arXiv subjects

Vítor Gomes Chagas

Publications and source records attributed to Vítor Gomes Chagas.

3 recordsLinked to original sources

arXiv2026

The Quadratic Bin Packing Problem: Exact Formulations and Algorithm

In this article, we introduce and study the Quadratic Bin Packing Problem (QBPP), which generalizes the classical bin packing problem by introducing a fixed cost for each used bin and a pairwise cost (or profit) incurred whenever two items are packed together. Beyond its theoretical relevance, the QBPP is of practical interest due to its numerous real-world applications, mainly related to cluster analysis. To address the QBPP, we propose three compact mixed-integer linear programming (MILP) formulations, along with a set-partitioning formulation. For each compact model, we present an enhanced version with a strengthened continuous relaxation, while, for the set-partitioning formulation, we develop a tailored Branch-and-Price algorithm. Computational experiments on benchmark instances demonstrated that, while the enhanced compact formulations can be effectively solved by a standard MILP solver for small-sized instances, the Branch-and-Price approach delivered superior performance overall, especially on larger and more challenging instances.

math.OC↗

arXiv2025

Realizing Graphs with Cut Constraints

Given a finite non-decreasing sequence $d=(d_1,\ldots,d_n)$ of natural numbers, the Graph Realization problem asks whether $d$ is a graphic sequence, i.e., there exists a labeled simple graph such that $(d_1,\ldots,d_n)$ is the degree sequence of this graph. Such a problem can be solved in polynomial time due to the Erdős and Gallai characterization of graphic sequences. Since vertex degree is the size of a trivial edge cut, we consider a natural generalization of Graph Realization, where we are given a finite sequence $d=(d_1,\ldots,d_n)$ of natural numbers (representing the trivial edge cut sizes) and a list of nontrivial cut constraints $\mathcal{L}$ composed of pairs $(S_j,\ell_j)$ where $S_j\subset \{v_1,\ldots,v_n\}$, and $\ell_j$ is a natural number. In such a problem, we are asked whether there is a simple graph with vertex set $V=\{v_1,\ldots,v_n\}$ such that $v_i$ has degree $d_i$ and $\partial(S_j)$ is an edge cut of size $\ell_j$, for each $(S_j,\ell_j)\in \mathcal{L}$. We show that such a problem is polynomial-time solvable whenever each $S_j$ has size at most three. Conversely, assuming P $\neq$ NP, we prove that it cannot be solved in polynomial time when $\mathcal{L}$ contains pairs with sets of size four, and our hardness result holds even assuming that each $d_i$ of $d$ equals $1$.

cs.CC↗

arXiv2024

A Framework for Efficient Approximation Schemes on Geometric Packing Problems of $d$-dimensional Fat Objects

We investigate approximation algorithms for several fundamental optimization problems on geometric packing. The geometric objects considered are very generic, namely $d$-dimensional convex fat objects. Our main contribution is a versatile framework for designing efficient approximation schemes for classic geometric packing problems. The framework effectively addresses problems such as the multiple knapsack, bin packing, multiple strip packing, and multiple minimum container problems. Furthermore, the framework handles additional problem features, including item multiplicity, item rotation, and additional constraints on the items commonly encountered in packing contexts. The core of our framework lies in formulating the problems as integer programs with a nearly decomposable structure. This approach enables us to obtain well-behaved fractional solutions, which can then be efficiently rounded. By modeling the problems in this manner, our framework offers significant flexibility, allowing it to address a wide range of problems and incorporate additional features. To the best of our knowledge, prior to this work, the known results on approximation algorithms for packing problems were either highly fixed for one problem or restricted to one class of objects, mainly polygons and hypercubes. In this sense, our framework is the first result with a general toolbox flavor in the context of approximation algorithms for geometric packing problems. Thus, we believe that our technique is of independent interest, being possible to inspire further work on geometric packing.

cs.CG↗