The $p$-center problem (pCP) is a fundamental problem in location science, where we are given customer demand points and possible facility locations, and we want to choose $p$ of these locations to open a facility such that the maximum distance of any customer demand point to its closest open facility is minimized. State-of-the-art solution approaches of pCP use its connection to the set cover problem to solve pCP in an iterative fashion by repeatedly solving set cover problems. The classical textbook integer programming (IP) formulation of pCP is usually dismissed due to its size and bad linear programming (LP)-relaxation bounds.
We present a novel solution approach that works on a new IP formulation that can be obtained by a projection from the classical formulation. The formulation is solved by means of branch-and-cut, where cuts for demand points are iteratively generated. Moreover, the formulation can be strengthened with combinatorial information to obtain a much tighter LP-relaxation. In particular, we present a novel way to use lower bound information to obtain stronger cuts. We show that the LP-relaxation bound of our strengthened formulation has the same strength as the best known bound in literature, which is based on a semi-relaxation.
Finally, we also present a computational study on instances from the literature with up to more than 700000 customers and locations. Our solution algorithm is competitive with highly sophisticated set-cover-based solution algorithms, which depend on various components and parameters.
We study the complexity of cutting planes and branching schemes from a theoretical point of view. We give some rigorous underpinnings to the empirically observed phenomenon that combining cutting planes and branching into a branch-and-cut framework can be orders of magnitude more efficient than employing these tools on their own. In particular, we give general conditions under which a cutting plane strategy and a branching scheme give a provably exponential advantage in efficiency when combined into branch-and-cut. The efficiency of these algorithms is evaluated using two concrete measures: nu…
Given an undirected graph $G=(V,E)$, the longest induced path problem (LIPP) consists of obtaining a maximum cardinality subset $W\subseteq V$ such that $W$ induces a simple path in $G$. In this paper, we propose two new formulations with an exponential number of constraints for the problem, together with effective branch-and-cut procedures for its solution. While the first formulation (cec) is based on constraints that explicitly eliminate cycles, the second one (cut) ensures connectivity via cutset constraints. We compare, both theoretically and experimentally, the newly proposed approaches …
The incorporation of cutting planes within the branch-and-bound algorithm, known as branch-and-cut, forms the backbone of modern integer programming solvers. These solvers are the foremost method for solving discrete optimization problems and thus have a vast array of applications in machine learning, operations research, and many other fields. Choosing cutting planes effectively is a major research topic in the theory and practice of integer programming. We conduct a novel structural analysis of branch-and-cut that pins down how every step of the algorithm is affected by changes in the parame…
Many relevant applications from diverse areas such as marketing, wildlife conservation, or defending critical infrastructure can be modeled as interdiction games. In this work, we introduce interdiction games whose objective is a monotone and submodular set function. Given a ground set of items, the leader interdicts the usage of some of the items of the follower in order to minimize the objective value achievable by the follower, who seeks to maximize a submodular set function over the uninterdicted items subject to knapsack constraints. We propose an exact branch-and-cut algorithm for these …
The paper investigates a dial-a-ride problem focusing on the residents of large cities. These individuals have the opportunity to use a wide variety of transportation modes. Because of this, ridepooling providers have to solve the tradeoff between a high pooling rate and a small detour for customers to be competitive. We provide a Branch-and-Cut algorithm for this problem setting and introduce a new technique using information about already fixed paths to identify infeasible solutions ahead of time and to improve lower bounds on the arrival times at customer locations. By this, we are able to …
In this paper, we describe a comprehensive algorithmic framework for solving mixed integer bilevel linear optimization problems (MIBLPs) using a generalized branch-and-cut approach. The framework presented merges features from existing algorithms (for both traditional mixed integer linear optimization and MIBLPs) with new techniques to produce a flexible and robust framework capable of solving a wide range of bilevel optimization problems. The framework has been fully implemented in the open-source solver MibS. The paper describes the algorithmic options offered by MibS and presents computatio…
A Branch-and-Cut Algorithm for the Optimal Design of Parking Lots with One-way and Two-way Lanes
Indian Institute of Science (IISc)（印度科学学院）；Centre for infrastructure, Sustainable Transportation, and Urban Planning (CiSTUP), Indian Institute of Science (IISc)（印度科学学院基础设施、可持续交通与城市规划中心）；Robert Bosch Centre for Cyber Physical Systems (RBCCPS), Indian Institute of Science (IISc)（印度科学学院罗伯特博世网络物理系统中心）
On solving integer bilevel optimization problems with a non-convex quadratic follower objective function using disjunctive cuts
Koç University（博阿兹奇大学）；University of Augsburg（奥格斯堡大学）；University of Michigan（密歇根大学）；Vienna University of Economics and Business（维也纳经济与商业大学）；Johannes Kepler University Linz（林茨约翰·开普勒大学）
Mixed Integer Programs (MIPs) model many optimization problems of interest in Computer Science, Operations Research, and Financial Engineering. Solving MIPs is NP-Hard in general, but several solvers have found success in obtaining near-optimal solutions for problems of intermediate size. Branch-and-Cut algorithms, which combine Branch-and-Bound logic with cutting-plane routines, are at the core of modern MIP solvers. Montanaro proposed a quantum algorithm with a near-quadratic speedup compared to classical Branch-and-Bound algorithms in the worst case, when every optimal solution is desired. …
Branch-and-cut is the most widely used algorithm for solving integer programs, employed by commercial solvers like CPLEX and Gurobi. Branch-and-cut has a wide variety of tunable parameters that have a huge impact on the size of the search tree that it builds, but are challenging to tune by hand. An increasingly popular approach is to use machine learning to tune these parameters: using a training set of integer programs from the application domain at hand, the goal is to find a configuration with strong predicted performance on future, unseen integer programs from the same domain. If the train…
A Pricing-for-Integrality Primal Heuristic for Branch-and-Price: An Application to the Electric Vehicle Routing Problem with Time Windows and Charging Time Slots
This paper studies the benefits of autonomous vehicles in ride-sharing platforms dedicated to serving commuting needs. It considers the Commute Trip Sharing Problem with Autonomous Vehicles (CTSPAV), the optimization problem faced by a reservation-based platform that receives daily commute-trip requests and serves them with a fleet of autonomous vehicles. The CTSPAV can be viewed as a special case of the Dial- A-Ride Problem (DARP). However, this paper recognizes that commuting trips exhibit special spatial and temporal properties that can be exploited in a branch and cut algorithm that levera…
This paper introduces a computational method for generating metric Travelling Salesman Problem (TSP) instances having a large integrality gap. The method is based on the solution of an integer programming problem, called IH-OPT, that takes as input a fractional solution of the Subtour Elimination Problem (SEP) on a TSP instance and computes a TSP instance having an integrality gap larger than or equal to the integrality gap of the first instance. The decision variables of IH-OPT are the entries of the TSP cost matrix, and the constraints are defined by the intersection of the metric cone with …
We investigate the theoretical complexity of branch-and-bound (BB) and cutting plane (CP) algorithms for mixed-integer optimization. In particular, we study the relative efficiency of BB and CP, when both are based on the same family of disjunctions. We extend a result of Dash to the nonlinear setting which shows that for convex 0/1 problems, CP does at least as well as BB, with variable disjunctions. We sharpen this by giving instances of the stable set problem where we can provably establish that CP does exponentially better than BB. We further show that if one moves away from 0/1 sets, this…
The orienteering problem is a route optimization problem which consists in finding a simple cycle that maximizes the total collected profit subject to a maximum distance limitation. In the last few decades, the occurrence of this problem in real-life applications has boosted the development of many heuristic algorithms to solve it. However, during the same period, not much research has been devoted to the field of exact algorithms for the orienteering problem. The aim of this work is to develop an exact method which is able to obtain optimality certification in a wider set of instances than wi…
How hard is learning to cut? Trade-offs and sample complexity
School of Operations Research and Information Engineering（运营研究与信息工程学院）；Cornell Tech, Cornell University（康奈尔科技与康奈尔大学）；Jacobs Technion-Cornell Institute（雅各布斯技术-康奈尔研究所）
We consider the vehicle routing problem with stochastic demands (VRPSD), a stochastic variant of the well-known VRP in which demands are only revealed upon arrival of the vehicle at each customer. Motivated by the significant recent progress on VRPSD research, we begin this paper by summarizing the key new results and methods for solving the problem. In doing so, we discuss the main challenges associated with solving the VRPSD under the chance-constraint and the restocking-based perspectives. Once we cover the current state-of-the-art, we introduce two major methodological contributions. First…
In contrast to the many continuous global optimization methods that assume the objective function and constraints are factorable, we study how to find globally maximal solutions to problems that are not factorable, focusing on a particular class of problems. Specifically, we develop a method for non-decreasing continuous submodular functions subject to constraints. We characterize the hypograph of such functions and develop a cutting plane algorithm that finds approximate solutions and bounds using an approximation of the convex hull of the hypograph. We also test a spatial branch-and-bound ap…
Solving Max-Cut to Global Optimality via Feasibility-Preserving Graph Neural Networks
Davidson School of Chemical Engineering, Purdue University（普渡大学化学工程大卫逊学院）；Faculty of Computer Science, RWTH Aachen University（亚琛工业大学计算机科学学院）；Jacobs Technion-Cornell Institute, Cornell Tech（技术学院-康奈尔研究所）
