arXiv subjects

Fabio Ciccarelli

Publications and source records attributed to Fabio Ciccarelli.

5 recordsLinked to original sources

arXiv2026

The colored knapsack problem: structural properties and exact algorithms

We introduce and study a novel generalization of the classical Knapsack Problem (KP), called the Colored Knapsack Problem (CKP). In this problem, the items are partitioned into classes of colors and the packed items need to be ordered such that no consecutive items are of the same color. We establish that the problem is weakly NP-hard and propose two exact dynamic programming algorithms with time complexities of $\mathcal{O}(bn^4)$ and $\mathcal{O}(b^2n^3)$, respectively. To enhance practical performance, we derive various dominance and fathoming rules for both approaches. From a theoretical perspective, we analyze the linear programming relaxation of the natural CKP formulation, proving that an optimal solution exists with at most two fractional items. We also show that the relaxation can be solved in $\mathcal{O}(n)$ time, matching the complexity of the classical KP. Finally, we establish a comprehensive benchmark of CKP instances, derived from the Colored Bin Packing Problem. Extensive computational experiments demonstrate that the proposed dynamic programming algorithms significantly outperform state-of-the-art MIP solvers on most of these instances.

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arXiv2026

Branch-and-price strikes back for the k-vertex cut problem

Given an undirected graph, the k-vertex cut problem (k-VCP) asks for a minimum-cost set of vertices whose removal yields at least k connected components in the resulting graph. The k-VCP is an important problem in network optimization, with applications in infrastructure protection and epidemic containment. We present a new extended integer linear programming (ILP) formulation that unifies and strengthens existing models and serves as the foundation for a new branch-and-price algorithm for the k-VCP. An in-depth theoretical study enables us to devise algorithmic components such as tailored branching rules that preserve the structure of the pricing problems, as well as valid inequalities and symmetry-handling techniques. We also show that our new model dominates all previous ILP formulations of the k-VCP in terms of their linear relaxations, which theoretically justifies the computational effectiveness of our approach. Extensive computational experiments against state-of-the-art methods demonstrate substantially improved performance, both in terms of instances solved to proven optimality and running times.

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arXiv2025

A Tight 2-Approximation Algorithm for the Bin Packing Problem with Setups

We study approximation algorithms for the Bin Packing Problem with Setups (BPPS), a generalization of the classical Bin Packing Problem (BPP) in which items are partitioned into classes and activating a class in a bin consumes a setup weight and incurs a setup cost. We show that direct adaptations of Next Fit (NF), First Fit (FF), Best Fit (BF), and Worst Fit (WF), as well as their decreasing-order variants, have unbounded absolute worst-case performance ratios, even with unit-weight items and zero setup costs. We then introduce a two-phase algorithm, $\mathrm{TP}_{\mathcal{A}}$ , that packs each class independently with a BPP algorithm and subsequently merges compatible packing patterns. We prove that the solution returned by $\mathrm{TP}_{\mathcal{A}}$ has cost at most twice the optimum under the assumption that produces pairwise merge-maximal solutions, i.e., such that no two packing patterns in the class-wise solution can be feasibly merged. If also runs in polynomial time, this yields a 2-approximation algorithm for the BPPS. The factor is tight: the absolute worst-case performance ratio of $\mathrm{TP}_{\mathcal{A}}$ is exactly 2, even when solves every class-wise BPP instance optimally. Since every Any Fit algorithm returns pairwise merge-maximal solutions, it follows that $\mathrm{TP}_{\mathrm{FF}}$ , $\mathrm{TP}_{\mathrm{BF}}$ , $\mathrm{TP}_{\mathrm{WF}}$ , and their decreasing-order variants all have an absolute worst-case performance ratio exactly 2. If, in addition, is an $\alpha$-approximation algorithm with $\alpha \leq 2$, we obtain a finer, component-wise guarantee with factor 2 for the bin-opening cost and factor $\alpha$ for the setup-cost component.

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arXiv2025

The Bin Packing Problem with Setups: Formulations, Structural Properties and Computational Insights

We introduce the Bin Packing Problem with Setups (BPPS), a generalization of the classical Bin Packing Problem with applications in production planning and logistics. In this problem, the items are partitioned into classes, and packing items of a class in a bin incurs a setup weight and cost. We propose a natural Integer Linear Programming (ILP) formulation for the BPPS and analyze its Linear Programming relaxation. We show that the resulting lower bound can be arbitrarily weak and introduce the Minimum Classes Inequalities (MCIs), which guarantee a worst-case ratio of 1/2 with respect to the optimal objective function value of the BPPS. We also derive the Minimum Bins Inequality (MBI) and an upper bound on the number of bins in any optimal solution, substantially reducing the formulation size. We further develop an arc-flow formulation for the BPPS based on a tailored graph construction and compression procedure. Its LP relaxation dominates that of the natural formulation, and both the MCIs and the MBI are extended to the arc-flow model. Finally, we introduce a benchmark comprising 576 randomly generated instances and 36 real-world instances derived from a vehicle-routing application, and conduct extensive computational experiments. Results show that the natural formulation performs best on instances with small or medium item weights, whereas the arc-flow formulation is more effective for large item weights and on the real-world testbed.

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arXiv2025

Strength of the Upper Bounds for the Edge-Weighted Maximum Clique Problem

We theoretically and computationally compare the strength of the three main upper bounds from the literature on the optimal value of the Edge-Weighted Maximum Clique Problem (EWMCP). We provide a set of instances for which the ratio between any of the three upper bounds and the optimal value of the EWMCP is unbounded, showing that none of them can give a performance guarantee. We further analyze the relative strength among the three upper bounds by determining, for every choice of a ratio between any two of them, the largest values it can attain and providing families of instances for which such values can be reached. Our results show that, for each pair of upper bounds, there exist appropriately chosen instances on which either bound is tighter than the other. Our theoretical analysis is complemented by extensive computational experiments on two benchmark datasets: the standard DIMACS instances and randomly generated instances, providing practical insights into the empirical strength of the upper bounds.

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