arXiv subjects

Fabio Furini

Publications and source records attributed to Fabio Furini.

11 recordsLinked to original sources

arXiv2026

An Exact Algorithm for the Max-Min Covering Location Blocker Problem

We introduce the Max-Min Covering Location Blocker Problem, a bilevel optimization problem in which a leader blocks a minimum-cost set of candidate locations so that the optimal coverage of a budget-constrained follower does not exceed a prescribed target. Each customer's coverage is determined by the least favorable open facility that can serve it. The induced follower coverage set function is nonmonotone and nonsubmodular, so the validity of interdiction cuts cannot be inherited from existing frameworks. We develop an exact nested decomposition that exploits the max-min coverage structure at both levels. At the outer level, we establish the validity of interdiction cuts from this structure and strengthen them with facility-specific coefficients bounding the coverage lost when a critical facility is removed. We solve the NP-hard follower problem by branch-and-Benders-cut after projecting out the customer coverage variables, and characterize an optimal dual solution of the separation problem in closed form, generating Benders optimality cuts in linear time without solving a linear program. Computational experiments on benchmark instances show that both ingredients substantially improve the performance of the method. With a moderate number of candidate locations, instances with 300,000 customers are solved to optimality within one hour. A comparison with two general-purpose bilevel solvers from the literature shows the proposed method to be one to three orders of magnitude faster, and to solve to optimality instances that neither of them closes. The model applies wherever the planner cannot control which facility serves a customer, as in public automated external defibrillator (AED) networks. A case study on a network using real AED data from Virginia Beach illustrates the model.

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arXiv2026

Parametric maximum closure on precedence forests

The Maximum Closure Problem asks for a maximum-weight closed subset of a precedence-constrained set of vertices; when vertex weights depend affinely on a scalar parameter $λ$, as in open-pit mine scheduling, the goal becomes computing the optimal closed set for every value of $λ$ at once. We address this problem when the precedence graph is a directed forest. We present two algorithms (Peel-and-Contract, PaC, and its dual DPaC) that compute the full parametric solution -- a canonical sequence of disjoint closure layers inducing the nested optimal closures -- by iteratively aggregating vertices along precedence arcs, with time-complexity $\mathcal{O}(n^2)$ for general forests and space-complexity $\mathcal{O}(n)$; heap-based implementations (HPaC, DHPaC) have a worse time-complexity of $\mathcal{O}(n^2\log n)$, but on random instances their computing times empirically grow as $n\log n$. When the forest is restricted to in-trees or out-trees, specialized heap-based variants (HIPaC, HOPaC) achieve time-complexity $\mathcal{O}(n\log n)$. We then present a further aggregation-based algorithm (Rake-and-Compress, RaC) with time- and space-complexity $\mathcal{O}(n\log n)$ also for general forests, which matches the $Ω(n\log n)$ lower bound for comparison-based algorithms. Computational experiments on random and structured forests with up to $10^5$ vertices characterize the practical behavior of the proposed algorithms, identify the topologies on which HPaC and RaC are preferable, and compare HPaC with the public implementation of the fully parametric pseudoflow algorithm, the state of the art for arbitrary precedence graphs, which it outperforms on forests by one to more than three orders of magnitude.

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arXiv2026

A tight 1/3-approximation algorithm and fully polynomial-time approximation schemes for the Colored Knapsack Problem

The $\textit{Colored Knapsack Problem}$ (ColKP) generalizes the classical Knapsack Problem by partitioning the items into color classes and requiring the selected items to admit an ordering in which consecutive items have different colors. The problem is weakly $\mathcal{NP}$-hard and admits two pseudo-polynomial dynamic programming (DP) algorithms proposed in the literature. These two DP algorithms have worst-case running times $O(b \, n^4)$ and $O(b^2 \, n^3)$, respectively, where $b$ is the knapsack capacity and $n$ is the number of items. We develop the first approximation algorithm for the ColKP. By rounding an optimal basic solution of the linear programming relaxation of its natural integer programming formulation and repairing color feasibility, we obtain a linear-time approximation-algorithm whose worst-case performance ratio is $1/3$. We then reformulate both DP algorithms so that profit, rather than knapsack capacity, indexes their pseudo-polynomial dimension, and combine them with profit scaling to obtain two fully polynomial-time approximation schemes (FPTASs). The first FPTAS runs in $O(n^5/\varepsilon)$ time for nonnegative profits and in $O(n^6/\varepsilon)$ time for arbitrary integer profits. The second FPTAS runs instead in $O(n^5/\varepsilon^2)$ and $O(n^7/\varepsilon^2)$ time, respectively. The approximation guarantee, along with new structural insights, provides the bounds needed to control the scaled profit range and establish these running times.

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arXiv2026

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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arXiv2026

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 $α$-approximation algorithm with $α\leq 2$, we obtain a finer, component-wise guarantee with factor 2 for the bin-opening cost and factor $α$ for the setup-cost component.

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arXiv2026

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

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 avoids both cliques. We develop an integer programming framework for certifying such lower bounds, restricting the search to circulant graphs, whose rotational symmetry lets us reformulate the problem in a projected distance space, reducing the number of binary variables from quadratic to linear in the graph order. We strengthen this projected model through coefficient reduction and solve it with a branch-and-cut algorithm whose separation routine exploits the common neighborhood structure of circulant graphs, combining heuristic and exact maximum-clique algorithms. In an extensive computational campaign on circulant graphs with up to 410 vertices, we improve the best lower bounds previously obtained by other methods by up to 11 points for 25 values of $R(3,n)$ with $24\le n\le49$ and $n\neq27$, each backed by an explicit graph certificate that can be independently verified with a stand-alone exact clique solver. To the best of our knowledge, our method also provides the first reproducible optimization-based procedure for certifying circulant Ramsey numbers $R_C(m,n)$, which we use to establish eight new values of $R_C(3,n)$ with $13\le n\le20$. Our framework, graph certificates, and stand-alone checker are provided as supplementary material to support independent verification and reuse.

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arXiv2026

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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arXiv2026

Advancing Branch-and-Price for Graph Coloring: New Pricing Strategies and Benchmark Results

This paper proposes BPCOL+, an exact branch-and-price algorithm for the Graph Coloring Problem. The algorithm integrates enhanced pricing strategies based on Zero-Suppressed Binary Decision Diagrams (ZDDs) for solving 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 procedures and column generation, BPCOL+ reduces the size of the ZDD by applying maximal stable set reduction techniques that exploit alternative dual vectors. Computational experiments on the 137 DIMACS benchmark instances and on 5,000 recently proposed Erdős--Rényi instances show that BPCOL+ outperforms existing exact branch-and-price algorithms and remains highly competitive with state-of-the-art SAT-based exact solvers. In particular, BPCOL+ solves 96 DIMACS instances within one hour and proves optimality for 4,641 of the 5,000 Erdős--Rényi 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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arXiv2026

Hardness of some optimization problems over correlation polyhedra

We prove the \textbf{NP}-hardness, using Karp reductions, of some problems related to the correlation polytope and its corresponding cone, spanned by all of the $n\times n$ rank-one matrices over $\{0,1\}$. The problems are: membership, rank of the decomposition, and a ``relaxed rank'' obtained from relaxing the zero-norm expression for the rank to an $\ell_1$ norm. While membership and rank are natural problems for any matrix cone, the relaxed rank problem occurs in some signal processing and statistical applications.

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arXiv2019

A Branch-and-Price Algorithm for the Temporal Bin Packing Problem

We study an extension of the classical Bin Packing Problem, where each item consumes the bin capacity during a given time window that depends on the item itself. The problem asks for finding the minimum number of bins to pack all the items while respecting the bin capacity at any time instant. A polynomial-size formulation, an exponential-size formulation, and a number of lower and upper bounds are studied. A branch-and-price algorithm for solving the exponential-size formulation is introduced. An overall algorithm combining the different methods is then proposed and tested trough extensive computational experiments.

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