{"id":69,"date":"2018-03-27T16:56:46","date_gmt":"2018-03-27T07:56:46","guid":{"rendered":"http:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=69"},"modified":"2018-03-27T16:56:46","modified_gmt":"2018-03-27T07:56:46","slug":"knapsack-with-integer-weights","status":"publish","type":"post","link":"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/03\/27\/knapsack-with-integer-weights\/","title":{"rendered":"\u6574\u6570\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u554f\u984c\u306eIsing\u8868\u73fe\u30cf\u30df\u30eb\u30c8\u30cb\u30a2\u30f3\uff1aKnapsack with Integer Weights"},"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\/03\/27\/knapsack-with-integer-weights\/#%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\/03\/27\/knapsack-with-integer-weights\/#%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\/03\/27\/knapsack-with-integer-weights\/#%E3%83%8A%E3%83%83%E3%83%97%E3%82%B5%E3%83%83%E3%82%AF%E5%95%8F%E9%A1%8C%E3%81%A8%E3%81%AF\" >\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u554f\u984c\u3068\u306f<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/03\/27\/knapsack-with-integer-weights\/#%E5%B0%8E%E5%85%A5\" >\u5c0e\u5165<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/03\/27\/knapsack-with-integer-weights\/#QUBO%E8%A1%A8%E7%8F%BE%E3%81%AB%E3%82%88%E3%82%8B%E5%AE%9A%E5%BC%8F%E5%8C%96\" >QUBO\u8868\u73fe\u306b\u3088\u308b\u5b9a\u5f0f\u5316<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/2018\/03\/27\/knapsack-with-integer-weights\/#%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\">\n<li>\u30bf\u30a4\u30c8\u30eb: Ising formulations of many NP problems<\/li>\n\n\n\n<li>\u8457\u8005: Andrew Lucas (Google Scholar)<\/li>\n\n\n\n<li>\u66f8\u8a8c\u60c5\u5831: Frontiers in Physics 2, 5 (2014). (doi: 10.3389\/fphy.2014.00005)<\/li>\n<\/ul>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u89aa\u30da\u30fc\u30b8: <a href=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/?p=43\" target=\"_blank\" rel=\"noopener\">\u201cIsing formulations of many NP problems\u201d by Andrew Lucas (2014)<\/a><\/li>\n\n\n\n<li>\u7ae0\u30bf\u30a4\u30c8\u30eb: 5.2. Knapsack with Integer Weights<\/li>\n<\/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>\u7d44\u5408\u305b\u6700\u9069\u5316\u554f\u984c\u306e\u985e\u578b\u306e\u3072\u3068\u3064\u306b\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u554f\u984c\u304c\u3042\u308b\u3002\u3053\u308c\u3092Ising\u6a21\u578b\uff08\u5909\u6570\u304c1 or -1\uff09\u3068\u7b49\u4fa1\u306a\u5f62\u5f0f\u3067\u3042\u308bQUBO\u5f62\u5f0f\uff08\u5909\u6570\u304c1 or 0\uff09\u306b\u3088\u3063\u3066\u8868\u73fe\u3057\u305f\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E3%83%8A%E3%83%83%E3%83%97%E3%82%B5%E3%83%83%E3%82%AF%E5%95%8F%E9%A1%8C%E3%81%A8%E3%81%AF\"><\/span>\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u554f\u984c\u3068\u306f<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"%E5%B0%8E%E5%85%A5\"><\/span>\u5c0e\u5165<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u554f\u984c\u306b\u306f\u3044\u304f\u3064\u304b\u306e\u7a2e\u985e\u304c\u3042\u308b\u304c\u3001\u4eca\u56de\u6271\u3046\u306e\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u5f62\u5f0f\u306e\u554f\u984c\u3067\u3042\u308b\u3002<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2018\/03\/d4487243f4f3230cffbb6a221e36b06f.png\" alt=\"\" class=\"wp-image-75\"\/><\/figure>\n<\/div>\n\n\n<p>\u3044\u307e\u3001N\u500b\u306e\u8377\u7269\uff08\u8377\u72691\u3001\u8377\u72692\u3001\u30fb\u30fb\u30fb\u8377\u7269N\uff09\u3092\u304c\u3042\u308a\u3001\u305d\u308c\u305e\u308c\u306e\u8377\u7269\u306b\u3064\u3044\u3066\u305d\u306e\u8377\u7269\u306e\u300c\u91cd\u3055\u300d\u3068\u300c\u4fa1\u5024\u300d\u304c\u5206\u304b\u3063\u3066\u3044\u308b\u3068\u3059\u308b\u3002\u3053\u308c\u3089\u306e\u8377\u7269\u3092\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u306b\u8a70\u3081\u8fbc\u3093\u3067\u904b\u3073\u305f\u3044\u304c\u3001\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u306b\u306f\u5bb9\u91cf\u304c\u3042\u308b\u306e\u3067\u6301\u3063\u3066\u3044\u304f\u8377\u7269\u306e\u91cd\u3055\u306e\u5408\u8a08\u306f\u3053\u306e\u5bb9\u91cf\u3092\u8d8a\u3048\u3066\u306f\u306a\u3089\u306a\u3044\u3002\u3053\u306e\u6761\u4ef6\u306e\u4e0b\u3001\u6301\u3063\u3066\u3044\u304f\u8377\u7269\u306e\u4fa1\u5024\u306e\u5408\u8a08\u3092\u53ef\u80fd\u306a\u9650\u308a\u5927\u304d\u304f\u3059\u308b\u306b\u306f\u3069\u306e\u8377\u7269\u3092\u6301\u3063\u3066\u3044\u304f\u306e\u304c\u826f\u3044\u304b\uff1f<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"QUBO%E8%A1%A8%E7%8F%BE%E3%81%AB%E3%82%88%E3%82%8B%E5%AE%9A%E5%BC%8F%E5%8C%96\"><\/span>QUBO\u8868\u73fe\u306b\u3088\u308b\u5b9a\u5f0f\u5316<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p><strong>\u9069\u5316\u5909\u6570<\/strong>: $x_{\\alpha} \\in \\{0,1\\}$ (\u8377\u7269$\\alpha$\u3092\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u306b\u5165\u308c\u308c\u3070$1$,\u5165\u308c\u306a\u3051\u308c\u3070$0$)<br><strong>\u88dc\u52a9\u5909\u6570<\/strong>: $y_n$ (\u81ea\u7136\u6570$m$\u3092\u8868\u3059\u3068\u304d\u5de6\u304b\u3089$m$\u756a\u76ee\u306e$y_m$\u304c$1$,\u305d\u308c\u4ee5\u5916\u306e\u5168\u3066\u306e$y_{n (n\\neq m)}$\u306f$0$)<br><strong>\u76ee\u7684\u95a2\u6570<\/strong>: <\/p>\n\n\n\n$$\n\\mathrm{Obj}\n=\n-\\sum_{\\alpha = 1}^N\n{c_{\\alpha}x_{\\alpha}}\n+\nA(1-\\sum_{n=1}^W {y_n})^2\n+\nA(\\sum_{n=1}^W {ny_n}\n&#8211;\n\\sum_{\\alpha}w_{\\alpha}x_{\\alpha})^2\n$$\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2018\/03\/b0a045d6699d5e6be7f9a1d6b025e179.png\" alt=\"\" class=\"wp-image-123\"\/><\/figure>\n<\/div>\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter\"><img decoding=\"async\" src=\"https:\/\/qard.is.tohoku.ac.jp\/T-Wave\/wp-content\/uploads\/2018\/03\/9cd9577e66e3934e1334401aacba650e.png\" alt=\"\" class=\"wp-image-124\"\/><\/figure>\n<\/div>\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>\u9234\u6728\u6d77\u6e21<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6587\u732e\u60c5\u5831 \u6982\u8981 \u7d44\u5408\u305b\u6700\u9069\u5316\u554f\u984c\u306e\u985e\u578b\u306e\u3072\u3068\u3064\u306b\u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u554f\u984c\u304c\u3042\u308b\u3002\u3053\u308c\u3092Ising\u6a21\u578b\uff08\u5909\u6570\u304c1 or -1\uff09\u3068\u7b49\u4fa1\u306a\u5f62\u5f0f\u3067\u3042\u308bQUBO\u5f62\u5f0f\uff08\u5909\u6570\u304c1 or 0\uff09\u306b\u3088\u3063\u3066\u8868\u73fe\u3057\u305f\u3002 \u30ca\u30c3\u30d7\u30b5\u30c3\u30af\u554f\u984c\u3068\u306f \u5c0e\u5165 \u30ca\u30c3 [&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":[21,74,98,112],"class_list":["post-69","post","type-post","status-publish","format-standard","hentry","category-review","tag-karp21np","tag-74","tag-98","tag-112"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ 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