Simulated Annealing is a Polynomial-Time Approximation Scheme for the Minimum Spanning Tree Problem

Abstract

We prove that Simulated Annealing with an appropriate cooling schedule computes arbitrarily tight constant-factor approximations to the minimum spanning tree problem in polynomial time. This result was conjectured by Wegener (Automata, Languages and Programming, ICALP, Berlin, 2005). More precisely, denoting by n, m, w_{max }, and w_{min } the number of vertices and edges as well as the maximum and minimum edge weight of the MST instance, we prove that simulated annealing with initial temperature T_0 ge w_{max } and multiplicative cooling schedule with factor 1-1/ell , where ell = omega (mnln (m)), with probability at least 1-1/m computes in time O(ell (ln ln (ell ) + ln (T_0/w_{min }) )) a spanning tree with weight at most 1+kappa times the optimum weight, where 1+kappa = frac{(1+o(1))ln (ell m)}{ln (ell ) -ln (mnln (m))}. Consequently, for any epsilon >0, we can choose ell in such a way that a (1+epsilon )-approximation is found in time O((mnln (n))^{1+1/epsilon +o(1)}(ln ln n + ln (T_0/w_{min }))) with probability at least 1-1/m. In the special case of so-called (1+epsilon )-separated weights, this algorithm computes an optimal solution (again in time O( (mnln (n))^{1+1/epsilon +o(1)}(ln ln n + ln (T_0/w_{min })))), which is a significant speed-up over Wegener’s runtime guarantee of O(m^{8 + 8/epsilon }). Our tighter upper bound also admits the result that in some situations a hybridization of simulated annealing and the {(1 + 1)} EA can lead to stronger runtime guarantees than either algorithm alone.