A local search algorithm for binary maximum 2-path partitioning

Let G be a complete (undirected) graph with 3l vertices. Given a binary weight function on the edges of G, the BINARY MAXIMUM 2-PATH PARTITIONING PROBLEM is to compute a set of l vertex-disjoint simple 2-edge paths with maximum total edge weight. The problem is NP-hard (Garey and Johnson 1979) [1]. In this paper we propose a simple local search algorithm with polynomial running time for the problem and analyze its performance for several search depths. For depth 2, we show that the algorithm is a 0.3333-approximation, and that the bound is tight. For depth 3, we show that the algorithm is a 0.4-approximation. For depth 9, we show that the algorithm is a 0.55-approximation, improving on the best-known 0.5265 bound for the problem. We also consider the special case where G is subcubic, that is, the maximum degree in its subgraph induced by the unit-weight edges is 3. In this case we show that the algorithm is a 0.375-approximation for depth 2 and a 0.5-approximation for depth 3. In addition, we show that depth 7 is sufficient for the 0.55 bound guarantee. Finally we give, by means of bad instances, upper bounds on the performance guarantees of the algorithm. For depth 2 we show a 0.4 upper bound in the subcubic case. For depth 3 we show a 0.6 upper bound, as well as a 0.7 upper bound in the subcubic case. For the general (non-negative) weight problem we show a 0.5556 upper bound for depth 3 (for depth 2, the tight 0.3333 ratio holds for this problem as well).