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This result was conjectured by Wegener\u00a0(Automata, Languages and Programming, ICALP, Berlin, 2005). More precisely, denoting by $$n, m, w_{\\max }$$<\/jats:tex-math>\n \n n<\/mml:mi>\n ,<\/mml:mo>\n m<\/mml:mi>\n ,<\/mml:mo>\n \n w<\/mml:mi>\n max<\/mml:mo>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and $$w_{\\min }$$<\/jats:tex-math>\n \n w<\/mml:mi>\n min<\/mml:mo>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> 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 $$T_0 \\ge w_{\\max }$$<\/jats:tex-math>\n \n \n T<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n \u2265<\/mml:mo>\n \n w<\/mml:mi>\n max<\/mml:mo>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and multiplicative cooling schedule with factor $$1-1\/\\ell $$<\/jats:tex-math>\n \n 1<\/mml:mn>\n -<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n \u2113<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where $$\\ell = \\omega (mn\\ln (m))$$<\/jats:tex-math>\n \n \u2113<\/mml:mi>\n =<\/mml:mo>\n \u03c9<\/mml:mi>\n (<\/mml:mo>\n m<\/mml:mi>\n n<\/mml:mi>\n ln<\/mml:mo>\n (<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, with probability at least $$1-1\/m$$<\/jats:tex-math>\n \n 1<\/mml:mn>\n -<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> computes in time $$O(\\ell (\\ln \\ln (\\ell ) + \\ln (T_0\/w_{\\min }) ))$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n \u2113<\/mml:mi>\n \n (<\/mml:mo>\n ln<\/mml:mo>\n ln<\/mml:mo>\n \n (<\/mml:mo>\n \u2113<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n +<\/mml:mo>\n ln<\/mml:mo>\n \n (<\/mml:mo>\n \n T<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n \/<\/mml:mo>\n \n w<\/mml:mi>\n min<\/mml:mo>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> a spanning tree with weight at most $$1+\\kappa $$<\/jats:tex-math>\n \n 1<\/mml:mn>\n +<\/mml:mo>\n \u03ba<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> times the optimum weight, where $$1+\\kappa = \\frac{(1+o(1))\\ln (\\ell m)}{\\ln (\\ell ) -\\ln (mn\\ln (m))}$$<\/jats:tex-math>\n \n 1<\/mml:mn>\n +<\/mml:mo>\n \u03ba<\/mml:mi>\n =<\/mml:mo>\n \n \n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n o<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n )<\/mml:mo>\n ln<\/mml:mo>\n (<\/mml:mo>\n \u2113<\/mml:mi>\n m<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n ln<\/mml:mo>\n (<\/mml:mo>\n \u2113<\/mml:mi>\n )<\/mml:mo>\n -<\/mml:mo>\n ln<\/mml:mo>\n (<\/mml:mo>\n m<\/mml:mi>\n n<\/mml:mi>\n ln<\/mml:mo>\n (<\/mml:mo>\n m<\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mfrac>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Consequently, for any $$\\epsilon >0$$<\/jats:tex-math>\n \n \u03f5<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we can choose $$\\ell $$<\/jats:tex-math>\n \u2113<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in such a way that a $$(1+\\epsilon )$$<\/jats:tex-math>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n \u03f5<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-approximation is found in time $$O((mn\\ln (n))^{1+1\/\\epsilon +o(1)}(\\ln \\ln n + \\ln (T_0\/w_{\\min })))$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \n (<\/mml:mo>\n m<\/mml:mi>\n n<\/mml:mi>\n ln<\/mml:mo>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \n 1<\/mml:mn>\n +<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n \u03f5<\/mml:mi>\n +<\/mml:mo>\n o<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n \n (<\/mml:mo>\n ln<\/mml:mo>\n ln<\/mml:mo>\n n<\/mml:mi>\n +<\/mml:mo>\n ln<\/mml:mo>\n \n (<\/mml:mo>\n \n T<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n \/<\/mml:mo>\n \n w<\/mml:mi>\n min<\/mml:mo>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with probability at least\u00a0$$1-1\/m$$<\/jats:tex-math>\n \n 1<\/mml:mn>\n -<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n m<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In the special case of so-called $$(1+\\epsilon )$$<\/jats:tex-math>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n \u03f5<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-separated weights, this algorithm computes an optimal solution (again in time $$O( (mn\\ln (n))^{1+1\/\\epsilon +o(1)}(\\ln \\ln n + \\ln (T_0\/w_{\\min })))$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n \n \n (<\/mml:mo>\n m<\/mml:mi>\n n<\/mml:mi>\n ln<\/mml:mo>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \n 1<\/mml:mn>\n +<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n \u03f5<\/mml:mi>\n +<\/mml:mo>\n o<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n \n (<\/mml:mo>\n ln<\/mml:mo>\n ln<\/mml:mo>\n n<\/mml:mi>\n +<\/mml:mo>\n ln<\/mml:mo>\n \n (<\/mml:mo>\n \n T<\/mml:mi>\n 0<\/mml:mn>\n <\/mml:msub>\n \/<\/mml:mo>\n \n w<\/mml:mi>\n min<\/mml:mo>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>), which is a significant speed-up over Wegener\u2019s runtime guarantee of $$O(m^{8 + 8\/\\epsilon })$$<\/jats:tex-math>\n \n O<\/mml:mi>\n (<\/mml:mo>\n \n m<\/mml:mi>\n \n 8<\/mml:mn>\n +<\/mml:mo>\n 8<\/mml:mn>\n \/<\/mml:mo>\n \u03f5<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Our tighter upper bound also admits the result that in some situations a hybridization of simulated annealing and the $${(1 + 1)}$$<\/jats:tex-math>\n \n (<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>\u00a0EA can lead to stronger runtime guarantees than either algorithm alone.<\/jats:p>","DOI":"10.1007\/s00453-023-01135-x","type":"journal-article","created":{"date-parts":[[2023,7,22]],"date-time":"2023-07-22T05:01:36Z","timestamp":1690002096000},"page":"64-89","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Simulated Annealing is a Polynomial-Time Approximation Scheme for the Minimum Spanning Tree Problem"],"prefix":"10.1007","volume":"86","author":[{"given":"Benjamin","family":"Doerr","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Amirhossein","family":"Rajabi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carsten","family":"Witt","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,7,22]]},"reference":[{"key":"1135_CR1","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1145\/42282.46160","volume":"35","author":"GH Sasaki","year":"1988","unstructured":"Sasaki, G.H., Hajek, B.: The time complexity of maximum matching by simulated annealing. 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