Minimum concave cost flows in capacitated grid networks

We study the minimum concave cost flow problem over a two-dimensional grid network (CFG), where one dimension represents time periods and the other dimension represents echelons. The concave function over each arc is given by a value oracle. We give a characterization of the computational complexity of CFG based on the grid size (T periods and L echelons), the distribution of sources and sinks over the grid, and arc capacity values. For the capacitated case, we give a polynomial-time algorithm with fixed L, one echelon of sources and one echelon of sinks, and O(1) different capacity values, and a polynomial-time algorithm with two echelons, O(1) capacity values on arcs connecting echelons and arbitrary capacity on other arcs; These are likely the most general polynomially solvable cases since we show the problem becomes NP-hard if any conditions on the parameters is relaxed. For the uncapacitated case, we give a polynomial-time algorithm with fixed L and one echelon of sources (or sinks); we show that the problem is NP-hard if L is an input parameter or there are two echelons of sources and two echelons of sinks. Our algorithms and hardness results generalize complexity results for many variants of the lot-sizing problem, and answer several open questions on serial supply chains.