On Solving the Maximum Flow Problem with Conflict Constraints

TL;DR

This work tackles the Maximum Flow Problem with Conflicts (MFPC), which adds pairwise conflict constraints to the classical maximum flow and is NP-hard. It introduces a MILP model with flow variables $f_{ij}$, activation variables $x_{ij}$, and a total flow variable $z$, linking activity to capacity via $f_{ij}\le u_{ij}x_{ij}$ and enforcing conflicts with $x_{ij}+x_{kl}\le 1$. Solving the model with the open-source CP-SAT solver from Google OR-Tools, the authors evaluate on the standard 160-instance benchmark set, achieving complete optimality proofs for 28 instances for the first time and improving six previously best-known heuristics. On average, exact solutions occur in about 200 seconds, with minimal risk of timeouts, demonstrating that CP-SAT provides a competitive and accessible tool for MFPC. The results substantially advance the state-of-the-art in MFPC, offering stronger benchmarks and practical implications for conflict-constrained network optimization.

Abstract

The Maximum Flow Problem with Conflict Constraints is a generalization that adds conflict constraints to a classical optimization problem on networks used to model several real-world applications. In the last few years several approaches, both heuristic and exact, have been proposed to attack the problem. In this paper we consider a mixed integer linear program and solve it with an open-source solver. Computational results on the benchmark instances commonly used in the literature of the problem are reported. All the 160 instances benchmark instances normally used in the literature are solved to optimality for the first time, with 28 instances closed for the first time. Moreover, in the process, 6 improvements to the best-known heuristic solutions are also found.

Figures (1)

• Figure 1: On the left an example of a small MFPC instance is shown, where the arc capacities are placed by the arcs. Arcs in black are not affected by conflicts while any two arcs sharing the same red, blue or green color are in conflict. On the right a feasible solution carrying the maximal possible flow from $s$ to $t$ without violating any conflict is shown.