Item Novel Techniques for the Zero-Forcing and p-Median Graph Location Problems(2017-03-31) Fast, Caleb C; Hicks, Illya VThis thesis presents new methods for solving two graph location problems, the p-Median problem and the zero-forcing problem. For the p-median problem, I present a branch decomposition based method that finds the best p-median solution that is limited to some input support graph. The algorithm can be used to either find an integral solution from a fractional linear programming solution, or it can be used to improve on the solutions given by a pool of heuristics. In either use, the algorithm compares favorably in running time or solution quality to state-of-the-art heuristics. For the zero-forcing problem, this thesis gives both theoretical and computational results. In the theoretical section, I show that the branchwidth of a graph is a lower bound on its zero-forcing number, and I introduce new bounds on the zero-forcing iteration index for cubic graphs. This thesis also introduces a special type of graph structure, a zero-forcing fort, that provides a powerful tool for the analysis and modeling of zero-forcing problems. In the computational section, I introduce multiple integer programming models for finding minimum zero-forcing sets and integer programming and combinatorial branch and bound methods for finding minimum connected zero-forcing sets. While the integer programming methods do not perform better than the best combinatorial method for the basic zero-forcing problem, they are easily adapted to the connected zero-forcing problem, and they are the best methods for the connected zero-forcing problem.Item The Maximum Anchored k-core Problem: Mixed Integer Programming Formulations(2022-04-14) Kroger, Samuel; Hicks, Illya VThe maximum anchored k-core problem plays an important role in marketing, network architecture, and social media; the problem allows network designers and influencers to find the most pivotal vertices which increase the size of the network. In this thesis, we investigate two mixed integer programming (MIP) formulations for the maximum anchored k-core problem: (i) a naive model and (ii) a strong model. We examine the MIP formulations analytically and computationally. We also compare the computational performance of the MIP models with two existing heuristic algorithms: Residual Core Maximization (RCM) and Onion-Layer based Anchored k-core (OLAK). Furthermore, we propose valid inequalities and fixing procedures to improve the computational performance of the MIP models. Finally, we conduct experiments on a set of benchmark instances. Our computational experiments show the superiority of the strong model against the naive model, and the heuristics.