Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting

Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer programs. In particular, prior to this work, it was open whether some classical problems, like the minimum odd-cut problem, can be expressed by a compact mixed-integer program with few (even constantly many) integer variables. This is in stark contrast to linear formulations, where recent breakthroughs in the field of extended formulations have shown that many polytopes associated to classical combinatorial optimization problems do not even admit approximate extended formulations of sub-exponential size. We provide a general framework for lifting inapproximability results of extended formulations to the setting of mixed-integer extended formulations, and obtain almost tight lower bounds on the number of integer variables needed to describe a variety of classical combinatorial optimization problems. Among the implications we obtain, we show that any mixed-integer extended formulation of sub-exponential size for the matching polytope, cut polytope, traveling salesman polytope or dominant of the odd-cut polytope, needs $\Omega(n/\log n)$ many integer variables, where $n$ is the number of vertices of the underlying graph. Conversely, the above-mentioned polyhedra admit polynomial-size mixed-integer formulations with only $O(n)$ or $O(n \log n)$ (for the traveling salesman polytope) many integer variables. Our results build upon a new decomposition technique that, for any convex set $C$, allows for approximating any mixed-integer description of $C$ by the intersection of $C$ with the union of a small number of affine subspaces.Comment: A conference version of this paper will be presented at SODA 201

Layered graph approaches for combinatorial optimization problems

Extending the concept of time-space networks, layered graphs associate information about one or multiple resource state values with nodes and arcs. While integer programming formulations based on them allow to model complex problems comparably easy, their large size makes them hard to solve for non-trivial instances. We detail and classify layered graph modeling techniques that have been used in the (recent) scientific literature and review methods to successfully solve the resulting large-scale, extended formulations. Modeling guidelines and important observations concerning the solution of layered graph formulations by decomposition methods are given together with several future research directions

Reformulation and decomposition of integer programs

In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm

AN INTEGER PROGRAMMING APPROACH FOR SINGLE TRUCK ROUTING-AND-SCHEDULING PROBLEMS TO ISLANDS WITH TIME-VARYING FERRY SCHEDULES

This study aims to develop a solving model for the single trucks routing-and-scheduling problems to islands with variations in ferry schedules. In this problem, the travel time is asymmetric and the truck routing is based on the sequence of island visits, known and unknown. The models are developed using an integer programming approach. Integer non-linear programming is formulated to solve problems where the sequence is unknown, whereas integer linear programming for the sequence is known. Besides, a delivery day scenario is built to determine the optimal route and schedule with minimum total travel time on each departure day. Numerical experiments were carried out on the case of a small distribution of a small industry in Central Moluccas, Indonesia. The results showed that the model developed could provide solutions to solve problems

Solving weighted and counting variants of connectivity problems parameterized by treewidth deterministically in single exponential time

It is well known that many local graph problems, like Vertex Cover and Dominating Set, can be solved in 2^{O(tw)}|V|^{O(1)} time for graphs G=(V,E) with a given tree decomposition of width tw. However, for nonlocal problems, like the fundamental class of connectivity problems, for a long time we did not know how to do this faster than tw^{O(tw)}|V|^{O(1)}. Recently, Cygan et al. (FOCS 2011) presented Monte Carlo algorithms for a wide range of connectivity problems running in time $c^{tw}|V|^{O(1)} for a small constant c, e.g., for Hamiltonian Cycle and Steiner tree. Naturally, this raises the question whether randomization is necessary to achieve this runtime; furthermore, it is desirable to also solve counting and weighted versions (the latter without incurring a pseudo-polynomial cost in terms of the weights). We present two new approaches rooted in linear algebra, based on matrix rank and determinants, which provide deterministic c^{tw}|V|^{O(1)} time algorithms, also for weighted and counting versions. For example, in this time we can solve the traveling salesman problem or count the number of Hamiltonian cycles. The rank-based ideas provide a rather general approach for speeding up even straightforward dynamic programming formulations by identifying "small" sets of representative partial solutions; we focus on the case of expressing connectivity via sets of partitions, but the essential ideas should have further applications. The determinant-based approach uses the matrix tree theorem for deriving closed formulas for counting versions of connectivity problems; we show how to evaluate those formulas via dynamic programming.Comment: 36 page

A Multicriteria Analysis for the Green VRP: A Case Discussion for the Distribution Problem of a Spanish Retailer

[EN] This research presents the group of green vehicle routing problems with environmental costs translated into money versus production of noise, pollution and fuel consumption. This research is focused on multi-objective green logistics optimization. Optimality criteria are environmental costs: minimization of amount of money paid as externality cost for noise, pollution and costs of fuel versus minimization of noise, pollution and fuel consumption themselves. Some mixed integer programming formulations of multi-criteria vehicle routing problems have been considered. Mathematical models were formulated under assumption of existence of asymmetric distance-based costs and use of homogeneous fleet. The exact solution methods are applied for finding optimal solutions. The software used to solve these models is the CPLEX solver with AMPL programming language. The researchers were able to use real data from a Spanish company of groceries. Problems deal with green logistics for routes crossing the Spanish regions of Navarre, Basque Country and La Rioja. Analyses of obtained results could help logistics managers to lead the initiative in area of green logistics by saving money paid for environmental costs as well as direct cost of fuel and minimization of pollution and noise.This work has been partially supported by the National Research Center (NCN), Poland (DEC-2013/11/B/ST8/04458), by AGH, and by the Spanish Ministry of Economy and Competitiveness (TRA2013-48180-C3-P and TRA2015-71883-REDT), and the Ibero-American Program for Science and Technology for Development (CYTED2014-515RT0489). Likewise, we want to acknowledge the support received by the CAN Foundation in Navarre, Spain (Grants CAN2014-3758 and CAN2015-70473)Sawik, B.; Faulin, J.; Pérez Bernabeu, E. (2017). A Multicriteria Analysis for the Green VRP: A Case Discussion for the Distribution Problem of a Spanish Retailer. Transportation Research Procedia. 22:305-313. https://doi.org/10.1016/j.trpro.2017.03.037S3053132