Local Search for the Minimum Fundamental Cycle Basis Problem

Let G=(V,E) be a bi-connected graph with non-negative weights on its m edges. Let n be the number of vertices of G. With respect to any spanning tree T of G we have a fundamental cycle basis of G formed by the m-n+1 fundamental cycles, each one corresponding to one of the co-tree edges. The problem of determining one such basis minimising the sum of the length of its cycles (MinFCB) is known to be NPhard. Applications exist in organic chemistry, periodic scheduling, and electrical networks. A recent result proves that no polynomial time approximation scheme can exist for MinFCB unless P=NP. The problem of determining a tree with respect to which the longest fundamental cycle is minimum is also NP-hard. When fundamentality is not required, that is when one looks for a basis of the cycle space of the graph, both problems are solvable in polynomial time. Constructive heuristics have been proposed in the literature, which are fast and an obvious first choice for very large instances. However they tend to produce solutions which are far from optimal. We report about on-going research on Local Search approaches for MinFCB. A formulation as Mixed Integer Programming problem is presented and proved effective for obtaining bounds. Preliminary computational results are reported. The final part of this work outlines some future research directions.