Theory Of Integer Programming Research Articles

Integer programming in parameterized complexity: Five miniatures

Powerful results from the theory of integer programming have recently led to substantial advances in parameterized complexity. However, our perception is that, except for Lenstra’s algorithm for solving integer linear programming in fixed dimension, there is still little understanding in the parameterized complexity community of the strengths and limitations of the available tools. This is understandable: it is often difficult to infer exact runtimes or even the distinction between FPT and XP algorithms, and some knowledge is simply unwritten folklore in a different community. We wish to make a step in remedying this situation. To that end, we first provide an easy to navigate quick reference guide of integer programming algorithms from the perspective of parameterized complexity. Then, we show their applications in three case studies, obtaining FPT algorithms with runtime f(k)poly(n). We focus on: •Modeling: since the algorithmic results follow by applying existing algorithms to new models, we shift the focus from the complexity result to the modeling result, highlighting common patterns and tricks which are used.•Optimality program: after giving an FPT algorithm, we are interested in reducing the dependence on the parameter; we show which algorithms and tricks are often useful for speed-ups.•Minding the poly(n): reducing f(k) often has the unintended consequence of increasing poly(n); so we highlight the common trade-offs and show how to get the best of both worlds. Specifically, we consider graphs of bounded neighborhood diversity which are in a sense the simplest of dense graphs, and we show several FPT algorithms for Capacitated Dominating Set, Sum Coloring, Max-q-Cut, and certain other coloring problems by modeling them as convex programs in fixed dimension, n-fold integer programs, bounded dual treewidth programs, indefinite quadratic programs in fixed dimension, parametric integer programs in fixed dimension, and 2-stage stochastic integer programs.

Dual-feasible functions for integer programming and combinatorial optimization: Algorithms, characterizations, and approximations

Within the framework of the superadditive duality theory of integer programming, we study two types of dual-feasible functions of a single real variable (Alves et al., 2016). We introduce software that automates testing piecewise linear functions for maximality and extremality, enabling a computer-based search. We build a connection to cut-generating functions in the Gomory–Johnson and related models, complete the characterization of maximal functions, and prove analogues of the Gomory–Johnson 2-slope theorem and the Basu–Hildebrand–Molinaro approximation theorem.

Sharp Bounds for the Second-Order General Connectivity Index of Hexagonal Chains

Given H, a hexagonal chain, we determine an expression of its second-order general connectivity index, denoted by χ2α(H), in terms of inlet features of H. Moreover, by applying the method in integer programming theory, we completely determine the extremal chains with the minimal or maximal χ2α(H) over the set of hexagonal chains.

Size-change termination with difference constraints

This article considers an algorithmic problem related to the termination analysis of programs. More specifically, we are given bounds on differences in sizes of data values before and after every transition in the program's control-flow graph. Our goal is to infer program termination via the following reasoning (“the size-change principle”): if in any infinite (hypothetic) execution of the program, some size must descend unboundedly, the program must always terminate, since infinite descent of a natural number is impossible. The problem of inferring termination from such abstract information is not the halting problem for programs and may well be decidable. If this is the case, the decision algorithm forms a “back end” of a termination verifier, and it is interesting to find out the computational complexity of the problem. A restriction of the problem described above, which only uses monotonicity information (but not difference bounds), is already known to be decidable. We prove that the unrestricted problem is undecidable, which gives a theoretical argument for studying restricted cases. We consider a case where the termination proof is allowed to make use of at most one bound per target variable in each transition. For this special case, which we claim is practically significant, we give (for the first time) an algorithm and show that the problem is in PSPACE, in fact that it is PSPACE-complete. The algorithm is based on combinatorial arguments and results from the theory of integer programming not previously used for similar problems. The algorithm has interesting connections to other work in termination, in particular to methods for generating linear ranking functions or invariants.

A survey on graphs which have equal domination and closed neighborhood packing numbers

Vizing's conjecture on the Cartesian product G□H of two graphs, γ(G□H) ≥ γ(G)γ(H), will hold for all graphs G if the domination and (closed neighborhood) packing number of G are equal. By using the theory of integer programming, we state known sufficient conditions for when the packing number and the domination number of a graph are equal. We also give a sufficient condition using the theory of efficient domination. For regular graphs we find a necessary and sufficient condition for equlity to hold.

THE FACILITY LOCATION PROBLEM: MODEL, ALGORITHM, AND APPLICATION TO COMPRESSOR ALLOCATION

Unlike other problems found in the oil industry, the allocation of compressors to gas-lifted oil wells lacks formal models and algorithms capable of delivering globally optimal assignments. To this end, this paper casts the compressor allocation problem as a facility location problem. Owing to the special structure of the target problem, an efficient (polynomial time) algorithm is conceived by applying the framework of dynamic programming. An example illustrates the inner-workings of the algorithm. Connections between the polyhedron of feasible solutions and integer-programming theory are established.

Matrices Defining Gorenstein Lattice Ideals

We study a class of integer matrices that define Gorenstein lattice ideals. We call them Gorenstein matrices. We give a combinatorial characterization of those which are of size (n + 1) × n and we relate them to the Frobenius problem in integer programming theory. We also give a necessary and sufficient condition for Gorensteinness of generic matrices which are defined in integer programming theory.

Integrated differentiated survivability in IP over WDM networks

The problem of differentiated Multi-Layer Integrated Survivability (MLIS) in IP over WDM networks is studied, which is decomposed into three sub-problems: survivable strategies design (SSD), spare capacity dimensioning (SCD), and dynamic survivable routing (DSR). A related work of network survivability in IP over WDM networks is firstly provided, and adaptive survivable strategies are also designed. A new Integrated Shared Pool (ISP) approach for SCD is then proposed, which is formulated by using integer-programming theory. Moreover, a novel survivable routing scheme called Differentiated Integrated Survivability Algorithm (DISA) for DSR is developed. Simulation results show that the proposed integrated survivability scheme performs much better than other solutions (e.g., "highest layer recovery" and "lowest layer recovery" schemes) in terms of traffic blocking ratio, spare resource requirement, and average traffic recovery ratio in IP over WDM networks.

Decomposition of the permanganate/oxalic acid overall reaction to elementary steps based on integer programming theory

A systematic method to obtain chemically acceptable decompositions of the overall autocatalytic reaction permanganate/oxalic acid into elementary steps is presented. The decomposition process was based on integer programming theory and chemical evidence, and it can be applied to elucidate other mechanisms as well. As a result, we obtained mathematical justification of the well-known autocatalytic nature and of the less-known crucial role of radical CO2−. By inspecting the obtained decompositions, one can find reaction steps inaccessible to chemical intuition, and these point out the direction of further experimental investigations.

Hypergeometric polynomials and integer programming

We examine connections between A-hypergeometric differential equations and the theory of integer programming. In the first part, we develop a ’hypergeometric sensitivity analysis‘ for small variations of constraint constants with creation operators and b-functions. In the second part, we study the indicial polynomial (b-function) along the hyperplane xi=0 via a correspondence between the optimal value of an integer programming problem and the roots of the indicial polynomial. Gröbner bases are used to prove theorems and give counter examples.

Integer Programming Models for Sales Resource Allocation

A practical conceptual framework for sales resource allocation modeling is presented in this paper. A literature review of sales resource allocation models is described in terms of this framework. The conceptual framework also lends itself to several integer programming models which may be used to address the variety of sales resource allocation decisions faced by every sales organization. A general model for sales resource allocation is developed which incorporates multiple sales resources, multiple time periods and carryover effects, non-separability, and risk. Several actual model implementations are discussed which illustrate the practical application of the integer programming models. The model implementations utilize recent advances in integer programming theory which enables sales managers and sales representatives to quickly develop and evaluate alternative sales resource allocation strategies.

Some Basis Theorems for Integral Monoids

We consider sets of integer vectors containing the zero vector and closed under addition, the integral monoids, and provide conditions under which they contain a finite subset of integer vectors which generate the entire monoid as nonnegative integer combinations. The paper concludes with some applications to the theory of integer programming.

Solving fixed charge network problems with group theory‐based penalties

AbstractMany well‐known transportation, communication, and facilities location problems in operations research can be formulated as fixed charge network problems, i.e. as minimum cost flow problems on a capacitated network in one commodity where some arcs have both fixed and variable costs. One approach to solving such problems is to use group theoretic concepts from the theory of integer programming to provide bounds for a branch‐and‐bound procedure. This paper presents such a group‐theory based algorithm for exact solution of fixed charge network problems which exploits the special structures of network problems. Computational results are reported for problems with as many as 100 fixed charge arcs.