Algebraic structural analysis of a vehicle routing problem of heterogeneous trucks. Identification of the properties allowing an exact approach.

Although integer linear programming problems are typically difficult to solve, there exist some easier problems, where the linear programming relaxation is integer. This thesis sheds light on a drayage problem which is supposed to have this nice feature, after extensive computational experiments. This thesis aims to provide a theoretical understanding of these results by the analysis of the algebraic structures of the mathematical formulation. Three reformulations are presented to prove if the constraint matrix is totally unimodular. We will show which experimental conditions are necessary and sufficient (or only sufficient or only necessary) for total unimodularity

Polynomial solvability of cost-based abduction

AbstractIn recent empirical studies we have shown that many interesting cost-based abduction problems can be solved efficiently by considering the linear program relaxation of their integer program formulation. We tie this to the concept of total unimodularity from network flow analysis, a fundamental result in polynomial solvability. From this, we can determine the polynomial solvability of abduction problems and, in addition, present a new heuristic for branch and bound in the non-polynomial cases

A structural approach to kernels for ILPs: Treewidth and Total Unimodularity

Kernelization is a theoretical formalization of efficient preprocessing for NP-hard problems. Empirically, preprocessing is highly successful in practice, for example in state-of-the-art ILP-solvers like CPLEX. Motivated by this, previous work studied the existence of kernelizations for ILP related problems, e.g., for testing feasibility of Ax <= b. In contrast to the observed success of CPLEX, however, the results were largely negative. Intuitively, practical instances have far more useful structure than the worst-case instances used to prove these lower bounds. In the present paper, we study the effect that subsystems with (Gaifman graph of) bounded treewidth or totally unimodularity have on the kernelizability of the ILP feasibility problem. We show that, on the positive side, if these subsystems have a small number of variables on which they interact with the remaining instance, then we can efficiently replace them by smaller subsystems of size polynomial in the domain without changing feasibility. Thus, if large parts of an instance consist of such subsystems, then this yields a substantial size reduction. We complement this by proving that relaxations to the considered structures, e.g., larger boundaries of the subsystems, allow worst-case lower bounds against kernelization. Thus, these relaxed structures can be used to build instance families that cannot be efficiently reduced, by any approach.Comment: Extended abstract in the Proceedings of the 23rd European Symposium on Algorithms (ESA 2015

A solution scheme of satisfiability problem by active usage of totally unimodularity property.

by Mei Long.Thesis (M.Phil.)--Chinese University of Hong Kong, 2003.Includes bibliographical references (leaves 93-98).Abstracts in English and Chinese.Table of Contents --- p.vAbstract --- p.viiiAcknowledgements --- p.xChapter 1 --- Introduction --- p.1Chapter 1.1 --- Satisfiability Problem --- p.1Chapter 1.2 --- Motivation of the Research --- p.1Chapter 1.3 --- Overview of the Thesis --- p.2Chapter 2 --- Satisfiability Problem --- p.4Chapter 2.1 --- Satisfiability Problem --- p.5Chapter 2.1.1 --- Basic Definition --- p.5Chapter 2.1.2 --- Phase Transitions --- p.5Chapter 2.2 --- History --- p.6Chapter 2.3 --- The Basic Search Algorithm --- p.8Chapter 2.4 --- Some Improvements to the Basic Algorithm --- p.9Chapter 2.4.1 --- Satz by Chu-Min Li --- p.9Chapter 2.4.2 --- Heuristics and Local Search --- p.12Chapter 2.4.3 --- Relaxation --- p.13Chapter 2.5 --- Benchmarks --- p.14Chapter 2.5.1 --- Specific Problems --- p.14Chapter 2.5.2 --- Randomly Generated Problems --- p.14Chapter 2.6 --- Software and Internet Information for SAT solving --- p.16Chapter 2.6.1 --- Stochastic Local Search Algorithms (incomplete) --- p.16Chapter 2.6.2 --- Systematic Search Algorithms (complete) --- p.16Chapter 2.6.3 --- Some useful Links to SAT Related Sites --- p.17Chapter 3 --- Integer Programming Formulation for Logic Problem --- p.18Chapter 3.1 --- SAT Problem --- p.19Chapter 3.2 --- MAXSAT Problem --- p.19Chapter 3.3 --- Logical Inference Problem --- p.19Chapter 3.4 --- Weighted Exact Satisfiability Problem --- p.20Chapter 4 --- Integer Programming Formulation for SAT Problem --- p.22Chapter 4.1 --- From 3-CNF SAT Clauses to Zero-One IP Constraints --- p.22Chapter 4.2 --- Integer Programming Model for 3-SAT --- p.23Chapter 4.3 --- The Equivalence of the SAT and the IP --- p.23Chapter 4.4 --- Example --- p.24Chapter 5 --- Integer Solvability of Linear Programs --- p.27Chapter 5.1 --- Unimodularity --- p.27Chapter 5.2 --- Totally Unimodularity --- p.28Chapter 5.3 --- Some Results on Recognition of Linear Solvability of IP --- p.32Chapter 6 --- TU Based Matrix Research Results --- p.33Chapter 6.1 --- 2x2 Matrix's TU Property --- p.33Chapter 6.2 --- Extended Integer Programming Model for SAT --- p.34Chapter 6.3 --- 3x3 Matrix's TU Property --- p.35Chapter 7 --- Totally Unimodularity Based Branching-and-Bound Algorithm --- p.38Chapter 7.1 --- Introduction --- p.38Chapter 7.1.1 --- Enumeration Trees --- p.39Chapter 7.1.2 --- The Concept of Branch and Bound --- p.42Chapter 7.2 --- TU Based Branching Rule --- p.43Chapter 7.2.1 --- How to sort variables based on 2x2 submatrices --- p.43Chapter 7.2.2 --- How to sort the rest variables --- p.45Chapter 7.3 --- TU Based Bounding Rule --- p.46Chapter 7.4 --- TU Based Branch-and-Bound Algorithm --- p.47Chapter 7.5 --- Example --- p.49Chapter 8 --- Numerical Result --- p.57Chapter 8.1 --- Experimental Result --- p.57Chapter 8.2 --- Statistical Results of ILOG CPLEX --- p.59Chapter 9 --- Conclusions --- p.61Chapter 9.1 --- Contributions --- p.61Chapter 9.2 --- Future Work --- p.62Chapter A --- The Coefficient Matrix A for Example in Chapter 7 --- p.64Chapter B --- The Detailed Numerical Information of Solution Process for Exam- ple in Chapter 7 --- p.66Chapter C --- Experimental Result --- p.67Chapter C.1 --- "# of variables: 20, # of clauses: 91" --- p.67Chapter C.2 --- "# of variables: 50, # of clauses: 218" --- p.70Chapter C.3 --- # of variables: 75，# of clauses: 325 --- p.73Chapter C.4 --- "# of variables: 100, # of clauses: 430" --- p.76Chapter D --- Experimental Result of ILOG CPLEX --- p.80Chapter D.1 --- # of variables: 20´ة # of clauses: 91 --- p.80Chapter D.2 --- # of variables: 50，#of clauses: 218 --- p.83Chapter D.3 --- # of variables: 75，# of clauses: 325 --- p.86Chapter D.4 --- "# of variables: 100, # of clauses: 430" --- p.89Bibliography --- p.9

On a linear programming approach to the discrete Willmore boundary value problem and generalizations

We consider the problem of finding (possibly non connected) discrete surfaces spanning a finite set of discrete boundary curves in the three-dimensional space and minimizing (globally) a discrete energy involving mean curvature. Although we consider a fairly general class of energies, our main focus is on the Willmore energy, i.e. the total squared mean curvature Our purpose is to address the delicate task of approximating global minimizers of the energy under boundary constraints. The main contribution of this work is to translate the nonlinear boundary value problem into an integer linear program, using a natural formulation involving pairs of elementary triangles chosen in a pre-specified dictionary and allowing self-intersection. Our work focuses essentially on the connection between the integer linear program and its relaxation. We prove that: - One cannot guarantee the total unimodularity of the constraint matrix, which is a sufficient condition for the global solution of the relaxed linear program to be always integral, and therefore to be a solution of the integer program as well; - Furthermore, there are actually experimental evidences that, in some cases, solving the relaxed problem yields a fractional solution. Due to the very specific structure of the constraint matrix here, we strongly believe that it should be possible in the future to design ad-hoc integer solvers that yield high-definition approximations to solutions of several boundary value problems involving mean curvature, in particular the Willmore boundary value problem

The family constrained network problem

AbstractAn extension of the classical fixed charge transportation problem is developed that allows a wide variety of practical production, distribution, and inventory planning models to be addressed. Computational results are presented for problems with up to one thousand network constraints and five thousand network variables

MODELS AND SOLUTION ALGORITHMS FOR EQUITABLE RESOURCE ALLOCATION IN AIR TRAFFIC FLOW MANAGEMENT

Population growth and economic development lead to increasing demand for travel and pose mobility challenges on capacity-limited air traffic networks. The U.S. National Airspace System (NAS) has been operated near the capacity, and air traffic congestion is expected to remain as a top concern for the related system operators, passengers and airlines. This dissertation develops a number of model reformulations and efficient solution algorithms to address resource allocation problems in air traffic flow management, while explicitly accounting for equitable objectives in order to encourage further collaborations by different stakeholders. This dissertation first develops a bi-criteria optimization model to offload excess demand from different competing airlines in the congested airspace when the predicted traffic demand is higher than available capacity. Computationally efficient network flow models with side constraints are developed and extensively tested using datasets obtained from the Enhanced Traffic Management System (ETMS) database (now known as the Traffic Flow Management System). Representative Pareto-optimal tradeoff frontiers are consequently generated to allow decision-makers to identify best-compromising solutions based on relative weights and systematical considerations of both efficiency and equity. This dissertation further models and solves an integrated flight re-routing problem on an airspace network. Given a network of airspace sectors with a set of waypoint entries and a set of flights belonging to different air carriers, the optimization model aims to minimize the total flight travel time subject to a set of flight routing equity, operational and safety requirements. A time-dependent network flow programming formulation is proposed with stochastic sector capacities and rerouting equity for each air carrier as side constraints. A Lagrangian relaxation based method is used to dualize these constraints and decompose the original complex problem into a sequence of single flight rerouting/scheduling problems. Finally, within a multi-objective utility maximization framework, the dissertation proposes several practically useful heuristic algorithms for the long-term airport slot assignment problem. Alternative models are constructed to decompose the complex model into a series of hourly assignment sub-problems. A new paired assignment heuristic algorithm is developed to adapt the round robin scheduling principle for improving fairness measures across different airlines. Computational results are presented to show the strength of each proposed modeling approach