Combinatorial Optimization

Symmetry-dependence in Rounding of a Convex Body

Categories Convex Optimization, Linear, Cone and Semidefinite Programming, Polyhedra Tags , , , ,

The symmetry measure of a convex body \(S\subset\mathbb{R}^n\) is given by: \(\mathrm{sym}(S):=\max\{\alpha\ge0:\ \mathrm{there\ exists}\ x\in S\ \mathrm{such\ that}\ -\alpha(S-x)\subseteq S-x\}\), where such an \(x\) is called a Minkowski center. We prove that every convex body \(S\) admits a \(\sqrt{\frac{n}{\mathrm{sym}(S)}}\)-rounding of \(S\), namely, there exists an origin-centered ellipsoid \(E\) and a center \(c\) such that the … Read more

Variable Selection for Feature-Based Newsvendor

Categories Applications - OR and Management Sciences, Combinatorial Optimization, Optimization in Data Science

Feature-based newsvendor models use observable covariates to tailor inventory decisions, aiming to balance holding and shortage costs under demand uncertainty. However, high-dimensional feature sets often hinder interpretability and inflate data collection and implementation costs. This paper studies variable selection for the feature-based newsvendor problem under a hard cardinality constraint on the number of selected features. … Read more

The subtle behavior of the facial distance

Categories Combinatorial Optimization, Convex and Nonsmooth Optimization, Polyhedra

We give a simple example that disproves the following 2015 conjecture of Lacoste-Julien and Jaggi concerning the pyramidal width (aka facial distance): The pyramidal width of a set of vertices is non-increasing when another vertex is added (assuming that all previous points remain vertices). In contrast to the recent example by Zhao (arXiv:2607.29555), our counterexample … Read more

The Integrality Gap of the Traveling Salesman Problem is 4/3 if the LP Solution Has at Most n+8 Non-Zero Components

Categories 0-1 Programming, Linear Programming, Polyhedra Tags , ,

We address the classical Dantzig – Fulkerson – Johnson formulation of the symmetric metric Traveling Salesman Problem and study the integrality gap of its linear relaxation, namely the Subtour Elimination Problem (SEP). This integrality gap is conjectured to be 4/3. We prove that, when solving a problem on n nodes, if the optimal SEP solution … Read more

An Integer Programming Approach to Compute Lower Bounds for Ramsey Numbers Using Circulant Graphs

Categories Branch and Cut Algorithms, Combinatorial Optimization, Integer Programming Tags , , , ,

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

A guide to inexact contiguity constraints

Categories Combinatorial Optimization, Integer Programming, Network Optimization Tags , , , ,

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

Indicator Cuts for Benders Decomposition with Mixed-Integer Subproblems

Categories (Mixed) Integer Linear Programming, Energy, Polyhedra Tags , , , ,

Classical Benders decomposition fails when the subproblem is a mixed-integer program, due to the absence of strong duality. We propose a novel class of dual-free indicator cuts that are applicable to all Benders-decomposable problems with a pure-integer master problem and mixed-integer linear programming (MILP) subproblems. These cuts are derived from the monotonicity property of the … Read more

An Exact Branch-and-Cut Method for the Asymmetric Orienteering Problem with Terrain-Driven Fatigue

Categories (Mixed) Integer Linear Programming, Branch and Cut Algorithms, Transportation

We present an end-to-end computational framework for provably optimal route planning in orienteering races on real terrain. The framework consists of a GIS pipeline that combines International Orienteering Federation (IOF) map symbols, a digital elevation model (DEM), and the Minetti metabolic model into a fully asymmetric cost matrix, where directional slope costs and cumulative fatigue … Read more

A Quantum Optimization Framework for Data-Assimilation-Augmented Parameter Estimation

Categories Combinatorial Optimization, Nonlinear Optimization, Unconstrained Optimization Tags , , , ,

Parameter estimation is a fundamental challenge in the calibration of ordinary differential equation (ODE) models, where repeated numerical integration can lead to high computational cost. In this work, we investigate whether quantum algorithms can be leveraged to assist parameter estimation in nonlinear dynamical systems. We develop a hybrid classical–quantum framework that reformulates a data-assimilation-augmented parameter … Read more

Second shortest simple paths in directed graphs: a crossing decomposition and a span-adaptive exact algorithm

Categories Combinatorial Optimization, Graphs and Matroids, Network Optimization Tags , , , ,

We study the computation of a second shortest simple \(s\)–\(t\) path (2-SP) in a directed graph with \(n\) nodes, \(m\) arcs and nonnegative integer arc costs bounded by \(C\). Working with reduced costs and a depth-first search that gives priority to a fixed shortest path \(P_{st}\), every candidate second path is a prefix of \(P_{st}\), … Read more