On the approximability of minmax (regret) network optimization problems

In this paper the minmax (regret) versions of some basic polynomially solvable deterministic network problems are discussed. It is shown that if the number of scenarios is unbounded, then the problems under consideration are not approximable within $\log^{1-\epsilon} K$ for any $\epsilon>0$ unless NP $\subseteq$ DTIME$(n^{\mathrm{poly} \log n})$, where $K$ is the number of scenarios.

💡 Research Summary

The paper investigates the approximability of min‑max (also called min‑regret) versions of several classic network optimization problems when the underlying data are uncertain and modeled by multiple scenarios. In the deterministic setting, problems such as shortest‑path, minimum‑spanning‑tree, maximum‑flow, and minimum‑cost‑flow are solvable in polynomial time. The authors consider the robust counterpart in which a set of $K$ cost scenarios ${c^1,\dots,c^K}$ is given. For a feasible solution $x$ (e.g., a path, a tree, a flow), the regret under scenario $k$ is defined as $r^k(x)=c^k(x)-\min_{y\in\mathcal{F}}c^k(y)$, i.e., the excess cost compared to the optimal solution for that scenario. The objective of the min‑max regret problem is to choose $x$ that minimizes the worst‑case regret $\max_{k} r^k(x)$.

The central question is whether, when the number of scenarios $K$ is unbounded, any polynomial‑time algorithm can achieve a non‑trivial approximation ratio. The authors prove a strong negative result: unless $NP\subseteq DTIME(n^{\mathrm{poly}\log n})$ (a widely believed complexity‑theoretic collapse), no polynomial‑time algorithm can guarantee an approximation factor better than $\log^{1-\epsilon}K$ for any constant $\epsilon>0$. In other words, the approximation ratio deteriorates essentially as a logarithmic function of the number of scenarios, and this bound is tight up to lower‑order terms.

The proof proceeds via a gap‑preserving reduction from the classic Set‑Cover problem, which is known to be hard to approximate within a factor of $(1-o(1))\log n$ unless $P=NP$. The reduction constructs a network and a family of scenarios such that each element of the universe corresponds to a scenario, and each set in the cover corresponds to a selectable edge or sub‑structure in the network. Costs are assigned so that a solution with low worst‑case regret corresponds exactly to a small set cover, while any solution with large regret implies that any set cover must be large. The reduction preserves the approximation gap, yielding a lower bound of $\log^{1-\epsilon}K$ for the min‑max regret version of each considered network problem.

The authors apply this construction uniformly to several problems: shortest‑path, minimum‑spanning‑tree, maximum‑flow, and minimum‑cost‑flow. For each, they show that the robust min‑max regret formulation inherits the same hardness. Consequently, when $K$ grows polynomially with the input size, the robust problems become essentially inapproximable.

Beyond the theoretical contribution, the paper discusses practical implications. Since real‑world applications often involve many scenarios (e.g., demand forecasts, failure states, price variations), the results suggest that exact or even moderately good robust solutions are computationally infeasible without additional structure. Practitioners must therefore either limit the number of scenarios (e.g., via sampling or scenario reduction), exploit special properties of the underlying network (planarity, bounded treewidth, etc.), or resort to heuristic and meta‑heuristic methods that lack provable guarantees but work well in practice.

In summary, the paper establishes that the min‑max (regret) versions of several polynomially solvable network problems become dramatically harder when the scenario set is unrestricted. The logarithmic inapproximability bound ties the difficulty directly to the number of scenarios and aligns with known hardness results for Set‑Cover. This work clarifies the theoretical limits of robust network optimization and points to the necessity of scenario management or problem‑specific algorithmic tricks for any practical deployment.