Tensions are functions \(p: E\to {\mathbb{R}}\) defined on the arc set E of a digraph G satisfying \(p(C^+)=p(C^-)\) for all cycles C of G, where \(C^+\) and \(C^-\) are the sets of forward arcs and backward arcs of C, respectively. Given upper bounds \(b_ e\) and costs \(a_ e\) on the edges \(e\in E\), a min cost tension minimizes the linear costs \(\sum p_ ea_ e\), while satisfying \(0\leq p_ e\leq b_ e\), \(e\in E\). Two algorithms to find such a tension, the negative cut algorithm and the shortest augmenting cut algorithm, are given. They represent analogues for the min cost tension problem of the algorithms of Klein (negative cycles) and of Busacker and Gowen (shortest augmenting paths) for the min cost flow problem. Both negative cuts and shortest augmenting cuts can be found by means of capacitated network flows. The author also sketches the use of the min cost tension problem for solving certain totally unimodular linear programs.