The Ramsey number R(m,n) is the smallest order at which every red-blue edge coloring of a complete graph must contain a blue clique (a complete subgraph) of size m or a red clique of size n. Determining these numbers exactly is extremely hard, and even certifying a lower bound requires exhibiting an explicit coloring that … Read more
For decades, researchers have complained that contiguity constraints make problems like districting much harder than other combinatorial optimization problems. This has prompted the use of integer programming models with inexact contiguity constraints that may be fast in practice but are invalid in the sense that they cut off or “overlook” some contiguous solutions. Examples include … Read more
The binary reflected Gray code orders the vertices of the unit hypercube along a Hamiltonian path in which consecutive vertices differ in exactly one coordinate. While Gray codes have been extensively studied from a combinatorial perspective, much less is known about the polyhedral structure of convex hulls of contiguous subpaths of this order. This paper … Read more
Many large-scale optimization problems decompose into a master problem and scenario subproblems, a structure that can be exploited by Benders decomposition. In Benders decomposition, each iteration may generate many cuts from scenario subproblems, and adding all of them as constraints then causes the master problem to grow rapidly. These are constraints that may need to … Read more
We present a machine learning (ML) component designed to enhance the well-known branch-and-cut (B&C) framework for the symmetric traveling salesman problem (TSP) in which only the subtour elimination constraints (SECs) violated by previously found integer solutions are considered. The objective of the ML component is to identify which SECs, from a pool of candidates, will … Read more
Two-Stage Vehicle Routing Problems with Stochastic Demands (VRPSDs) form a class of stochastic combinatorial optimization problems where routes are planned in advance, demands are revealed upon vehicle arrival, and recourse actions are triggered whenever capacity is exceeded. Following recent works, we consider VRPSDs where demands are given by an empirical probability distribution of scenarios. Existing … Read more
Convex maximization encompasses a broad class of optimization problems and is generally NP-hard, even for low-rank objectives. This paper investigates structural conditions under which convex maximization becomes polynomially solvable. From a geometric perspective, we introduce comonotonicity, a structural property of the feasible region crucial for problem tractability, and establish mathematical characterizations of this property. Under comonotonicity and … Read more
In Tower Defense (TD) games, the objective is to defend a specific point on the game map from mobile units by constructing towers with offensive capabilities. In this work, we focus on Bloons Tower Defense (Bloons TD), one of the earliest and most prominent TD games. We show that the problem of finding tower configurations … Read more
The goal of this survey is to provide a road-map for exploring the growing area of stochastic mixed-integer programming (SMIP) models and algorithms. We provide a comprehensive overview of existing decomposition algorithms for two-stage SMIPs, including Dantzig-Wolfe decomposition, dual decomposition, Lagrangian cuts, and decomposition approaches using parametric cutting planes and scaled cuts. Moreover, we explicitly … Read more
We consider a fundamental question in political districting: How many counties can be kept whole (i.e., not split across multiple districts), while satisfying basic criteria like contiguity and population balance? To answer this question, we propose integer programming techniques based on combinatorial Benders decomposition. The main problem decides which counties to keep whole, and the … Read more