Approximation Algorithms for Embedding a Weighted Directed Hypergraph on a Mixed Cycle∗

Given a weighted directed hypergraph H = (V,EH ;w), where w : EH → R+, we consider the problem of embedding all weighted directed hyperedges on a mixed cycle, which consists of undirected and directed links. The objective is to minimize the maximum congestion of any undirected or directed link in the mixed cycle. In this paper, we first formulate this new problem as an integer linear program, and by utilizing a nontrivial LP-rounding technique, we design a 2approximation algorithm. Then, we design a combinatorial algorithm with approximation ratio 3 for the problem, whose running time is O(nm). Finally, we present a polynomial time approximation scheme (PTAS) for the special version where each directed hyperedge only contains one sink.