Merging Combinatorial Design and Optimization: the Oberwolfach Problem

The Oberwolfach Problem $OP(F)$, posed by Gerhard Ringel in 1967, is a paradigmatic Combinatorial Design problem asking whether the complete graph $K_v$ decomposes into edge-disjoint copies of a $2$-regular graph $F$ of order $v$. In Combinatorial Design Theory, so-called difference methods represent a well-known solution technique and construct solutions in infinitely many cases exploiting symmetric and balanced structures. This approach reduces the problem to finding a well-structured $2$-factor which allows us to build solutions that we call $1$- or $2$-rotational according to their symmetries. We tackle $OP$ by modeling difference methods with Optimization tools, specifically Constraint Programming ($CP$) and Integer Programming ($IP$), and correspondingly solve instances with up to $v=120$ within $60s$. In particular, we model the $2$-rotational method by solving in cascade two subproblems, namely the binary and group labeling, respectively. A polynomial-time algorithm solves the binary labeling, while $CP$ tackles the group labeling. Furthermore, we prov ide necessary conditions for the existence of some $1$-rotational solutions which stem from computational results. This paper shows thereby that both theoretical and empirical results may arise from the interaction between Combinatorial Design Theory and Operation Research.