A polynomial-time approximation scheme for embedding hypergraph in a cycle

We consider the problem of embedding hyperedges of a hypergraph as paths in a cycle such that the maximum congestion, namely the maximum number of paths that use any single edge in a cycle, is minimized. The minimum congestion hypergraph embedding in a cycle problem is known to be NP-hard and its graph version, the minimum congestion graph embedding in a cycle, is solvable in polynomial-time. Furthermore, for the graph problem, a polynomial-time approximation scheme for the weighted version is known. For the hypergraph model, several approximation algorithms with a ratio of two have been previously published. A recent paper reduced the approximation ratio to 1.5. We present a polynomial-time approximation scheme in this article, settling the debate regarding whether the problem is polynomial-time approximable.