Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms

Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical programming used in the context of decoding include linear, integer, and nonlinear programming, network flows, notions of duality as well as matroid and polyhedral theory. This survey article reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory. Published July 201

Adaptive Cut Generation Algorithm for Improved Linear Programming Decoding of Binary Linear Codes

Linear programming (LP) decoding approximates maximum-likelihood (ML) decoding of a linear block code by relaxing the equivalent ML integer programming (IP) problem into a more easily solved LP problem. The LP problem is defined by a set of box constraints together with a set of linear inequalities called "parity inequalities" that are derived from the constraints represented by the rows of a parity-check matrix of the code and can be added iteratively and adaptively. In this paper, we first derive a new necessary condition and a new sufficient condition for a violated parity inequality constraint, or "cut," at a point in the unit hypercube. Then, we propose a new and effective algorithm to generate parity inequalities derived from certain additional redundant parity check (RPC) constraints that can eliminate pseudocodewords produced by the LP decoder, often significantly improving the decoder error-rate performance. The cut-generating algorithm is based upon a specific transformation of an initial parity-check matrix of the linear block code. We also design two variations of the proposed decoder to make it more efficient when it is combined with the new cut-generating algorithm. Simulation results for several low-density parity-check (LDPC) codes demonstrate that the proposed decoding algorithms significantly narrow the performance gap between LP decoding and ML decoding

Algorithms for solving linear integer progamming problems

This thesis discusses several methods used to solve linear integer programming problems. Chapters 1-3 give the necessary linear programming background. Chapter 4 introduces integer programming and describes, in general, the two classes of solution methods -enumeration and cutting planes. Chapter 5 details two specific cutting plane methods, one all-integer approach and one fractional approach.Chapter 6 describes the branch-and-bound method, one of the enumeration methods. In Chapter 7 the additive algorithm for 0-1 programming problems is discussed. Chapter 8 describes the branch and-cut method, a combination of the cutting plane and branch-and bound approaches. Chapter 9 presents the plant location problem as an example of the integer programming problem. Examples of each method are included in the thesis

A Study On The Split Delivery Vehicle Routing Problem

This dissertation examines the Split Delivery Vehicle Routing Problem (SDVRP), a relaxed version of classical capacitated vehicle routing problem (CVRP) in which the demand of any client can be split among the vehicles that visit it. We study both scenarios of the SDVRP in this dissertation. For the SDVRP with a fixed number of the vehicles, we provide a Two-Stage algorithm. This approach is a cutting-plane based exact method called Two-Stage algorithm in which the SDVRP is decomposed into two stages of clustering and routing. At the first stage, an assignment problem is solved to obtain some clusters that cover all demand points and get the lower bound for the whole problem; at the second stage, the minimal travel distance of each cluster is calculated as a traditional Traveling Salesman Problem (TSP), and the upper bound is obtained. Adding the information obtained from the second stage as new cuts into the first stage, we solve the first one again. This procedure stops when there are no new cuts to be created from the second stage. Several valid inequalities have been developed for the first stage to increase the computational speed. A valid inequality is developed to completely solve the problem caused by the index of vehicles. Another strong valid inequality is created to provide a valid distance lower bound for each set of demand points. This algorithm can significantly outperform other exact approaches for the SDVRP in the literature. If the number of the vehicles in the SDVRP is a variable, we present a column generation based branch and price algorithm. First, a restricted master problem (RMP) is presented, which is composed of a finite set of feasible routes. Solving the linear relaxation of the RMP, values of dual variables are thus obtained and passed to the sub-problem, the pricing problem, to generate a new column to enter the base of the RMP and solve the new RMP again. This procedure repeats until the objective function value of the pricing problem is greater than or equal to zero (for minimum problem). In order to get the integer feasible (optimal) solution, a branch and bound algorithm is then performed. Since after branching, it is not guaranteed that the possible favorable column will appear in the master problem. Therefore, the column generation is performed again in each node after branching. The computational results indicate this approach is promising in solving the SDVRP in which the number of the vehicles is not fixed

On SOCP-based disjunctive cuts for solving a class of integer bilevel nonlinear programs

We study a class of integer bilevel programs with second-order cone constraints at the upper-level and a convex-quadratic objective function and linear constraints at the lower-level. We develop disjunctive cuts (DCs) to separate bilevel-infeasible solutions using a second-order-cone-based cut-generating procedure. We propose DC separation strategies and consider several approaches for removing redundant disjunctions and normalization. Using these DCs, we propose a branch-and-cut algorithm for the problem class we study, and a cutting-plane method for the problem variant with only binary variables. We present an extensive computational study on a diverse set of instances, including instances with binary and with integer variables, and instances with a single and with multiple linking constraints. Our computational study demonstrates that the proposed enhancements of our solution approaches are effective for improving the performance. Moreover, both of our approaches outperform a state-of-the-art generic solver for mixed-integer bilevel linear programs that is able to solve a linearized version of our binary instances.Comment: arXiv admin note: substantial text overlap with arXiv:2111.0682

Engineering Art Galleries

The Art Gallery Problem is one of the most well-known problems in Computational Geometry, with a rich history in the study of algorithms, complexity, and variants. Recently there has been a surge in experimental work on the problem. In this survey, we describe this work, show the chronology of developments, and compare current algorithms, including two unpublished versions, in an exhaustive experiment. Furthermore, we show what core algorithmic ingredients have led to recent successes