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

TL;DR

This work investigates the worst-case behavior of minimum cost flow problems under interval uncertainty in arc capacities. It establishes strong NP-hardness for computing the worst-case optimum, even on series-parallel graphs, and provides an exact mixed-integer linear programming formulation along with a pseudopolynomial algorithm for series-parallel graphs. A key structural result shows that, at an extremal worst case, the arcs with interior capacities form a forest, bounding the number of such arcs by \(n-1\) with tightness demonstrated. The paper also analyzes the more-for-less paradox, offering a path-based characterization and a stronger result for complete graphs, and studies the computational hardness of deciding immunity of cost matrices to this paradox.

Abstract

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.

Figures (6)

• Figure 1: Reduction from the knapsack problem to the minimum cost flow with interval capacities. Numbers above the arcs are the capacities, numbers below the arcs are the costs.
• Figure 2: Minimum cost flow problem with interval capacities shown in \ref{['exa:lower:bound:n-1']}.
• Figure 3: An instance of the minimum cost flow problem with interval capacities admitting the more-for-less paradox (see \ref{['exa:paradox:simple']}).
• Figure 4: The more-for-less paradox in \ref{['exa:paradox:simple']}. For the requested flow $f=1$ we have $c_w = 12$(a) , while for $f=2$ we obtain $c_w = 4$(b).
• Figure 5: A more complex instance of the minimum cost flow problem with interval capacities admitting the more-for-less paradox (see \ref{['exa:paradox:complex']}).

Theorems & Definitions (45)

• Proposition 1
• proof
• Lemma 2
• proof
• Theorem 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof