arXiv subjects

Roberto Baldacci

Publications and source records attributed to Roberto Baldacci.

7 recordsLinked to original sources

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 Erd\H{o}s--R\'enyi 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 Erd\H{o}s--R\'enyi instances.

math.OC↗

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.

math.OC↗

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.

math.CO↗

arXiv2025

On a Variant of the Minimum Path Cover Problem in Acyclic Digraphs: Computational Complexity Results and Exact Method

The Minimum Path Cover (MPC) problem consists of finding a minimum-cardinality set of node-disjoint paths that cover all nodes in a given graph. We explore a variant of the MPC problem on acyclic digraphs (DAGs) where, given a subset of arcs, each path within the MPC should contain at least one arc from this subset. We prove that the feasibility problem is strongly NP-hard on arbitrary DAGs, but the problem can be solved in polynomial time when the DAG is the transitive closure of a path. Given that the problem may not always be feasible, our solution focuses on covering a maximum number of nodes with a minimum number of node-disjoint paths, such that each path includes at least one arc from the predefined subset of arcs. This paper introduces and investigates two integer programming formulations for this problem. We propose several valid inequalities to enhance the linear programming relaxations, employing them as cutting planes in a branch-and-cut approach. The procedure is implemented and tested on a wide range of instances, including real-world instances derived from an airline crew scheduling problem, demonstrating the effectiveness of the proposed approach.

cs.DM↗

arXiv2018

The Electric Two-echelon Vehicle Routing Problem

Two-echelon distribution systems are attractive from an economical standpoint and help to keep large vehicles out of city centers. Large trucks can be used to deliver goods to intermediate facilities in accessible locations, whereas smaller vehicles allow to reach the final customers. Due to their reduced size and emissions, companies consider using an electric fleet of terrestrian or aerial vehicles for last mile deliveries. Route planning in multi-tier logistics leads to notoriously difficult problems. This difficulty is accrued in the presence of an electric fleet, since each vehicle operates on a smaller range, and may require visits to charging stations. To study these challenges, we introduce the Electric Two-echelon Vehicle Routing Problem as a prototypical problem. We propose a large neighbourhood search metaheuristic as well as an exact mathematical programming algorithm, which uses decomposition techniques to enumerate promising first-level solutions, in conjunction with bounding functions and route enumeration for the second-level routes. These algorithms produce optimal or near-optimal solutions for the problem, and allow us to evaluate the impact of several defining features of optimized battery-powered distribution networks. We created representative E2EVRP benchmark instances to simulate realistic metropolitan areas. In particular, we observe that the detour miles due to recharging decrease proportionally to $1/ρ^x$ with $x \approx 5/4$ as a function of the charging stations density $ρ$; e.g., in a scenario where the density of charging stations is doubled, recharging detours are reduced by 58\%. Finally, we evaluate the trade-off between battery capacity and detour miles. This estimate is critical for strategic fleet-acquisition decisions, in a context where large batteries are generally more costly and less environment-friendly.

cs.DS↗

arXiv2018

Optimal Solution of Vehicle Routing Problems with Fractional Objective Function

This work proposes a first extensive analysis of the Vehicle Routing Problem with Fractional Objective Function (vrpfof). We investigate how the principal techniques used either in the context of fractional programming or in the context of vehicle routing problems interact. We present new dual and primal bounding procedures which have been incorporated in an exact method. The method proposed allows to extend specific variants of \vrp\ to their counterpart with a fractional objective function. Extensive numerical experiments prove the validity of our approach.

math.OC↗

arXiv2018

A Euclidean Algorithm for Binary Cycles with Minimal Variance

The problem is considered of arranging symbols around a cycle, in such a way that distances between different instances of a same symbol be as uniformly distributed as possible. A sequence of moments is defined for cycles, similarly to the well-known praxis in statistics and including mean and variance. Mean is seen to be invariant under permutations of the cycle. In the case of a binary alphabet of symbols, a fast, constructive, sequencing algorithm is introduced, strongly resembling the celebrated Euclidean method for greatest common divisor computation, and the cycle returned is characterized in terms of symbol distances. A minimal variance condition is proved, and the proposed Euclidean algorithm is proved to satisfy it, thus being optimal. Applications to productive systems and information processing are briefly discussed.

cs.DS↗