On an Ordering Problem in Weighted Hypergraphs

Abstract

We consider the problem of mapping the n vertices of an edge-weighted hypergraph to the points $1,\ldots ,n$ on the real line, so as to minimize the weighted sum of the coordinates of right ends of all edges. This problem naturally appears in warehouse logistics: n shelves are arranged in one row, every shelf can host one type of items, the edges are sets of items requested together, their weights are the request frequencies, and items must be picked from the shelves and brought to a collection point at the left end of the row. The problem is to place all items so as to minimize the average length of the collection tours. It is NP-complete even for graphs, but it can be solved in $O^*(2^n)$ time by dynamic programming on subsets. In the present work we focus on hypergraphs with small connected components, which also has a practical motivation: Typical requests comprise related items from only one of many small disjoint groups. As a first result we solve, in polynomial time, an auxiliary problem with prescribed ordering in every component. For the unrestricted problem we conclude some worst-case time bounds that beat $O^*(2^n)$ for components of sizes up to 6. Some simple preprocessing can further reduce the time in many instances. Furthermore, the case of star graphs can be solved via bipartite matchings. Finally, there remain various interesting open problems.