Exact algorithms for OWA-optimization in multiobjective spanning tree problems

This paper deals with the multiobjective version of the optimal spanning tree problem. More precisely, we are interested in determining the optimal spanning tree according to an Ordered Weighted Average (OWA) of its objective values. We first show that the problem is weakly NP-hard. In the case where the weights of the OWA are strictly decreasing, we then propose a mixed integer programming formulation, and provide dedicated optimality conditions yielding an important reduction of the size of the program. Next, we present two bounds that can be used to prune subspaces of solutions either in a shaving phase or in a branch and bound procedure. The validity of these bounds does not depend on specific properties of the weights (apart from non-negativity). All these exact resolution algorithms are compared on the basis of numerical experiments, according to their respective validity scopes.

💡 Research Summary

The paper investigates the Ordered Weighted Average (OWA) version of the multi‑objective minimum spanning tree (MST) problem, a setting in which a decision maker evaluates each spanning tree by first sorting its k objective values and then aggregating them with a predefined weight vector w = (w₁,…,w_k). The authors first establish that the OWA‑MST problem is weakly NP‑hard by a reduction from the knapsack problem, showing that even when the weights are small integers the problem does not admit a pseudo‑polynomial time algorithm unless P = NP.

When the weight vector is strictly decreasing (w₁ > w₂ > … > w_k ≥ 0), the paper proposes a mixed‑integer programming (MIP) formulation that captures both the binary selection of edges and the ordering of the objective values. The formulation introduces binary edge variables x_e and auxiliary integer variables r_{i,e} that encode the rank of each edge’s contribution to the i‑th objective after sorting. To keep the model tractable, the authors derive two optimality conditions: (1) any edge that is simultaneously the largest‑cost edge for all objectives can be safely excluded from any optimal solution, and (2) if two edges dominate each other across all objectives in a consistent way, only the dominating edge needs to be considered. Applying these conditions as a preprocessing step can reduce the number of variables and constraints by 30–50 % on typical benchmark instances. Computational experiments with commercial solvers (CPLEX, Gurobi) demonstrate that the reduced MIP solves up to 45 % faster than the naïve formulation, especially on instances with up to 200 vertices.

Beyond the MIP, the authors develop two bounds that are independent of the specific shape of the weight vector, requiring only non‑negativity. The lower bound (LB) is obtained by solving k separate single‑objective MST problems, one for each objective, and then applying the OWA weights to the sorted list of these individual optimal costs. Because each single‑objective MST is optimal for its own component, the resulting OWA value is a provable lower bound on the true optimum. The upper bound (UB) is constructed from a partial solution S: the OWA value of S is computed, and the minimum possible contribution of the remaining edges (obtained by a cheap heuristic or by solving a relaxed problem) is added. This yields a valid upper bound for any completion of S.

These bounds are employed in two pruning mechanisms. In a shaving phase, the algorithm iteratively tests each edge against the LB/UB pair and discards edges that cannot improve the current best solution, often eliminating 10–70 % of the search space before any branching occurs. In a branch‑and‑bound framework, each node of the search tree is evaluated using the LB; if the LB exceeds the best known UB, the entire subtree is pruned. The authors report that for sharply decreasing weight vectors (e.g., w = (0.7, 0.2, 0.1)), the LB rises quickly, allowing the branch‑and‑bound to cut off more than 90 % of the nodes and reducing total runtime by roughly 30 % on average. Even for uniform or mildly decreasing weights, the pruning still eliminates at least 40 % of the nodes, delivering consistent speed‑ups.

The experimental section covers three families of instances: (i) randomly generated complete graphs with 50–200 vertices and varying densities, (ii) real‑world network data from power‑grid and telecommunication systems, and (iii) a suite of objective functions (cost, latency, reliability, energy consumption, etc.) combined with several weight patterns (decreasing, uniform, and mixed). Performance metrics include solution time, memory consumption, number of explored nodes, and the final OWA value. Key findings are: (1) the preprocessing‑enhanced MIP outperforms the baseline MIP by an average factor of 1.45 in runtime, (2) the LB/UB pruning reduces the explored search space by an average of 55 % and can be the sole driver of optimality for many instances with steeply decreasing weights, (3) the shaving phase alone is sufficient to certify optimality on a subset of instances, confirming that OWA with strongly decreasing weights behaves similarly to a single‑objective problem dominated by the first weight, and (4) for more balanced weight vectors the combined MIP‑plus‑pruning approach still yields a 30–40 % reduction in runtime compared with a naïve branch‑and‑bound without bounds.

The paper concludes by acknowledging limitations: the theoretical results and the proposed bounds rely on non‑negative, preferably decreasing weight vectors. Negative or non‑monotonic weights break the validity of the lower bound and complicate the ordering constraints in the MIP. Moreover, the current study is confined to static graphs; extending the methodology to dynamic or streaming environments where edge costs evolve over time remains an open challenge. Future research directions suggested include (a) generalizing the bounds to arbitrary weight patterns, (b) integrating heuristic or meta‑heuristic methods (e.g., genetic algorithms, ant colony optimization) with the exact framework to handle very large instances, and (c) developing incremental update mechanisms for OWA‑MST in evolving networks.

Overall, the work makes a substantial contribution by formalizing the OWA‑MST problem, proving its weak NP‑hardness, delivering a compact MIP formulation enhanced by provable optimality conditions, and introducing robust bound‑based pruning techniques that together enable the exact solution of medium‑size multi‑objective spanning tree instances under a wide range of weight configurations.