The minimum spanning tree problem with conflict constraints and its variations

We consider the minimum spanning tree problem with conflict constraints (MSTC). It is observed that computing an $\epsilon$-optimal solution to MSTC is NP-hard for any $\epsilon >0$. For a general conflict graph, computing even a feasible solution is NP-hard. When the underlying graph is a cactus, we show that the feasibility problem is polynomially bounded whereas the optimization version is still NP-hard. When the conflict graph is a collection of disjoint cliques, (equivalently, when the conflict relation is transitive) we observe that MSTC is a matroid intersection problem and hence can be solved in polynomial time. We also identify other special cases where MSTC can be solved in polynomial time. These include instances where the conflict graph is a collection of disjoint cliques and a fixed number of stars. Exploiting these polynomially solvable special cases, we derive strong lower bounds and develop an exact branch and bound algorithm. Also various heuristic algorithms are discussed to solve the problem along with preliminary experimental results. As a byproduct of this investigation, we obtained a polynomial time approximation algorithm for the maximum edge clique partitioning problem with performance ratio $O\left(% \frac{n(\log \log n)^2}{\log^3 n}\right)$, improving the previously best known bound of $O(n)$.