Integer Programming Formulation Research Articles

A New Binary Programming Formulation and Social Choice Property for Kemeny Rank Aggregation

Rank aggregation is widely used in group decision making and many other applications, where it is of interest to consolidate heterogeneous ordered lists. Oftentimes, these rankings may involve a large number of alternatives, contain ties, and/or be incomplete, all of which complicate the use of robust aggregation methods. In particular, these characteristics have limited the applicability of the aggregation framework based on the Kemeny-Snell distance, which satisfies key social choice properties that have been shown to engender improved decisions. This work introduces a binary programming formulation for the generalized Kemeny rank aggregation problem—whose ranking inputs may be complete and incomplete, with and without ties. Moreover, it leverages the equivalence of two ranking aggregation problems, namely, that of minimizing the Kemeny-Snell distance and of maximizing the Kendall-τ correlation, to compare the newly introduced binary programming formulation to a modified version of an existing integer programming formulation associated with the Kendall-τ distance. The new formulation has fewer variables and constraints, which leads to faster solution times. Moreover, we develop a new social choice property, the nonstrict extended Condorcet criterion, which can be regarded as a natural extension of the well-known Condorcet criterion and the Extended Condorcet criterion. Unlike its parent properties, the new property is adequate for handling complete rankings with ties. The property is leveraged to develop a structural decomposition algorithm, through which certain large instances of the NP-hard Kemeny rank aggregation problem can be solved exactly in a practical amount of time. To test the practical implications of the new formulation and social choice property, we work with instances constructed from a probabilistic distribution and with benchmark instances from PrefLib, a library of preference data.

An efficient algorithm for st-Connected Vertex Separator problem

In this paper, we give some basic notions concerning the st-connected vertex separator problem(st-CVS problem), then we give mathematical formula for st-CVS problem, then some special cases on some types of graphs, after then introduce an integer programming formulations for this problem. Also, we introduce a heuristic to solve this problem.

Multi-trip heterogeneous vehicle routing problem coordinated with production scheduling: Memetic algorithm and simulated annealing approaches

This paper studies a joint production scheduling and outbound distribution planning problem in which orders are processed by several machines in series with the same sequence known as permutation flow shop and are then distributed to destined customers in a batch by multiple heterogeneous capacitated vehicles, all of which are allowed to take multiple trips. When order is placed by a customer due date is prescribed as well. Due to the limited number of vehicles, orders need to be combined in a trip to decrease the total travelling time needed by the vehicles but in which case some of the customer orders can be tardy resulting from late deliveries. The objective of the problem is therefore to determine the minimum total tour time traversed by the vehicles and tardiness that might result from late deliveries. We present a mixed integer programming (MIP) formulation. But, due to intractability matters, optimal solution of the MIP formulation can only be obtained for small sized instances. Therefore, we also propose a Memetic Algorithm (MA) with the use of a new splitting procedure based on Prins’ algorithm for considering both total tardiness and total tour time and Simulated Annealing (SA) in order to offer favorable solutions in a reasonable time. Computational experiments are performed to assess the effectiveness of the proposed algorithms on a large number of test instances, which are randomly generated. Computational results indicate that both MA and SA can find optimal solutions for small sized instances in quite short time while MA outperforms SA in terms of solution quality for medium and big sized instances.

The Star Degree Centrality Problem: A Decomposition Approach

We consider the problem of identifying the induced star with the largest cardinality open neighborhood in a graph. This problem, also known as the star degree centrality (SDC) problem, is shown to be [Formula: see text]-complete. In this work, we first propose a new integer programming (IP) formulation, which has a smaller number of constraints and nonzero coefficients in them than the existing formulation in the literature. We present classes of networks in which the problem is solvable in polynomial time and offer a new proof of [Formula: see text]-completeness that shows the problem remains [Formula: see text]-complete for both bipartite and split graphs. In addition, we propose a decomposition framework that is suitable for both the existing and our formulations. We implement several acceleration techniques in this framework, motivated by techniques used in Benders decomposition. We test our approaches on networks generated based on the Barabási–Albert, Erdös–Rényi, and Watts–Strogatz models. Our decomposition approach outperforms solving the IP formulations in most of the instances in terms of both solution time and quality; this is especially true for larger and denser graphs. We then test the decomposition algorithm on large-scale protein–protein interaction networks, for which SDC is shown to be an important centrality metric.Summary of Contribution: In this study, we first introduce a new integer programming (NIP) formulation for the star degree centrality (SDC) problem in which the goal is to identify the induced star with the largest open neighborhood. We then show that, although the SDC can be efficiently solved in tree graphs, it remains [Formula: see text]-complete in both split and bipartite graphs via a reduction performed from the set cover problem. In addition, we implement a decomposition algorithm motivated by Benders decomposition together with several acceleration techniques to both the NIP formulation and the existing formulation in the literature. Our experimental results indicate that the decomposition implementation on the NIP is the best solution method in terms of both solution time and quality.

College admissions with ties and common quotas: Integer programming approach

Admission to universities is organised in a centralised scheme in Hungary. In this paper we investigate two major specialities of this application: ties and common quotas. A tie occur when some students have the same score at a programme. If not enough seats are available for the last tied group of applicants at a programme then there are three reasonable policies used in practice: 1) all must be rejected, as in Hungary 2) all can be accepted, as in Chile 3) a lottery decides which students are accepted from this group, as in Ireland. Even though student-optimal stable matchings can be computed efficiently for each of the above three cases, we developed (mixed) integer programming (IP) formulations for solving these problems, and compared the solutions obtained by the three policies for a real instance of the Hungarian application from 2008. In the case of Hungary common quotas arise from the faculty quotas imposed on their programmes and from the national quotas set for state-financed students in each subject. The overlapping structure of common quotas makes the computational problem of finding a stable solution NP-hard, even for strict rankings. In the case of ties and common quotas we propose two reasonable stable solution concepts for the Hungarian and Chilean policies. We developed (mixed) IP formulations for solving these stable matching problems and tested their performance on the large scale real instance from 2008 and also for one from 2009 under two different assumptions. We demonstrate that the most general case is also solvable in practice by IP technique.

An Analysis Framework for Examination Timetabling

An examination timetabling problem taken from real world universities was proposed at the International Timetabling Competition (ITC2007). The aim was to establish a common base for comparing different solution approaches. This paper presents new preprocessing methods that disclose hidden constraints and significantly increase the number of new edges that can be added to the conflict graph. Results show that the size of the maximum clique of the obtained conflict graph has been more than doubled for two instances as a result of our preprocessing. These larger cliques mean that instances can be analyzed in advance of a solution and end users gain useful information for making decisions. In addition, we have looked at the different criteria that compose the objective function, in order to provide more useful insights into the difficulty of problems in practice. We propose new integer programming formulations using clique inequalities to compute optimal solutions for 4 criteria and to obtain lower bounds for the 3 others. Results are presented and discussed for all the benchmark instances.

Combinatorial Heuristics for Inventory Routing Problems

We consider the deterministic inventory routing problem over a discrete finite time horizon. Given clients on a metric, each with daily demands that must be delivered from a depot and holding costs over the planning horizon, an optimal solution selects a set of daily tours through a subset of clients to deliver all demands before they are due and minimizes the total holding and tour routing costs over the horizon. In the capacitated case, a limited number of vehicles are available, where each vehicle makes at most one trip per day. Each trip from the depot is allowed to carry a limited amount of supply to deliver. We develop fast heuristics for both cases by solving a family of prize-collecting Steiner tree instances. Computational experiments show our heuristics can find near-optimal solutions for both cases and substantially reduce the runtime compared with a pure mixed integer programming formulation approach.

Parliament seating assignment problems

Motivated by evidence that parliament seatings are relevant for decision making, we consider the problem to assign seats in a parliament to members of parliament. We prove that the resulting seating assignment problem is strongly NP-hard in several restricted settings. We present a Mixed Integer Programming formulation of the problem, we describe two families of valid inequalities and we discuss symmetry-breaking constraints. Further, we design a heuristic. Finally, we compare the outcomes of the Mixed Integer Programming formulation with the outcomes of the heuristic in a computational study.

A New Formulation for the Dial-a-Ride Problem

This paper proposes a new mixed integer programming formulation and branch and cut (BC) algorithm to solve the dial-a-ride problem (DARP). The DARP is a route-planning problem where several vehicles must serve a set of customers, each of which has a pickup and delivery location, and includes time window and ride time constraints. We develop “restricted fragments,” which are select segments of routes that can represent any DARP route. We show how to enumerate these restricted fragments and prove results on domination between them. The formulation we propose is solved with a BC algorithm, which includes new valid inequalities specific to our restricted fragment formulation. The algorithm is benchmarked on existing and new instances, solving nine existing instances to optimality for the first time. In comparison with current state-of-the-art methods, run times are reduced between one and two orders of magnitude on large instances.

Reentrant flow shop with identical jobs and makespan criterion

The article focuses on the problem of a reentrant flow shop with identical jobs and makespan minimisation objective. After presenting an optimal integer programming formulation for the problem, we consider the case of unit processing times. We correct two results published a few decades ago regarding two greedy heuristic algorithms. In some cases, these algorithms lead to an optimal solution, but they are generally not optimal. We point out the advantages of each of the two heuristic algorithms and, by machine learning analysis, discuss features that differentiate between them.

The Bipartite Boolean Quadric Polytope

We consider the Bipartite Boolean Quadratic Programming Problem (BQP01), which generalizes the well-known Boolean quadratic programming problem (QP01). The model has applications in graph theory, matrix factorization and bioinformatics, among others. The primary focus of this paper is on studying the structure of the Bipartite Boolean Quadric Polytope (BQPm,n) resulting from a linearization of a quadratic integer programming formulation of BQP01.We present some basic properties and partial relaxations of BQPm,n, as well as some families of facets and valid inequalities. We find facet-defining inequalities including a family of odd-cycle inequalities. We discuss various approaches to obtain a valid inequality and facets from those of the related Boolean quadric polytope. The key strategy is based on rounding coefficients, and it is applied to the families of clique and cut inequalities in BQPm,n.

The Graphical Traveling Salesperson Problem has no integer programming formulation in the original space

The Graphical Traveling Salesperson Problem is the problem of assigning, for a given weighted graph, a nonnegative number xe to each edge e such that the induced multi-subgraph is of minimum weight among those that are spanning, connected and Eulerian. Known MIP formulations are based on integer variables xe. Carr and Simonetti (IPCO 2021) showed that unless NP=coNP, no (reasonable) formulation can have integrality constraints only on x-variables, a challenge posed by Denis Naddef. We establish the same result unconditionally.

Constraint programming approaches for the discretizable molecular distance geometry problem

AbstractThe Distance Geometry Problem (DGP) seeks to find positions for a set of points in geometric space when some distances between pairs of these points are known. The so‐called discretization assumptions allow us to discretize the search space of DGP instances. In this paper, we focus on a key subclass of DGP, namely the Discretizable Molecular DGP, and study its associated graph vertex ordering problem, the Contiguous Trilateration Ordering Problem (CTOP), which helps solve DGP. We propose the first constraint programming formulations for CTOP, as well as a set of checks for proving infeasibility, domain reduction techniques, symmetry breaking constraints, and valid inequalities. Our computational results on random and pseudo‐protein instances indicate that our formulations outperform the state‐of‐the‐art integer programming formulations.

A tailored Benders decomposition approach for last-mile delivery with autonomous robots

This work addresses an operational problem of a logistics service provider that consists of finding an optimal route for a vehicle carrying customer parcels from a central depot to selected facilities, from where autonomous devices like robots are launched to perform last-mile deliveries. The objective is to minimize a tardiness indicator based on the customer delivery deadlines. This article provides a better understanding of how three major tardiness indicators can be used to improve the quality of service by minimizing the maximum tardiness, the total tardiness, or the number of late deliveries. We study the problem complexity, devise a unifying Mixed Integer Programming formulation and propose an efficient branch-and-Benders-cut scheme to deal with instances of realistic size. Numerical results show that this novel Benders approach with a tailored combinatorial algorithm for generating Benders cuts largely outperforms all other alternatives. In our managerial study, we vary the number of available facilities, the coverage radius of autonomous robots and their speed, to assess their impact on the quality of service and environmental costs.

Novel centrality metrics for studying essentiality in protein‐protein interaction networks based on group structures

AbstractIn this work, we introduce centrality metrics based on group structures, and we show their performance in estimating importance in protein‐protein interaction networks (PPINs). The centrality metrics introduced are extensions of well‐known nodal metrics. However, instead of focusing on a single node, we focus on that node and the set of nodes around it. Furthermore, we require the set of nodes to induce a specific pattern or structure. The structures investigated range from the “stricter“ induced stars and cliques, to a “looser” definition of a representative structure. We derive the computational complexity of all metrics and provide mixed integer programming formulations; due to the problem complexity and the size of PPINs, using commercial solvers is not always viable. Hence, we propose a combinatorial branch‐and‐bound solution approach. We conclude by showing the effectiveness of the proposed metrics in identifying essential proteins inHelicobacter pyloriand comparing them to nodal metrics.

Multistage robust discrete optimization via quantified integer programming

Decision making needs to take an uncertain environment into account. Over the last decades, robust optimization has emerged as a preeminent method to produce solutions that are immunized against uncertainty. The main focus in robust discrete optimization has been on the analysis and solution of one- or two-stage problems, where the decision maker has limited options in reacting to additional knowledge gained after parts of the solution have been fixed. Due to its computational difficulty, multistage problems beyond two stages have received less attention.In this paper we argue that multistage robust discrete problems can be seen through the lens of quantified integer programs, where powerful tools to reduce the search tree size have been developed. By formulating both integer and quantified integer programming formulations, it is possible to compare the performance of state-of-the-art solvers from both worlds. Using selection, assignment, lot-sizing and knapsack problems as a testbed, we show that problems with up to nine stages can be solved to optimality in reasonable time.

Electric bus fleet composition and scheduling

The low energy density of batteries and the long recharging times constitute a significant barrier for electrification of public transportation (PT) systems since electric buses (EB) require too heavy and expensive batteries to achieve the operational availability of their combustion engine counterparts. New recharging technologies such as fast chargers and dynamic wireless power transfer (DWPT) emerge as promising solutions to overcome these challenges. Optimizing the bus fleet composition and the schedules is essential to take advantage of these emerging technologies and achieve electrification of PT in a cost-efficient way. To address this challenge, this paper proposes an integer (binary) programming formulation to find the optimal electric bus fleet composition and scheduling that minimizes the total procurement cost of the buses and the operating cost of the schedules. A column generation (CG) approach is devised to obtain provably high-quality solutions, for large problem instances. The success of the approach is due to a novel dynamic programming algorithm we develop to solve the generalized resource-constrained shortest path problem that needs to be solved in each CG iteration to find out new schedules to include in the model. Extensive computational studies on large real-world PT networks attest to the efficacy of the suggested methodology and reveal valuable managerial insights from a system-wide perspective.

Mixed integer programming formulation for fault identification based on MicroPMUs

Mixed integer programming formulation for fault identification based on <scp>MicroPMUs</scp>

Solving the minimum convex partition of point sets with integer programming

The partition of a problem into smaller sub-problems satisfying certain properties is often a key ingredient in the design of divide-and-conquer algorithms. For questions related to location, the partition problem can be modeled, in geometric terms, as finding a subdivision of a planar map – which represents, say, a geographical area – into regions subject to certain conditions while optimizing some objective function. In this paper, we investigate one of these geometric problems known as the Minimum Convex Partition Problem (mcpp ). A convex partition of a point set P in the plane is a subdivision of the convex hull of P whose edges are segments with both endpoints in P and such that all internal faces are empty convex polygons. The mcpp is an NP-hard problem where one seeks to find a convex partition with the least number of faces.We present a novel polygon-based integer programming formulation for the mcpp, which leads to better dual bounds than the previously known edge-based model. Moreover, we introduce a primal heuristic, a branching rule and a pricing algorithm. The combination of these techniques leads to the ability to solve instances with twice as many points as previously possible while constrained to identical computational resources. A comprehensive experimental study is presented to show the impact of our design choices.

An integer programming approach for solving a generalized version of the Grundy domination number

A legal dominating sequence of a graph is an ordered dominating set of vertices where each element dominates at least another one not dominated by its predecessors in the sequence. The length of a largest legal dominating sequence is called Grundy domination number. In this work, we introduce a generalized version of the Grundy domination problem. We explicitly calculate the corresponding parameter for paths and web graphs. We propose integer programming formulations for the new problem, find families of valid inequalities and perform extensive computational experiments to compare the formulations as well as to test these inequalities as cuts in a branch-and-cut framework. We also design and evaluate the performance of a heuristic for finding good initial lower and upper bounds and a tabu search that improves the initial lower bound. The test instances include randomly generated graphs, structured graphs, classical benchmark instances and two instances from a real application. Our approach is exact for graphs with 20–50 vertices and provides good solutions for graphs up to 10000 vertices.