{"id":43,"date":"2018-05-07T12:00:33","date_gmt":"2018-05-07T03:00:33","guid":{"rendered":"http:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=43"},"modified":"2018-05-07T12:00:33","modified_gmt":"2018-05-07T03:00:33","slug":"ising-formulations-of-many-np-problems-by-andrew-lucas-2014","status":"publish","type":"post","link":"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/05\/07\/ising-formulations-of-many-np-problems-by-andrew-lucas-2014\/","title":{"rendered":"\u7d44\u5408\u305b\u6700\u9069\u5316\u554f\u984c\u306e\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u8868\u73fe &#8220;Ising formulations of many NP problems&#8221; by Andrew Lucas (2014)"},"content":{"rendered":"\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/05\/07\/ising-formulations-of-many-np-problems-by-andrew-lucas-2014\/#%E6%96%87%E7%8C%AE%E6%83%85%E5%A0%B1\" >\u6587\u732e\u60c5\u5831<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/05\/07\/ising-formulations-of-many-np-problems-by-andrew-lucas-2014\/#%E6%A6%82%E8%A6%81\" >\u6982\u8981<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/05\/07\/ising-formulations-of-many-np-problems-by-andrew-lucas-2014\/#%E5%8F%96%E3%82%8A%E6%89%B1%E3%82%8F%E3%82%8C%E3%81%A6%E3%81%84%E3%82%8B%E7%B5%84%E5%90%88%E3%81%9B%E6%9C%80%E9%81%A9%E5%8C%96%E5%95%8F%E9%A1%8C\" >\u53d6\u308a\u6271\u308f\u308c\u3066\u3044\u308b\u7d44\u5408\u305b\u6700\u9069\u5316\u554f\u984c<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/05\/07\/ising-formulations-of-many-np-problems-by-andrew-lucas-2014\/#%E8%A8%98%E4%BA%8B%E6%8B%85%E5%BD%93%E8%80%85%E3%81%8B%E3%82%89%E3%81%AE%E3%82%B3%E3%83%A1%E3%83%B3%E3%83%88\" >\u8a18\u4e8b\u62c5\u5f53\u8005\u304b\u3089\u306e\u30b3\u30e1\u30f3\u30c8<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/05\/07\/ising-formulations-of-many-np-problems-by-andrew-lucas-2014\/#%E6%9C%AC%E8%A8%98%E4%BA%8B%E3%81%AE%E6%8B%85%E5%BD%93%E8%80%85\" >\u672c\u8a18\u4e8b\u306e\u62c5\u5f53\u8005<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%96%87%E7%8C%AE%E6%83%85%E5%A0%B1\"><\/span>\u6587\u732e\u60c5\u5831<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul class=\"wp-block-list\"><li>\u30bf\u30a4\u30c8\u30eb: <a href=\"https:\/\/www.frontiersin.org\/articles\/10.3389\/fphy.2014.00005\/full\" target=\"_blank\" rel=\"noopener\">Ising formulations of many NP problems<\/a><\/li><li>\u8457\u8005: <a href=\"https:\/\/web.stanford.edu\/~ajlucas\/\">Andrew Lucas<\/a>&nbsp;(<a href=\"https:\/\/scholar.google.com\/citations?user=6ipiQb8AAAAJ\">Google Scholar<\/a>)<\/li><li>\u66f8\u8a8c\u60c5\u5831: Frontiers in Physics <strong>2<\/strong>, 5 (2014). (doi: <a href=\"https:\/\/doi.org\/10.3389\/fphy.2014.00005\" target=\"_blank\" rel=\"noopener\">10.3389\/fphy.2014.00005<\/a>)<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%A6%82%E8%A6%81\"><\/span>\u6982\u8981<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>\u8fd1\u5e74\u3001NP\u5b8c\u5168\u3001\u3042\u308b\u3044\u306fNP\u56f0\u96e3\u306a\u7d44\u5408\u305b\u6700\u9069\u5316\u554f\u984c\u306e\u65ad\u71b1\u91cf\u5b50\u6700\u9069\u5316\uff08AQO, Adiabatic Quantum Optimization\uff09\u3092\u7528\u3044\u305f\u89e3\u6cd5\u304c\u6ce8\u76ee\u3055\u308c\u3066\u3044\u308b\u3002\u5168\u3066\u306e\u7d44\u5408\u305b\u6700\u9069\u5316\u554f\u984c\u306f<strong>Ising\u30b9\u30d4\u30f3\u30b0\u30e9\u30b9\u306e\u57fa\u5e95\u72b6\u614b\u63a2\u7d22<\/strong>\u3078\u3068\u5b9a\u5f0f\u5316\u3059\u308b\u3053\u3068\u304c\u53ef\u80fd\u3067\u3042\u308a\u3001\u305d\u306e\u305f\u3081\u306e\u624b\u6cd5\u3068\u3057\u3066AQO\u3092\u7528\u3044\u308b\u3002<\/p>\n\n\n\n<p>\u672c\u8ad6\u6587\u3067\u306f\u3001\u7d44\u5408\u305b\u6700\u9069\u5316\u554f\u984c\u306e\u7d2f\u8a08\u30d1\u30bf\u30fc\u30f3\u3092\u96c6\u3081\u305f<strong>Karp\u306e21\u306eNP\u5b8c\u5168\u554f\u984c<\/strong> (<a href=\"https:\/\/en.wikipedia.org\/wiki\/Karp%27s_21_NP-complete_problems\" target=\"_blank\" rel=\"noopener\">Karp&#8217;s 21 NP-complete problems<\/a>)\u3092\u542b\u3080\u7d44\u5408\u305b\u6700\u9069\u5316\u554f\u984c\u306eIsing\u30b9\u30d4\u30f3\u30b0\u30e9\u30b9\u306b\u3088\u308b\u5b9a\u5f0f\u5316\u306e\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u3092\u7db2\u7f85\u3057\u3066\u3044\u308b\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%8F%96%E3%82%8A%E6%89%B1%E3%82%8F%E3%82%8C%E3%81%A6%E3%81%84%E3%82%8B%E7%B5%84%E5%90%88%E3%81%9B%E6%9C%80%E9%81%A9%E5%8C%96%E5%95%8F%E9%A1%8C\"><\/span>\u53d6\u308a\u6271\u308f\u308c\u3066\u3044\u308b\u7d44\u5408\u305b\u6700\u9069\u5316\u554f\u984c<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<ul class=\"wp-block-list\"><li>\u5206\u5272\u554f\u984c (Partitioning Problems)\n<ul>\n<li>\u6570\u5206\u5272 (Number Partitioning)<\/li>\n<li><a href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=122\">\u30b0\u30e9\u30d5\u5206\u5272 (Graph Partitioning)<\/a><\/li>\n<li><a href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=97\">\u6700\u5927\u30af\u30ea\u30fc\u30af\u554f\u984c (Cliques)<\/a><\/li>\n<\/ul>\n<\/li><li>\u4e8c\u5024\u6574\u6570\u7dda\u578b\u8a08\u753b (Binary Integer Linear Programming)<\/li><li>\u88ab\u8986\u30fb\u30d1\u30c3\u30ad\u30f3\u30b0\u554f\u984c (Covering and Packing Problems)\n<ul>\n<li>\u53b3\u5bc6\u88ab\u8986\u554f\u984c (Exact Cover)<\/li>\n<li>\u96c6\u5408\u30d1\u30c3\u30ad\u30f3\u30b0\u554f\u984c (Set Packing)<\/li>\n<li><a href=\"\/T-Wave\/?p=1136\">\u9802\u70b9\u88ab\u8986\u554f\u984c (Vertex Cover)<\/a><\/li>\n<li><a href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=651\">\u5145\u8db3\u53ef\u80fd\u6027\u554f\u984c (SAT; Satisfiability)<\/a><\/li>\n<li>\u6700\u5c0f\u91cd\u307f\u6700\u5927\u30de\u30c3\u30c1\u30f3\u30b0\u554f\u984c (Minimal Maximal Matching)<\/li>\n<\/ul>\n<\/li><li>\u4e0d\u7b49\u5f0f\u3092\u4f34\u3046\u554f\u984c\n<ul>\n<li>\u96c6\u5408\u88ab\u8986\u554f\u984c (Set Cover)<\/li>\n<li><a href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=69\">\u6574\u6570\u91cd\u307f\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u554f\u984c (Knapsack with Integer Weights)<\/a><\/li>\n<\/ul>\n<\/li><li>\u5f69\u8272\u554f\u984c (Coloring Problems)\n<ul>\n<li><a href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=418\">\u30b0\u30e9\u30d5\u5f69\u8272\u554f\u984c (Graph Coloring)<\/a><\/li>\n<li><a href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=434\">\u30af\u30ea\u30fc\u30af\u88ab\u8986\u554f\u984c (Clique Cover)<\/a><\/li>\n<li><a href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=133\">\u30b8\u30e7\u30d6\u30b9\u30b1\u30b8\u30e5\u30fc\u30ea\u30f3\u30b0\u554f\u984c (Job Sequencing with Integer Lengths)<\/a><\/li>\n<\/ul>\n<\/li><li>\u30cf\u30df\u30eb\u30c8\u30f3\u30b5\u30a4\u30af\u30eb (Hamiltonian Cycles)\n<ul>\n<li>\u30cf\u30df\u30eb\u30c8\u30f3\u30b5\u30a4\u30af\u30eb\u30fb\u30cf\u30df\u30eb\u30c8\u30f3\u8def\u63a2\u7d22 (Hamiltonian Cycles and Paths)<\/li>\n<li>\u5de1\u56de\u30bb\u30fc\u30eb\u30b9\u30de\u30f3\u554f\u984c (Traveling Salesman)<\/li>\n<\/ul>\n<\/li><li>\u6728\u554f\u984c (Tree Problems)\n<ul>\n<li>\u6700\u5927\u6b21\u6570\u5236\u7d04\u4ed8\u304d\u6700\u5c0f\u5168\u57df\u6728\uff08\u30b9\u30d1\u30cb\u30f3\u30b0\u30fb\u30c4\u30ea\u30fc\uff09\u554f\u984c (Minimal Spanning Tree with a Maximal Degree Constraint)<\/li>\n<li>\u30b7\u30e5\u30bf\u30a4\u30ca\u30fc\u6728 (Steiner Trees)<\/li>\n<li>\u6709\u5411\u30d5\u30a3\u30fc\u30c9\u30d0\u30c3\u30af\u9802\u70b9\u96c6\u5408\u554f\u984c (Directed Feedback Vertex Set)<\/li>\n<li>\u7121\u5411\u30d5\u30a3\u30fc\u30c9\u30d0\u30c3\u30af\u9802\u70b9\u96c6\u5408\u554f\u984c (Undirected Feedback Vertex Set)<\/li>\n<li>\u30d5\u30a3\u30fc\u30c9\u30d0\u30c3\u30af\u8fba\u96c6\u5408\u554f\u984c (Feedback Edge Set)<\/li>\n<\/ul>\n<\/li><li>\u30b0\u30e9\u30d5\u540c\u578b\u6027\u5224\u5b9a\u554f\u984c (Graph Isomorphism)<\/li><\/ul>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E8%A8%98%E4%BA%8B%E6%8B%85%E5%BD%93%E8%80%85%E3%81%8B%E3%82%89%E3%81%AE%E3%82%B3%E3%83%A1%E3%83%B3%E3%83%88\"><\/span>\u8a18\u4e8b\u62c5\u5f53\u8005\u304b\u3089\u306e\u30b3\u30e1\u30f3\u30c8<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>D-Wave\u30de\u30b7\u30f3\u306e\u30d7\u30ed\u30b0\u30e9\u30df\u30f3\u30b0\u306e\u6838\u5fc3\u3068\u306f\u3001\u554f\u984c\u3092\u8868\u73fe\u3059\u308bIsing\u30b9\u30d4\u30f3\u30b0\u30e9\u30b9\u306e\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u3092\u66f8\u304d\u4e0b\u3059\u3053\u3068\u306b\u4ed6\u306a\u3089\u306a\u3044\u3002\u672c\u8ad6\u6587\u3067\u306f\u3001\u985e\u578b\u5316\u3055\u308c\u305f\u591a\u304f\u306e\u7d44\u5408\u305b\u6700\u9069\u5316\u554f\u984c\u306b\u5bfe\u3059\u308b\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u3092\u7db2\u7f85\u3057\u3066\u304a\u308a\u3001\u5b9f\u554f\u984c\u306e\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\u3092\u8a2d\u8a08\u3059\u308b\u4e0a\u3067\u5927\u5909\u53c2\u8003\u306b\u306a\u308b\u3067\u3042\u308d\u3046\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E6%9C%AC%E8%A8%98%E4%BA%8B%E3%81%AE%E6%8B%85%E5%BD%93%E8%80%85\"><\/span>\u672c\u8a18\u4e8b\u306e\u62c5\u5f53\u8005<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u307f\u3084\u307e\u307e\u3055\u307f\u3061<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6587\u732e\u60c5\u5831 \u30bf\u30a4\u30c8\u30eb: Ising formulations of many NP problems \u8457\u8005: Andrew Lucas&nbsp;(Google Scholar) \u66f8\u8a8c\u60c5\u5831: Frontiers in Ph [&hellip;]<\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[21,74,112],"class_list":["post-43","post","type-post","status-publish","format-standard","hentry","category-review","tag-karp21np","tag-74","tag-112"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.3 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>\u7d44\u5408\u305b\u6700\u9069\u5316\u554f\u984c\u306e\u30a4\u30b8\u30f3\u30b0\u30e2\u30c7\u30eb\u8868\u73fe &quot;Ising formulations of many NP problems&quot; 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