Ordering a Sparse Graph to Minimize the Sum of Right Ends of Edges
Motivated by a warehouse logistics problem we study mappings of the vertices of a graph onto prescribed points on the real line that minimize the sum (or equivalently, the average) of the coordinates of the right ends of all edges. We focus on graphs whose edge numbers do not exceed the vertex numbers too much, that is, graphs with few cycles. Intuitively, dense subgraphs should be placed early in the ordering, in order to finish many edges soon. However, our main ""calculation trick"" is to compare the objective function with the case when (almost) every vertex is the right end of exactly one edge. The deviations from this case are described by ""charges"" that can form ""dipoles"". This reformulation enables us to derive polynomial algorithms and NP-completeness results for relevant special cases, and FPT results.