English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We prove that Simulated Annealing with an appropriate cooling schedulecomputes arbitrarily tight constant-factor approximations to the minimumspanning tree problem in polynomial time. This result was conjectured byWegener (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 weightof 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 $\ell = \omega (mn\ln(m))$, with probability at least $1-1/m$ computes intime $O(\ell (\ln\ln (\ell) + \ln(T_0/w_{\min}) ))$ a spanning tree with weightat most $1+\kappa$ times the optimum weight, where $1+\kappa =\frac{(1+o(1))\ln(\ell m)}{\ln(\ell) -\ln (mn\ln (m))}$. Consequently, for any$\epsilon>0$, we can choose $\ell$ in such a way that a$(1+\epsilon)$-approximation is found in time$O((mn\ln(n))^{1+1/\epsilon+o(1)}(\ln\ln n + \ln(T_0/w_{\min})))$ withprobability 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 $O( (mn\ln(n))^{1+1/\epsilon+o(1)}(\ln\ln n +\ln(T_0/w_{\min})))$), which is a significant speed-up over Wegener's runtimeguarantee of $O(m^{8 + 8/\epsilon})$.

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