Polynomial Number Of Variables Research Articles

Extended formulations for binary optimal control problems

AbstractExtended formulations are an important tool in polyhedral combinatorics. Many combinatorial optimization problems require an exponential number of inequalities when modeled as a linear program in the natural space of variables. However, by adding artificial variables, one can often find a small linear formulation, i.e., one containing a polynomial number of variables and constraints, such that the projection to the original space of variables yields a perfect linear formulation. Motivated by binary optimal control problems with switching constraints, we show that a similar approach can be useful also for optimization problems in function space, in order to model the closed convex hull of feasible controls in a compact way. More specifically, we present small extended formulations for switches with bounded variation and for dwell-time constraints. For general linear switching point constraints, we devise an extended model linearizing the problem, but show that a small extended formulation that is compatible with discretization cannot exist unless P = NP.

A Differentiable Path-Following Method with a Compact Formulation to Compute Proper Equilibria

The concept of proper equilibrium was established as a strict refinement of perfect equilibrium. This establishment has significantly advanced the development of game theory and its applications. Nonetheless, it remains a challenging problem to compute such an equilibrium. This paper develops a differentiable path-following method with a compact formulation to compute a proper equilibrium. The method incorporates square-root-barrier terms into payoff functions with an extra variable and constitutes a square-root-barrier game. As a result of this barrier game, we acquire a smooth path to a proper equilibrium. To further reduce the computational burden, we present a compact formulation of an ε-proper equilibrium with a polynomial number of variables and equations. Numerical results show that the differentiable path-following method is numerically stable and efficient. Moreover, by relaxing the requirements of proper equilibrium and imposing Selten’s perfection, we come up with the notion of perfect d-proper equilibrium, which approximates a proper equilibrium and is less costly to compute. Numerical examples demonstrate that even when d is rather large, a perfect d-proper equilibrium remains to be a proper equilibrium. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms-Continuous. Funding: This work was partially supported by General Research Fund (GRF) CityU 11306821 of Hong Kong SAR Government. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0148 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0148 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Maximum weighted induced forests and trees: new formulations and a computational comparative review

AbstractGiven a graph with a weight associated with each vertex , the maximum weighted induced forest problem (MWIF) consists of encountering a maximum weighted subset of the vertices such that induces a forest. This NP‐hard problem is known to be equivalent to the minimum weighted feedback vertex set problem, which has applicability in a variety of domains. The closely related maximum weighted induced tree problem (MWIT), on the other hand, requires that the subset induces a tree. We propose two new integer programming formulations with an exponential number of constraints and branch‐and‐cut procedures for MWIF. Computational experiments using benchmark instances are performed comparing several formulations, including the newly proposed approaches and those available in the literature, when solved by a standard commercial mixed integer programming solver. More specifically, five formulations are compared, two compact ones (i.e., with a polynomial number of variables and constraints) and the three others with an exponential number of constraints. The experiments show that a new formulation for the problem based on directed cutset inequalities for eliminating cycles (DCUT) offers stronger linear relaxation bounds earlier in the search process. The results also indicate that the other new formulation, denoted tree with cycle elimination (TCYC), outperforms those available in the literature when it comes to the average times for proving optimality for the benchmark instances with up to 81 vertices. Additionally, this formulation can achieve much lower average times for solving the larger random instances that can be optimally solved. Furthermore, we show how the formulations for MWIF can be easily extended for MWIT. Such extension allowed us to analyze the difference between the optimal solution values of the two problems when considering different classes of graphs.

Using multiflow formulations to solve the Steiner tree problem in graphs

We present three different mixed integer linear models with a polynomial number of variables and constraints for the Steiner tree problem in graphs. The linear relaxations of these models are compared to show that a good (strong) linear relaxation can be a good approximation for the problem. We present computational results for the STP OR-Library (J.E. Beasley) instances of type b, c, d and e.

Compact formulations and an iterated local search-based matheuristic for the minimum weighted feedback vertex set problem

Given a weighted graph G=(V,E), the minimum weighted feedback vertex set problem consists in obtaining a minimum weight subset F⊆V of the vertex set whose removal makes the graph acyclic. Differently from other approaches in the literature, in this work we tackle this problem via the maximum weighted induced forest problem. First, we propose two new compact mixed integer programming (MIP) formulations, using a polynomial number of variables and constraints. Next, we develop a matheuristic that hybridizes a multi-start iterated local search heuristic with a MIP-based local search procedure. Extensive computational experiments carried out on a set of benchmark instances show that the newly proposed MILS+-mtz matheuristic is extremely effective and outperforms or at least match the best heuristics available in the literature in terms of solution quality for 79 out of 99 (79.80%) large instance groups, with 55 (55.56%) of them being strictly better than the previously best known. Furthermore, the compact formulations were able to optimally solve 474 out of 810 test instances in less than 3600 seconds of running time.

A compact mixed integer linear formulation for safe set problems

The Safe Set Problem is a type of partitioning problem with particular constraints on the weight of adjacent partition components. Several theoretical results exist in the literature along with a mixed integer linear formulation. We propose a new, more compact, Mixed Integer Linear Program for the (Weighted) Safe Set Problem and Connected Safe Set Problem with a polynomial number of variables and constraints, based on a flow from an additional fictitious node. The model is enriched by symmetry-breaking considerations and a variable reduction subproblem. We test the model on a benchmark of small instances which underline the difficulty of solving the Safe Set Problem through mathematical programming approaches. We also compare it with the only existing formulation in the literature and find that our approach is usually more efficient on a set of benchmark instances.

Exact methods for the discrete multiple allocation (r|p) hub-centroid problem

In the (r|p) hub-centroid problem ((r|p)HCP), two noncooperative firms, leader and follower, locate hubs sequentially in order to maximize their own total flow being transported through system. The leader locates p hubs, knowing that the follower will react by locating r hubs. After location decisions, each flow is transported by one hub or a pair of hubs placed by either leader or follower according to the lowest cost criteria. The problem consists of optimizing the leader decision. For the discrete multiple allocation version of the (r|p)HCP, we prove that this version is ∑2p-hard, propose for the first time two mixed integer linear programming formulations with a polynomial number of variables and present exact branch-and-cut algorithms to solve the formulations. The algorithms are compared to the best exact one found in the literature, obtaining better results for almost all instances. Furthermore, we present the optimal solution for several open instances.

Coordinating resources in Stackelberg Security Games

In this work we formulate a Stackelberg Security Game that coordinates resources in a border patrol problem. In this security domain, resources from different precincts have to be paired to conduct patrols in the border due to logistic constraints. Given this structure, models that enumerate the pure defender strategies scale poorly. We describe the set of mixed strategies using a polynomial number of variables but exponentially many constraints that come from the matching polytope. We then include this description in a mixed integer formulation to compute the Strong Stackelberg Equilibrium efficiently with a branch and cut scheme. Since the optimal patrol solution is a probability distribution over the set of exponential size, we also introduce an efficient sampling method that can be used to deploy the security resources every shift. Our computational results evaluate the efficiency of the branch and cut scheme developed and the accuracy of the sampling method.

The probabilistic pickup-and-delivery travelling salesman problem

Transportation problems are essential in commercial logistics and have been widely studied in the literature during the last decades. Many of them consist in designing routes for vehicles to move commodities between locations. This article approaches a pickup-and-delivery single-vehicle routing problem where there is susceptibility to uncertainty in customer requests. The probability distributions of the requests are assumed to be known, and the objective is to design an a priori route with minimum expected length. The problem has already been approached in the literature, but through a heuristic method. This article proposes the first exact approach to the problem. Two mathematical formulations are proposed: one is a compact model (i.e. defined by a polynomial number of variables and constraints); the other one contains an exponential number of inequalities and is solved within a branch-and-cut framework. Computational results show the upsides as well as the breakdowns of both formulations.

ILP Models and Column Generation for the Minimum Sum Coloring Problem

We study two Integer Linear Programming (ILP) formulations for the Minimum Sum Coloring Problem (MSCP). The problem is an extension of the classical Vertex Coloring Problem in which each color is represented by a positive natural number. The MSCP asks to minimize the sum of the cardinality of subsets of vertices receiving the same color, weighted by the index of the color, while ensuring that vertices linked by an edge receive different colors. The first ILP formulation has a polynomial number of variables while the second one has an exponential number of variables and is tackled via column generation. Computational tests show that the linear programming relaxation of the second formulation provides tight lower bounds which allow us to solve to proven optimality some hard instances of the literature.

Integer programming formulations for three sequential discrete competitive location problems with foresight

We deal with three competitive location problems based on the classical Maximal Covering Location Problem. The environment of these problems consists of an open market with two firms (leader and follower), several customers and locations where facilities can be located. In order to capture the demand of the customers, the leader enters the market by locating a set of facilities knowing the potential locations where the follower can locate her facilities after the leader’s decision. We consider here three pairs of objective functions for the leader/follower previously studied in the literature: maximizing/minimizing the demand captured by the leader, minimizing/maximizing the regret of the leader, maximizing the demand captured by each firm (also known as Stackelberg). For each model, we propose an integer linear programming formulation with a polynomial number of variables and an exponential number of constraints. The formulations are solved by branch-and-cut algorithms where the constraints are generated on demand by solving appropriate separation problems. We report extensive computational experiments realized on instances inspired by those from the literature, comparing our algorithms with the exact and heuristic algorithms previously published for these problems.

A branch‐and‐cut algorithm for the Team Orienteering Problem

AbstractThe Team Orienteering Problem aims at maximizing the total amount of profit collected by a fleet of vehicles while not exceeding a predefined travel time limit on each vehicle. In the last years, several exact methods based on different mathematical formulations were proposed. In this paper, we present a new two‐index formulation with a polynomial number of variables and constraints. This compact formulation, reinforced by connectivity constraints, was solved by means of a branch‐and‐cut algorithm. The total number of instances solved to optimality is 327 of 387 benchmark instances, 26 more than any previous method. Moreover, 24 not previously solved instances were closed to optimality.

Segment-based alteration for container liner shipping network design

Container liner shipping companies only partially alter their shipping networks to cope with the changing demand, rather than entirely redesign and change the network. In view of the practice, this paper proposes an optimal container liner shipping network alteration problem based on an interesting idea of segment, which is a sequence of legs from a head port to a tail port that are visited by the same type of ship more than once in the existing shipping network. In segment-based network alteration, the segments are intact and each port is visited by the same type of ship and from the same previous ports. As a result, the designed network needs minimum modification before implementation. A mixed-integer linear programming model with a polynomial number of variables is developed for the proposed segmented-based liner shipping network alternation problem. The developed model is applied to an Asia–Europe–Oceania liner shipping network with a total of 46 ports and 11 ship routes. Results demonstrate that the problem could be solved efficiently and the optimized network reduces the total cost of the initial network considerably.

Web services composition: Complexity and models

A web service is a modular and self-described application callable with standard web technologies. A workflow describes how to combine the functionalities of different web services in order to create a new value added functionality resulting in composite web service. QoS-aware web service composition means to select a composite web service that maximizes a QoS objective function while satisfying several QoS constraints (e.g. price or duration). The workflow-based QoS-aware web service composition problem has received a lot of interest, mainly in web service community. This general problem is NP-hard since it is equivalent to the multidimensional multiple choice knapsack problem (MMKP). In this article, the theoretical complexity is analysed more precisely in regard to the property of the workflow structuring the composition. For some classes of workflows and some QoS models, the composition problem can be solved in polynomial time (since the workflow is a series–parallel directed graph). Otherwise, when there exist one or several QoS constraints to verify, the composition problem becomes NP-hard. In this case, we propose a new mixed integer linear program to represent the problem with a polynomial number of variables and constraints. Then, using CPLEX, we present some experimental results showing that our proposed model is able to solve big size instances.

A mixed integer linear programming model and variable neighborhood search for Maximally Balanced Connected Partition Problem

This paper deals with Maximally Balanced Connected Partition (MBCP) problem. It introduces a mixed integer linear programming (MILP) formulation of the problem with polynomial number of variables and constraints. Also, a variable neighborhood search (VNS) technique for solving this problem is presented. The VNS implements the suitable neighborhoods based on changing the component for an increasing number of vertices. An efficient implementation of the local search procedure yields a relatively short running time. The numerical experiments are made on instances known in the literature. Based on the MILP model, tests are run using two MILP solvers, CPLEX and Gurobi. It is shown that both solvers succeed in finding optimal solutions for all smaller and most of medium scale instances. Proposed VNS reaches most of the optimal solutions. The algorithm is also successfully tested on large scale problem instances for which optimal solutions are not known.

Enhanced compact models for the connected subgraph problem and for the shortest path problem in digraphs with negative cycles

We investigate the minimum-weight connected subgraph problem. The importance of this problem stems from the fact that it constitutes the backbone of many network design problems having applications in several areas including telecommunication, energy, and distribution planning. We show that this problem is NP-hard, and we propose a new polynomial-size nonlinear mixed-integer programming model. We apply the Reformulation-Linearization Technique (RLT) to linearize the proposed model while keeping a polynomial number of variables and constraints. Furthermore, we show how similar modelling techniques enable an enhanced polynomial size formulation to be derived for the shortest elementary path. This latter problem is known to be intractable and has many applications (in particular, within the context of column generation). We report the results of extensive computational experiments on graphs with up to 1000 nodes. These results attest to the efficacy of the proposed compact formulations. In particular, we show that the proposed formulations consistently outperform compact formulations from the literature.

On mixing sets arising in chance-constrained programming

The mixing set with a knapsack constraint arises in deterministic equivalent of chance-constrained programming problems with finite discrete distributions. We first consider the case that the chance-constrained program has equal probabilities for each scenario. We study the resulting mixing set with a cardinality constraint and propose facet-defining inequalities that subsume known explicit inequalities for this set. We extend these inequalities to obtain valid inequalities for the mixing set with a knapsack constraint. In addition, we propose a compact extended reformulation (with polynomial number of variables and constraints) that characterizes a linear programming equivalent of a single chance constraint with equal scenario probabilities. We introduce a blending procedure to find valid inequalities for intersection of multiple mixing sets. We propose a polynomial-size extended formulation for the intersection of multiple mixing sets with a knapsack constraint that is stronger than the original mixing formulation. We also give a compact extended linear program for the intersection of multiple mixing sets and a cardinality constraint for a special case. We illustrate the effectiveness of the proposed inequalities in our computational experiments with probabilistic lot-sizing problems.

A new approximation algorithm for the multilevel facility location problem

In this paper we propose a new integer programming formulation for the multilevel facility location problem and a novel 3-approximation algorithm based on LP-rounding. The linear program that we use has a polynomial number of variables and constraints, thus being more efficient than the one commonly used in the approximation algorithms for these types of problems.

Integral polyhedra related to integer multicommodity flows on a cycle

The integer multicommodity flow problem on a cycle (IMFC) is to find a feasible integral routing of given demands between κ pairs of nodes on a link-capacitated undirected cycle, which is known to be polynomially solvable. Along with integral polyhedra related to IMFC, this paper shows that there exists a linear program, with a polynomial number of variables and constraints, which solves IMFC. Using the results, we also present a compact polyhedral description of the convex hull of feasible solutions to a certain class of instances of IMFC whose number of variables and constraints is O ( κ ) , which in turn means that there exists a non-trivial special case for which a minimum cost integer multicommodity flow problem can be solved in polynomial time.

A New Mixed-Integer Programming Model for Harvest Scheduling Subject to Maximum Area Restrictions

Forest ecosystem management often requires spatially explicit planning because the spatial arrangement of harvests has become a critical economic and environmental concern. Recent research on exact methods has addressed both the design and the solution of forest management problems with constraints on the clearcut size, but where simultaneously harvesting two adjacent stands in the same period does not necessarily exceed the maximum opening size. Two main integer programming approaches have been proposed for this area restriction model. However, both encompass an exponential number of variables or constraints. In this work, we present a new integer programming model with a polynomial number of variables and constraints. Branch and bound is used to solve it. The model was tested with both real and hypothetical forests ranging from 45 to 1,363 polygons. Results show that the proposed model's solutions were within or slightly above 1% of the optimal solution and were obtained in a short computation time.