The integrality gap of the subtour elimination relaxation for the Traveling Salesman Problem is a longstanding open problem, epitomized by the \(\frac{4}{3}\)-conjecture. Understanding this gap requires a detailed analysis of the extreme points of the subtour elimination polytope. In this work, we introduce a new perspective for studying the integrality gap through what we call … Read more
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
Family medicine residency programs must balance educational and operational requirements while providing residents with consistent exposure to continuity clinics, a central principle of the Clinic First Model. We study the Family Medicine Residency Scheduling problem and develop a binary integer programming (BIP) framework that incorporates Clinic First principles through two criteria: Clinic Time Consistency (CTC), … Read more
We study cutting-plane generation for binary polynomial optimization (BPO), whose feasible region is the multilinear set of a hypergraph. Strong inequalities for this set—such as two-links, flowers, and odd $\beta$-cycles—are classically hand-derived for fixed support patterns. Instead, we propose a decision-diagram (DD) approach: for any chosen support, it separates a facet-defining cut in the local … Read more
We study the convex hull obtained after deleting prescribed vertices from the binary cube. The analysis separates three regimes according to the number of deleted vertices. When this number is fixed, both the original-space facet count and the linear extension complexity remain linear in the ambient dimension, up to constants depending only on the number … Read more
This paper proposes BPCOL+, an exact branch-and-price algorithm for the Graph Coloring Problem. The algorithm is both novel and highly effective, integrating enhanced pricing strategies within Zero-Suppressed Binary Decision Diagrams (ZDDs) to efficiently solve the pricing problem associated with the maximal-stable-set-based set-covering formulation. After computing upper and lower bounds at the root node using heuristic … Read more
We present an exact method for separating Chvátal-Gomory cuts from binary knapsack sets, consisting of two steps: i) enumerating a finite set of possible optimal multipliers for the knapsack constraint; ii) for each candidate, adjusting optimally the remaining multipliers. We prove that ii) can be formulated as a binary knapsack problem, leading to a pseudopolynomial-time … Read more
In this paper, we compare the strength of alternate formulations (polyhedra) of the binary knapsack set. We introduce a specific class of knapsack sets for which we prove that the polyhedra based on their minimal cover inequalities (together with the bounds on the variables) are strictly contained inside the polyhedra defined by their continuous knapsack … Read more
Probing is an important presolving technique in mixed-integer programming solvers. It selects binary variables, tentatively fixes them to 0 and 1, and performs propagation to deduce additional variable fixings, bound tightenings, substitutions, and implications. In this work, we propose clique probing instead of probing on individual variables, we select cliques, a set of binary variables … Read more
In this paper, we propose an extension of the classical weight reduction inequalities for the binary knapsack polytope for settings where the maximum-weight item in the associated pack is not unique. We derive sufficient conditions under which the extended inequalities are facet-defining and identify conditions under which they strictly dominate the original weight reduction inequalities. … Read more