# Capacity inverse minimum cost flow problem

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## Abstract

Given a directed graph *G*=(*N*,*A*) with arc capacities *u* _{ ij } and a minimum cost flow problem defined on *G*, the *capacity inverse minimum cost flow problem* is to find a new capacity vector
\(\hat{u}\)
for the arc set *A* such that a given feasible flow
\(\hat{x}\)
is optimal with respect to the modified capacities. Among all capacity vectors
\(\hat{u}\)
satisfying this condition, we would like to find one with minimum
\(\|\hat{u}-u\|\)
value.

We consider two distance measures for
\(\|\hat{u}-u\|\)
, rectilinear (*L* _{1}) and Chebyshev (*L* _{∞}) distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is
\(\mathcal{NP}\)
-hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.

## Keywords

Inverse problems Network flows Minimum cost flows## Preview

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