On the Hardness of Approximating Minimum Monopoly Problems

Abstract

We consider inapproximability for two graph optimisation problems called monopoly and partial monopoly. We prove that these problems cannot be approximated within a factor of ($\frac{1} {3}$ — ∈) ln n and ($\frac{1} {2}$ — ∈) ln n, unless NP ⫅ Dtime(n ^{O(log log n)}), respectively. We also show that, if Δ is the maximum degree in a graph G, then both problems cannot be approximated within a factor of lnΔO(ln lnΔ), unless P = NP, though both these problems can be approximated within a factor of ln(Δ) + O(1). Finally, for cubic graphs, we give a 1.6154 approximation algorithm for the monopoly problem and a $\frac{5} {3}$ approximation algorithm for partial monopoly problem, and show that they are APX-complete.