Francesco Carrabs | Searcharxiv

arXiv2026

Minimum cost network flow with interval capacities: The worst-case scenario

We study the problem of determining the worst optimal value and characterizing the corresponding worst-case scenarios in minimum cost network flow problems with interval uncertainty in arc capacities. In this setting, each capacity can take any value within its specified lower and upper bounds. We prove that computing the worst optimal value is a strongly NP-hard problem and remains NP-hard even when restricted to series-parallel graphs. Further, we propose a mixed-integer linear programming formulation that computes the exact worst optimal value, as well as a pseudopolynomial-time algorithm designed for the special case of series-parallel graphs. We also examine the structural properties of the most extremal worst-case scenarios and show that the arcs whose capacities are not fixed at their interval bounds form a forest. This result establishes an upper bound on the number of such arcs, which we show to be tight by constructing a class of instances in which the bound is attained. Finally, we investigate the more-for-less paradox in minimum cost network flow problems with interval capacities, which occurs in instances where increasing the required flow leads to a decrease in the worst-case optimal cost. We provide a general characterization of this phenomenon using augmenting paths and establish a stronger characterization for complete graphs. In addition, we discuss the properties of the cost matrices immune against the paradox and prove that deciding whether a given cost matrix has this property is a strongly co-NP-hard problem.

math.OC↗

arXiv2024

Advanced Kernel Search approach for the MST Problem with conflicts involving affinity detection and initial solution construction

The Minimum Spanning Tree Problem with Conflicts consists in finding the minimum conflict-free spanning tree of a graph, i.e., the spanning tree of minimum cost, including no pairs of edges that are in conflict. In this paper, we solve this problem using an enhanced Kernel Search method, which iteratively solves refined problem restrictions. Our approach addresses two central open questions in the kernel search literature: (1) how to determine the affinity between variables to ensure that the restricted problem contains variables that are as compatible as possible, meaning they are more likely to appear together in a feasible solution, and (2) how to construct an initial feasible solution quickly. To this end, we integrate the computation of independent sets from the conflict graph within the algorithm to detect affinities and effectively manage conflicts. Furthermore, we heuristically construct an initial starting point, significantly accelerating the computational process. Although our methodology is designed for MSTC, its principles could be extended to other combinatorial optimization problems with conflicts. Experimental results on benchmark instances demonstrate the efficiency and competitiveness of our approach compared to existing methods in the literature, achieving 17 new best-known values.

math.OC↗