Simulated Annealing is a Polynomial-Time Approximation Scheme for the Minimum Spanning Tree Problem
نویسندگان
چکیده
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 (2005). More precisely, denoting $n, m, w_{\max}$, and $w_{\min}$ number of vertices edges as well maximum edge weight MST instance, we simulated annealing initial temperature $T_0 \ge w_{\max}$ multiplicative factor $1-1/\ell$, where $\ell = \omega (mn\ln(m))$, probability at least $1-1/m$ time $O(\ell (\ln\ln (\ell) + \ln(T_0/w_{\min}) ))$ a most $1+\kappa$ times optimum weight, $1+\kappa \frac{(1+o(1))\ln(\ell m)}{\ln(\ell) -\ln (mn\ln (m))}$. Consequently, for any $\epsilon>0$, can choose $\ell$ such way $(1+\epsilon)$-approximation is found $O((mn\ln(n))^{1+1/\epsilon+o(1)}(\ln\ln n \ln(T_0/w_{\min})))$ $1-1/m$. In special case so-called $(1+\epsilon)$-separated weights, this algorithm optimal solution (again $O( (mn\ln(n))^{1+1/\epsilon+o(1)}(\ln\ln \ln(T_0/w_{\min})))$), which significant speed-up over Wegener's runtime guarantee $O(m^{8 8/\epsilon})$.
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ژورنال
عنوان ژورنال: Algorithmica
سال: 2023
ISSN: ['1432-0541', '0178-4617']
DOI: https://doi.org/10.1007/s00453-023-01135-x