Browsing by Author "Hicks, Illya V."
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Item A Branch Decomposition Algorithm for the p-Median Problem(INFORMS, 2017) Fast, Caleb C.; Hicks, Illya V.In this paper, we use a branch decomposition technique to improve approximations to the p-median problem. Starting from a support graph produced either by a combination of heuristics or by linear programming, we use dynamic programming guided by a branch decomposition of that support graph to find the best p-median solution on the support graph. Our results show that when heuristics are used to build the support graph and the support graph has branchwidth at most 7, our algorithm is able to provide a solution of lower cost than any of the heuristic solutions. When linear programming is used to build the support graph and the support graph has branchwidth at most 7, then our algorithm provides better solutions than popular heuristics and is faster than integer programming. Thus, our algorithm is a useful practical tool when support graphs have branchwidth at most 7.Item Beyond Interference Avoidance: Distributed Sun-network Scheduling in Wireless Networks with Local Views(2013-09-16) Santacruz, Pedro; Sabharwal, Ashutosh; Aazhang, Behnaam; Knightly, Edward W.; Hicks, Illya V.In most wireless networks, nodes have only limited local information about the state of the network, which includes connectivity and channel state information. With limited local information about the network, each node’s knowledge is mismatched; therefore, they must make distributed decisions. In this thesis, we pose the following question - if every node has network state information only about a small neighborhood, how and when should nodes choose to transmit? While link scheduling answers the above question for point-to-point physical layers which are designed for an interference-avoidance paradigm, we look for answers in cases when interference can be embraced by advanced code design, as suggested by results in network information theory. To make progress on this challenging problem, we propose two constructive distributed algorithms, one conservative and one aggressive, which achieve rates higher than link scheduling based on interference avoidance, especially if each node knows more than one hop of network state information. Both algorithms schedule sub-networks such that each sub-network can employ advanced interference-embracing coding schemes to achieve higher rates. Our innovation is in the identification, selection and scheduling of sub-networks, especially when sub-networks are larger than a single link. Using normalized sum-rate as the metric of network performance, we prove that the proposed conservative sub-network scheduling algorithm is guaranteed to have performance greater than or equal to pure coloring-based link scheduling. In addition, the proposed aggressive sub-network scheduling algorithm is shown, through simulations, to achieve better normalized sum-rate than the conservative algorithm for several network classes. Our results highlight the advantages of extending the design space of possible scheduling strategies to include those that leverage local network information.Item Biclique Partitions, Biclique Covers, and Disjunctive Constraints(2023-04-20) Lyu, Bochuan; Hicks, Illya V.This thesis focuses on disjunctive constraints and related combinatorial optimization problems: minimum biclique partition problem and minimum biclique cover problem. We introduce techniques to build small ideal mixed-integer programming (MIP) formulations of combinatorial disjunctive constraints (CDCs) via the independent branching scheme. We present a pairwise IB-representable class of CDCs, CDCs admitting junction trees, and provide a combinatorial procedure to build MIP formulations for those constraints. An NP-complete combinatorial optimization problem, the minimum biclique cover problem, needs to be solved in the combinatorial procedure. It motivates us to study minimum biclique partition and biclique cover problems on co-chordal graphs, which is the complementary graph of chordal. We provide heuristics to find biclique partitions on co-chordal graphs and show that the heuristics can provide biclique partitions close to optimal or even optimal if the graphs satisfy further assumptions on the structures. In addition, we provide a tighter bound for minimum biclique covers on a subclass of co-chordal graphs and present a lower bound for minimum biclique covers of general graphs. We also investigate the disjunctive constraints for piecewise linear relaxations of univariate and high-dimensional nonlinear functions that could appear in optimization problems. In order to study those piecewise linear relaxations, we introduce a new class of combinatorial disjunctive constraints: generalized n-dimensional ordered CDCs and present logarithmically sized ideal formulations under the independent branching scheme.Item Branch Decompositions and their Applications(2000-04) Hicks, Illya V.Many real-life problems can be modeled as optimization or decision problems on graphs. Also, many of those real-life problems are NP-hard. One traditional method to solve these problems is by branch and bound while another method is by graph decompositions. In the 1980's, Robertson and Seymour conceived of two new ways to decompose the graph in order to solve these problems. These ingenious ideas were only byproducts of their work proving Wagner's Conjecture. A branch decomposition is one of these ideas. A paper by Arnborg, Lagergren and Seese showed that many NP-complete problems can be solved in polynomial time using divide and conquer techniques on input graphs with bounded branchwidth, but a paper by Seymour and Thomas proved that computing an optimal branch decomposition is also NP-complete. Although computing optimal branch decompositions is NP-complete, there is a plethora of theory about branchwidth and branch decompositions. For example, a paper by Seymour and Thomas offered a polynomial time algorithm to compute the branchwidth and optimal branch decomposition for planar graphs. This doctoral research is concentrated on constructing branch decompositions for graphs and using branch decompositions to solve NP-complete problems modeled on graphs. In particular, a heuristic to compute near-optimal branch decompositions is presented and the heuristic is compared to previous heuristics in the subject. Furthermore, a practical implementation of an algorithm given in a paper by Seymour and Thomas for computing optimal branch decompositions of planar graphs is implemented with the addition of heuristics to give the algorithm a "divide and conquer" design. In addition, this work includes a theoretical result relating the branchwidth of planar graphs to their duals, characterizations of branchwidth for Halin and chordal graphs. Also, this work presents an algorithm for minor containment using a branch decomposition and a parallel implementation of the heuristic for general graphs using pthreads.Item Branch-decomposition heuristics for linear matroids(2010) Ma, Jing; Hicks, Illya V.This thesis present two new heuristics which utilize classification and max-flow algorithm respectively to derive near-optimal branch-decompositions for linear matroids. In the literature, there are already excellent heuristics for graphs, however, no practical branch-decomposition methods for general linear matroids have been addressed yet. Introducing a "measure" which compares the "similarity" of elements of a linear matroid, this work reforms the linear matroid into a similarity graph. Then, two different methods, classification method and max-flow method, both basing on the similarity graph are developed into heuristics. Computational results using the classification method and the max-flow method on linear matroid instances are shown respectively.Item Bridging the Gap between Operations Research and Machine Learning with Decision Trees and Neural Nets(2024-04-19) Alston, Brandon; Hicks, Illya V.This thesis focuses on bridging the overlap between the fields of Operations Research and Machine Learning. We do so by providing efficient Mixed Integer Linear Optimization (MILO) formulations that solve the optimal decision tree, both the univariate and multivariate cases, and binarized neural nets. We provide four MILO formulations for designing optimal binary classification trees: two flow-based formulations and two cut-based formulations. Given that an optimal binary can be obtained by solving a biobjective optimization problem that seeks to (i) maximize the number of correctly classified datapoints and (ii) minimize the number of branching vertices we also are the first to introduce using a biobjective approach that avoids the numerical issues associated with tuning hyperparameters of weighted objective functions. We use a unique fractional separation procedure to speed up our cut-based models given that MILO solvers often employ reductions only to flow-based models. A binarized neural net, which cannot be trained using gradient descent based backpropagation, can be implemented using Boolean operations and is fundamentally a discrete optimization problem. In particular the fixed network structures, discrete edge weights, and our definition of decision variables allow us to transfer some of the techniques used for decision trees into binarized neural nets. We efficiently model the non-linear properties of neural nets by choosing activation factions to be sign(X). We provide computational results of our proposed models against benchmark methods from the literature.Item Cocircuits of vector matroids(2012) Arellano, John David; Hicks, Illya V.In this thesis, I present a set covering problem (SCP) formulation of the matroid cogirth problem, finding the cardinality of the smallest cocircuit of a matroid. Addressing the matroid cogirth problem can lead to significantly enhancing the design process of sensor networks. The solution to the matroid cogirth problem provides the degree of redundancy of the corresponding sensor network, and allows for the evaluation of the quality of the network. I provide an introduction to matroids, and their relation to the degree of redundancy problem. I also discuss existing methods developed to solve the matroid cogirth problem and the SCP. Computational results are provided to validate a branch-and-cut algorithm that addresses the SCP formulation.Item Complexed Multifunctional Metallic and Chalcogenide Nanostructures as Theranostic Agents(2013-12-03) Young, Joseph; Drezek, Rebekah A.; Hicks, Illya V.; Kono, JunichiroNanostructures have attracted substantial attention due to their distinctive properties and various applications. Nanostructures consisting of multiple morphologies and/or materials have recently become the focus of intense study with particular attention being paid to their optical and magnetic properties and the enhanced role of the interface between materials. Of particular interest are metallic-based plasmonic nanostructures, structures that support surface plasmon resonances that are sensitive to the environment, and ferrimagnetic-based nanostructures, structures that exhibit strong magnetic properties when exposed to an external field. These nanostructures provide theranostic potential in the context of cancer photothermal therapies, diagnostics and imaging. Additionally, chalcogenide based nanostructure complexes are particularly interesting. Metallic chalcogenides offer the ability to combine different types of linear and nonlinear optical properties, enable design of nanostructure complexes with surface plasmon resonance effects in new wavelength ranges, and act as photo-emitting agents for novel theranostic applications. In this thesis an in depth analysis of plasmonic, magnetic and photo-emitting nanostructures as theranostic agents is presented. We have created several multifunctional nanostructures and the factors contributing to the functional properties of these nanostructures are explored systematically through experimentation, theory, and simulations. Both in vivo and in vitro testing demonstrates the applicability of these nanostructures as theranostic agents.Item Independence systems and stable set relaxations(2008) McClosky, Benjamin; Hicks, Illya V.Many fundamental combinatorial optimization problems involve the search for subsets of graph elements which satisfy some notion of independence. This thesis develops techniques for optimizing over a class of independence systems and focuses on systems having the vertex set of a finite graph as a ground set. The search for maximum stable sets in a graph offers a well-studied example of such a problem. More generally, for any integer k ≥ 1, the maximum co-k-plex problem fits into this framework as well. Co-k-plexes are defined as a relaxation of stable sets. This thesis studies co-k-plexes from polyhedral, algorithmic, and enumerative perspectives. The polyhedral analysis explores the relationship between the stable set polytope and co-k-plex polyhedra. Results include generalizations of odd holes, webs, wheels, and the claw. Sufficient conditions for the integrality of some related linear systems and results on the composition of stable set polyhedra are also given. The algorithmic analysis involves the development of heuristic and exact algorithms for finding maximum k-plexes. This problem is closely related to the search for co-k-plexes. The final chapter includes results on the enumerative structure of co-k-plexes in certain graphs.Item Integer Programming Techniques for Propagation and Rainbow Connection Problems in Graphs(2021-04-28) Smith, Logan A.; Hicks, Illya V.This thesis exhibits a collection of combinatorial optimization problems and the integer programs proposed to solve them based on new mathematical insights. In particular, graph propagation and graph throttling problems including the positive semidefinite zero forcing set problem and the minimum power dominating set problem are considered, as well as the graph connectivity problem known as the strong rainbow connection problem. A parallel treatment of the graph propagation problems is provided in which set cover problems are defined using problem specific blocking sets. These blocking sets are introduced, their structural properties are investigated, and computational methods for identifying them are proposed, providing a general recipe for developing integer programming approaches for graph propagation problems. The strong rainbow connection problem is also studied, and the first general computational method for the problem is introduced. New lower bounds, computational enhancements, and an alternative solution method based on iterative lower bound improvement are also proposed, the latter of which is shown to be highly effective in practice.Item Integrated Value Function Global Optimization Approaches for Two-Stage Stochastic Programs(2021-05-27) Antley, Eric M; Schaefer, Andrew J.; Hicks, Illya V.; Huchette, Joey; Ozaltın, Osman Y.We consider a class of nonnegative two-stage stochastic integer programs with discretely distributed right-hand sides. We develop a global optimization algorithm that exploits the structure of the superadditive dual for mixed-integer programs to search for the optimal solution. We demonstrate that our algorithm is capable of solving instances with large scenario counts to optimality.Item Low Complexity Detection and Precoding for Massive MIMO Systems: Algorithm, Architecture, and Application(2014-12-03) Yin, Bei; Cavallaro, Joseph R.; Aazhang, Behnaam; Hicks, Illya V.; Studer, ChristophMassive (or large-scale) MIMO is an emerging technology to improve the spectral efficiency of existing (small-scale) MIMO wireless communication systems. The main idea is to equip the base station (BS) with hundreds of antennas that serves a small number of users (in the orders of tens) simultaneously in the same frequency band. In such a system, the data detection and precoding are among the most challenging tasks in terms of computational complexity and performance. Although theoretical results show that simple detection and precoding algorithms are able to achieve optimal error rate performance when the number of BS antennas approaches infinity, the systems with realistic antenna configurations have to resort to computationally expensive algorithms to achieve near-optimal performance. In this research, we show that by utilizing the special property of massive MIMO systems, approximate linear detection and precoding can deliver near-optimal error rate performance with low complexity. We first propose approximate methods relying on Neumann series. This approach requires lower computational complexity than that of an exact inversion while delivering near-optimal results when there is a large ratio between BS and user antennas. We then develop a novel reconfigurable VLSI architecture to perform both the necessary Gram matrix computation and Neumann series based matrix inversion. The Neumann series approach, however, suffers from a considerable error-rate performance loss if the ratio of BS to user antennas is not large enough. To improve the performance, we investigate the conjugate gradient (CG) method (without explicitly computing matrix inversion) and conjugate gradient least square (CGLS) method (without explicitly computing Gram matrix and matrix inversion). Although CG and CGLS for precoding are rather straightforward, the necessary signal-to-interference-and-noise-ratio (SINR) for soft-output detection is not computed by CG and CGLS. To solve this problem, we propose an exact and an approximate method to compute the SINR within CG and CGLS algorithm with low complexity. We show that compared to exact and Neumann series based linear methods, CG based detection and precoding method is suitable for systems with small to medium number of users, while CGLS is suitable for systems with large number of users. A novel reconfigurable VLSI architecture is then proposed to support the both CG and CGLS.Item The Minimalᅠk-Core Problem for Modelingᅠ k-Assemblies(Springer, 2015) Wood, Cynthia I.; Hicks, Illya V.The concept of cell assembly was introduced by Hebb and formalized mathematically by Palm in the framework of graph theory. In the study of associative memory, a cell assembly is a group of neurons that are strongly connected and represent a "concept" of our knowledge. This group is wired in a specific manner such that only a fraction of its neurons will excite the entire assembly. We link the concept of cell assembly to the closure of a minimal k-core and study a particular type of cell assembly called k-assembly. The goal of this paper is to find all substructures within a network that must be excited in order to activate a k-assembly. Through numerical experiments, we confirm that fractions of these important subgroups overlap. To explore the problem, we present a backtracking algorithm to find all minimal k-cores of a given undirected graph, which belongs to the class of NP-hard problems. The proposed method is a modification of the Bron and Kerbosch algorithm for finding all cliques of an undirected graph. The results in the tested graphs offer insight in analyzing graph structure and help better understand how concepts are stored.Item Mixed Integer Linear Optimization Formulations for Learning Optimal Binary Classification Trees(2021-11-10) Alston, Brandon; Hicks, Illya V.Decision trees are powerful tools for classification and regression that attract many researchers working in the burgeoning area of machine learning. A classification decision tree has two types of vertices: (i) branching vertices at which datapoints are tested on a selection of discrete features, and (ii) leaf vertices at which datapoints are assigned classes. An optimal binary classification tree is a special type of classification tree in which each branching vertex has exactly two children and can be obtained by solving a biobjective mixed integer linear optimization problem that seeks to minimize the (i) number of misclassified datapoints and (ii) number of branching vertices. In this thesis we present two new multicommodity flow formulations and a new cut-based formulation to learn such optimal binary classification trees. We then provide a comparison of the formulations' strength, valid inequalities to strengthen all formulations, and accompanying computational results.Item Modeling Disjunctive Constraints via Junction Trees(2021-12-22) Lyu, Bochuan; Hicks, Illya V.; Huchette, Joseph A.In this thesis, we study the independent-branching (IB) framework of disjunctive constraints and identify a class of pairwise IB-representable disjunctive constraints: disjunctive constraints with junction trees. For this class of constraints, the existence of junction trees can be recognized in polynomial time. We also present a polynomial-time heuristic algorithm for the minimum biclique cover problem on the associated conflict graphs to build small and strong mixed-integer programming (MIP) formulations. Additionally, we apply the heuristic to find a smaller MIP formulation of generalized special ordered set with less variables and constraints than Huchette and Vielma [2019]. In computational experiments, we compare the proposed heuristic with other methods on a large set of artificially generated disjunctive constraints with junction trees. The new method significantly reduces the numbers of binary variables and constraints required for the MIP formulations than those of vertex cover approach.Item On the Integrality Gap of the Subtour Relaxation of the Traveling Salesman Problem for Certain Fractional 2-matching Costs(2014-04-11) Fast, Caleb; Hicks, Illya V.; Bixby, Robert E.; Cooper, Keith D.; Tapia, Richard A.This thesis provides new bounds on the strength of the subtour relaxation of the Traveling Salesman Problem (TSP) for fractional 2-matching cost instances whose support graphs have no fractional cycles larger than five vertices. This work provides insight for improving approximation heuristics for the TSP and into the structure of solutions produced by the subtour relaxation. Guided by a T-join derived from the subtour relaxation, I form a tour by adding edges to the subtour relaxation. By this constructive process, I prove that the optimal solution of the TSP is within 4/3, 17/12, or strictly less than 3/2 of the optimal solution of the subtour relaxation. Thus, this thesis takes a step towards proving the 4/3 conjecture for the TSP and the development of a 4/3 approximation algorithm for the TSP. These developments would provide improved approximations for applications such as DNA sequencing, route planning, and circuit board testing.Item The Closure of the Minimal k-core Problem for Modeling k-assemblies(2013-12-04) Wood, Cynthia; Hicks, Illya V.; Cox, Steven J.; Tapia, Richard A.In this thesis, I present a backtracking algorithm to find all minimal k-cores of a given undirected graph, which belongs to the class of NP-hard problems. The proposed method is a modification of the Bron and Kerbosch algorithm for finding all cliques of an undirected graph. The minimal k-core problem has applications in the area of neuroscience. For example, in the study of associative memory, a cell assembly is a group of neurons that are strongly connected and represent a “concept” of our knowledge. This group is wired in a specific manner such that only a fraction of its neurons will excite the entire assembly. Recent studies have linked the concept of a particular type of cell assembly called k-assembly to the closure of a minimal k-core. Therefore, the proposed method puts us a step closer to test its mathematical definition.Item Two mod-p Johnson filtrations(2014-04-17) Cooper, James Michael; Putman, Thomas Andrew; Wolf, Michael; Hicks, Illya V.We consider two mod-p central series of the free group given by Stallings and Zassenhaus. Applying these series to definitions of Dennis Johnson's filtration of the mapping class group we obtain two mod-p Johnson filtrations. Further, we adapt the definition of the Johnson homomorphisms to obtain mod-p Johnson homomorphisms. We calculate the image of the first of these homomorphisms. We give generators for the kernels of these homomorphisms as well. We restrict the range of our mod-p Johnson homomorphisms using work of Morita. We finally prove the announced result of Perron that a rational homology 3-sphere may be given as a Heegaard splitting with gluing map coming from certain members of our mod-p Johnson filtrations.