arXiv subjects

Matheus J. Ota

Publications and source records attributed to Matheus J. Ota.

6 recordsLinked to original sources

arXiv2026

On vehicle routing problems with stochastic demands -- Scenario-optimal recourse policies

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 approaches rely on integer L-shaped (ILS) cuts, whose coefficients are tailored for specific recourse policies. In contrast, we propose a framework that casts recourse policies as solutions of a higher-dimensional mixed-integer program, and we characterize its convex hull in the original lower-dimensional space via a new class of inequalities called scenario recourse inequalities (SRIs). We show that SRIs are valid for any recourse policy satisfying mild assumptions and are sufficient for formulating the VRPSD under a scenario-optimal recourse policy, where the recourse actions are chosen optimally for each scenario. Under this latter policy, we also demonstrate that SRIs dominate several known classes of ILS cuts. We conduct computational experiments on the VRPSD with scenarios under both the classical and the scenario-optimal recourse policies. By using the SRIs, our algorithm solves 329 more instances to optimality than the previous state-of-the-art ILS algorithm.

math.OC↗

arXiv2026

On vehicle routing problems with stochastic demands -- Generic disaggregated integer L-shaped formulations

We study the vehicle routing problem with stochastic demands (VRPSD), an important variant of the classical capacitated vehicle routing problem in which customer demands are modeled as random variables. We develop the first algorithm for the VRPSD in the case where the demands are given by an empirical probability distribution of scenarios -- a data-driven variant that tackles a significant challenge identified in the literature: dealing with correlations. Indeed, most previous exact algorithms for this problem relied on independence of the random variables. To address the VRPSD with scenarios, we introduce a unifying framework that generalizes existing integer L-shaped (ILS) formulations developed for other variants of the problem. This framework and subsequent analysis allow us to generalize previous ILS cuts and pinpoint which assumptions are needed to apply those generalizations. In particular, our results enable, for the first time, the combination of two previous types of inequalities: partial route and set cuts, which leads to significant computational improvements.

math.OC↗

arXiv2025

Representability of forests via generalized subtour elimination constraints

Generalized subtour elimination constraints (GSECs) are widely used in state-of-the-art exact algorithms for vehicle routing and network design problems, as their right-hand sides often capture problem-specific feasibility conditions of each solution component. In this work, we present the first characterization of the families of forests that can be represented as the integer points inside a polytope defined by GSECs. This result generalizes a recent framework developed for vehicle routing problems under uncertainty and broadens the applicability of GSEC-based formulations to a wider class of combinatorial problems. In particular, using our characterization, we recover vehicle routing formulations that could not be obtained with previous results. Additionally, we show that GSECs can naturally model a robust variant of the capacitated minimum spanning tree problem.

math.OC↗

arXiv2025

Approximating value functions via corner Benders' cuts

We introduce a novel technique to generate Benders' cuts from a conic relaxation ("corner") derived from a basis of a higher-dimensional polyhedron that we aim to outer approximate in a lower-dimensional space. To generate facet-defining inequalities for the epigraph associated to this corner, we develop a computationally-efficient algorithm based on a compact reverse polar formulation and a row generation scheme that handles the redundant inequalities. Via a known connection between arc-flow and path-flow formulations, we show that our method can recover the linear programming bound of a Dantzig-Wolfe formulation using multiple cuts in the projected space. In computational experiments, our generic technique enhances the performance of a problem-specific state-of-the-art algorithm for the vehicle routing problem with stochastic demands, a well-studied variant of the classic capacitated vehicle routing problem that accounts for customer demand uncertainty.

math.OC↗

arXiv2021

Approximation and parameterized algorithms to find balanced connected partitions of graphs

Partitioning a connected graph into $k$~vertex-disjoint connected subgraphs of similar (or given) orders is a classical problem that has been intensively investigated since late seventies. Given a connected graph $G=(V,E)$ and a weight function $w : V \to \mathbb{Q}_\geq$, a connected $k$-partition of $G$ is a partition of $V$ such that each class induces a connected subgraph. The balanced connected $k$-partition problem consists in finding a connected $k$-partition in which every class has roughly the same weight. To model this concept of balance, one may seek connected $k$-partitions that either maximize the weight of a lightest class $(\text{max-min BCP}_k)$ or minimize the weight of a heaviest class $(\text{min-max BCP}_k)$. Such problems are equivalent when $k=2$, but they are different when $k\geq 3$. In this work, we propose a simple pseudo-polynomial $\frac{k}{2}$-approximation algorithm for $\text{min-max BCP}_k$ which runs in time $\mathcal{O}(W|V||E|)$, where $W = \sum_{v \in V} w(v)$. Based on this algorithm and using a scaling technique, we design a (polynomial) $(\frac{k}{2} +\varepsilon)$-approximation for the same problem with running-time $\mathcal{O}(|V|^3|E|/\varepsilon)$, for any fixed $\varepsilon>0$. Additionally, we propose a fixed-parameter tractable algorithm based on integer linear programming for the unweighted $\text{max-min BCP}_k$ parameterized by the size of a vertex cover.

cs.DS↗

arXiv2019

Integer Programming Approaches to Balanced Connected $k$-Partition

We address the problem of partitioning a vertex-weighted connected graph into $k$ connected subgraphs that have similar weights, for a fixed integer $k\geq 2$. This problem, known as the \emph{balanced connected $k$-partition problem} ($BCP_k$), is defined as follows. Given a connected graph $G$ with nonnegative weights on the vertices, find a partition $\{V_i\}_{i=1}^k$ of $V(G)$ such that each class $V_i$ induces a connected subgraph of $G$, and the weight of a class with the minimum weight is as large as possible. It is known that $BCP_k$ is $NP$-hard even on bipartite graphs and on interval graphs. It has been largely investigated under different approaches and perspectives. On the practical side, $BCP_k$ is used to model many applications arising in police patrolling, image processing, cluster analysis, operating systems and robotics. We propose three integer linear programming formulations for the balanced connected $k$-partition problem. The first one contains only binary variables and a potentially large number of constraints that are separable in polynomial time. Some polyhedral results on this formulation, when all vertices have unit weight, are also presented. The other formulations are based on flows and have a polynomial number of constraints and variables. Preliminary computational experiments have shown that the proposed formulations outperform the other formulations presented in the literature.

cs.DM↗