背包约束下不可分割物品的近似最大份额分配

IF 1.1 4区 数学 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
Bin Deng, Weidong Li
{"title":"背包约束下不可分割物品的近似最大份额分配","authors":"Bin Deng, Weidong Li","doi":"10.1007/s10878-025-01331-1","DOIUrl":null,"url":null,"abstract":"<p>The maximin share (MMS) allocation problem under a knapsack constraint is to allocate a set of indivisible goods to a set of <i>n</i> heterogeneous agents, such that the total cost of the allocated goods does not exceed the given budget, and the approximation ratio of the MMS allocation is as large as possible. For any <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;&amp;#x03F5;&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"2.614ex\" role=\"img\" style=\"vertical-align: -0.706ex;\" viewbox=\"0 -821.4 3854.7 1125.3\" width=\"8.953ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-3F5\" y=\"0\"></use><use x=\"684\" xlink:href=\"#MJMAIN-2208\" y=\"0\"></use><use x=\"1629\" xlink:href=\"#MJMAIN-28\" y=\"0\"></use><use x=\"2019\" xlink:href=\"#MJMAIN-30\" y=\"0\"></use><use x=\"2519\" xlink:href=\"#MJMAIN-2C\" y=\"0\"></use><use x=\"2964\" xlink:href=\"#MJMAIN-31\" y=\"0\"></use><use x=\"3465\" xlink:href=\"#MJMAIN-29\" y=\"0\"></use></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>ϵ</mi><mo>∈</mo><mo stretchy=\"false\">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math></span></span><script type=\"math/tex\">\\epsilon \\in (0, 1)</script></span>, we prove that <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;93&lt;/mn&gt;&lt;mn&gt;95&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;&amp;#x03F5;&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"3.215ex\" role=\"img\" style=\"vertical-align: -1.006ex;\" viewbox=\"0 -950.8 3476.3 1384.1\" width=\"8.074ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMAIN-28\" y=\"0\"></use><g transform=\"translate(389,0)\"><g transform=\"translate(120,0)\"><rect height=\"60\" stroke=\"none\" width=\"827\" x=\"0\" y=\"220\"></rect><g transform=\"translate(60,407)\"><use transform=\"scale(0.707)\" xlink:href=\"#MJMAIN-39\"></use><use transform=\"scale(0.707)\" x=\"500\" xlink:href=\"#MJMAIN-33\" y=\"0\"></use></g><g transform=\"translate(60,-363)\"><use transform=\"scale(0.707)\" xlink:href=\"#MJMAIN-39\"></use><use transform=\"scale(0.707)\" x=\"500\" xlink:href=\"#MJMAIN-35\" y=\"0\"></use></g></g></g><use x=\"1679\" xlink:href=\"#MJMAIN-2B\" y=\"0\"></use><use x=\"2680\" xlink:href=\"#MJMATHI-3F5\" y=\"0\"></use><use x=\"3086\" xlink:href=\"#MJMAIN-29\" y=\"0\"></use></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mfrac><mn>93</mn><mn>95</mn></mfrac><mo>+</mo><mi>ϵ</mi><mo stretchy=\"false\">)</mo></math></span></span><script type=\"math/tex\">(\\frac{93}{95}+ \\epsilon )</script></span>-approximate MMS allocation does not always exist for two agents, while the MMS allocation problem without a knapsack constraint always has an MMS allocation for two agents. We propose a bag-filling based algorithm that can produce a <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mfrac&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"2.914ex\" role=\"img\" style=\"vertical-align: -1.106ex;\" viewbox=\"0 -778.3 2042.9 1254.7\" width=\"4.745ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><g transform=\"translate(120,0)\"><rect height=\"60\" stroke=\"none\" width=\"1802\" x=\"0\" y=\"220\"></rect><use transform=\"scale(0.707)\" x=\"974\" xlink:href=\"#MJMATHI-6E\" y=\"564\"></use><g transform=\"translate(60,-363)\"><use transform=\"scale(0.707)\" x=\"0\" xlink:href=\"#MJMAIN-33\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"500\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"1101\" xlink:href=\"#MJMAIN-2212\" y=\"0\"></use><use transform=\"scale(0.707)\" x=\"1879\" xlink:href=\"#MJMAIN-32\" y=\"0\"></use></g></g></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mi>n</mi><mrow><mn>3</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span></span><script type=\"math/tex\">\\frac{n}{3n-2}</script></span>-approximate MMS allocation. When <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"1.912ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -735.2 2435.1 823.4\" width=\"5.656ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use><use x=\"878\" xlink:href=\"#MJMAIN-3D\" y=\"0\"></use><use x=\"1934\" xlink:href=\"#MJMAIN-32\" y=\"0\"></use></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi><mo>=</mo><mn>2</mn></math></span></span><script type=\"math/tex\">n=2</script></span> and <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"1.912ex\" role=\"img\" style=\"vertical-align: -0.205ex;\" viewbox=\"0 -735.2 2435.1 823.4\" width=\"5.656ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><use x=\"0\" xlink:href=\"#MJMATHI-6E\" y=\"0\"></use><use x=\"878\" xlink:href=\"#MJMAIN-3D\" y=\"0\"></use><use x=\"1934\" xlink:href=\"#MJMAIN-33\" y=\"0\"></use></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi><mo>=</mo><mn>3</mn></math></span></span><script type=\"math/tex\">n=3</script></span>, by more careful analysis, we improve the approximation ratios to <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"3.209ex\" role=\"img\" style=\"vertical-align: -1.005ex;\" viewbox=\"0 -949.2 713.9 1381.8\" width=\"1.658ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><g transform=\"translate(120,0)\"><rect height=\"60\" stroke=\"none\" width=\"473\" x=\"0\" y=\"220\"></rect><use transform=\"scale(0.707)\" x=\"84\" xlink:href=\"#MJMAIN-32\" y=\"556\"></use><use transform=\"scale(0.707)\" x=\"84\" xlink:href=\"#MJMAIN-33\" y=\"-512\"></use></g></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mn>2</mn><mn>3</mn></mfrac></math></span></span><script type=\"math/tex\">\\frac{2}{3}</script></span> and <span><span style=\"\"></span><span data-mathml='&lt;math xmlns=\"http://www.w3.org/1998/Math/MathML\"&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;' role=\"presentation\" style=\"font-size: 100%; display: inline-block; position: relative;\" tabindex=\"0\"><svg aria-hidden=\"true\" focusable=\"false\" height=\"3.209ex\" role=\"img\" style=\"vertical-align: -1.005ex;\" viewbox=\"0 -949.2 713.9 1381.8\" width=\"1.658ex\" xmlns:xlink=\"http://www.w3.org/1999/xlink\"><g fill=\"currentColor\" stroke=\"currentColor\" stroke-width=\"0\" transform=\"matrix(1 0 0 -1 0 0)\"><g transform=\"translate(120,0)\"><rect height=\"60\" stroke=\"none\" width=\"473\" x=\"0\" y=\"220\"></rect><use transform=\"scale(0.707)\" x=\"84\" xlink:href=\"#MJMAIN-31\" y=\"556\"></use><use transform=\"scale(0.707)\" x=\"84\" xlink:href=\"#MJMAIN-32\" y=\"-513\"></use></g></g></svg><span role=\"presentation\"><math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mn>1</mn><mn>2</mn></mfrac></math></span></span><script type=\"math/tex\">\\frac{1}{2}</script></span>, respectively.</p>","PeriodicalId":50231,"journal":{"name":"Journal of Combinatorial Optimization","volume":"50 1","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Approximate maximin share allocation for indivisible goods under a knapsack constraint\",\"authors\":\"Bin Deng, Weidong Li\",\"doi\":\"10.1007/s10878-025-01331-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The maximin share (MMS) allocation problem under a knapsack constraint is to allocate a set of indivisible goods to a set of <i>n</i> heterogeneous agents, such that the total cost of the allocated goods does not exceed the given budget, and the approximation ratio of the MMS allocation is as large as possible. For any <span><span style=\\\"\\\"></span><span data-mathml='&lt;math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"&gt;&lt;mi&gt;&amp;#x03F5;&lt;/mi&gt;&lt;mo&gt;&amp;#x2208;&lt;/mo&gt;&lt;mo stretchy=\\\"false\\\"&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=\\\"false\\\"&gt;)&lt;/mo&gt;&lt;/math&gt;' role=\\\"presentation\\\" style=\\\"font-size: 100%; display: inline-block; position: relative;\\\" tabindex=\\\"0\\\"><svg aria-hidden=\\\"true\\\" focusable=\\\"false\\\" height=\\\"2.614ex\\\" role=\\\"img\\\" style=\\\"vertical-align: -0.706ex;\\\" viewbox=\\\"0 -821.4 3854.7 1125.3\\\" width=\\\"8.953ex\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g fill=\\\"currentColor\\\" stroke=\\\"currentColor\\\" stroke-width=\\\"0\\\" transform=\\\"matrix(1 0 0 -1 0 0)\\\"><use x=\\\"0\\\" xlink:href=\\\"#MJMATHI-3F5\\\" y=\\\"0\\\"></use><use x=\\\"684\\\" xlink:href=\\\"#MJMAIN-2208\\\" y=\\\"0\\\"></use><use x=\\\"1629\\\" xlink:href=\\\"#MJMAIN-28\\\" y=\\\"0\\\"></use><use x=\\\"2019\\\" xlink:href=\\\"#MJMAIN-30\\\" y=\\\"0\\\"></use><use x=\\\"2519\\\" xlink:href=\\\"#MJMAIN-2C\\\" y=\\\"0\\\"></use><use x=\\\"2964\\\" xlink:href=\\\"#MJMAIN-31\\\" y=\\\"0\\\"></use><use x=\\\"3465\\\" xlink:href=\\\"#MJMAIN-29\\\" y=\\\"0\\\"></use></g></svg><span role=\\\"presentation\\\"><math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>ϵ</mi><mo>∈</mo><mo stretchy=\\\"false\\\">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy=\\\"false\\\">)</mo></math></span></span><script type=\\\"math/tex\\\">\\\\epsilon \\\\in (0, 1)</script></span>, we prove that <span><span style=\\\"\\\"></span><span data-mathml='&lt;math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"&gt;&lt;mo stretchy=\\\"false\\\"&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mn&gt;93&lt;/mn&gt;&lt;mn&gt;95&lt;/mn&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;&amp;#x03F5;&lt;/mi&gt;&lt;mo stretchy=\\\"false\\\"&gt;)&lt;/mo&gt;&lt;/math&gt;' role=\\\"presentation\\\" style=\\\"font-size: 100%; display: inline-block; position: relative;\\\" tabindex=\\\"0\\\"><svg aria-hidden=\\\"true\\\" focusable=\\\"false\\\" height=\\\"3.215ex\\\" role=\\\"img\\\" style=\\\"vertical-align: -1.006ex;\\\" viewbox=\\\"0 -950.8 3476.3 1384.1\\\" width=\\\"8.074ex\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g fill=\\\"currentColor\\\" stroke=\\\"currentColor\\\" stroke-width=\\\"0\\\" transform=\\\"matrix(1 0 0 -1 0 0)\\\"><use x=\\\"0\\\" xlink:href=\\\"#MJMAIN-28\\\" y=\\\"0\\\"></use><g transform=\\\"translate(389,0)\\\"><g transform=\\\"translate(120,0)\\\"><rect height=\\\"60\\\" stroke=\\\"none\\\" width=\\\"827\\\" x=\\\"0\\\" y=\\\"220\\\"></rect><g transform=\\\"translate(60,407)\\\"><use transform=\\\"scale(0.707)\\\" xlink:href=\\\"#MJMAIN-39\\\"></use><use transform=\\\"scale(0.707)\\\" x=\\\"500\\\" xlink:href=\\\"#MJMAIN-33\\\" y=\\\"0\\\"></use></g><g transform=\\\"translate(60,-363)\\\"><use transform=\\\"scale(0.707)\\\" xlink:href=\\\"#MJMAIN-39\\\"></use><use transform=\\\"scale(0.707)\\\" x=\\\"500\\\" xlink:href=\\\"#MJMAIN-35\\\" y=\\\"0\\\"></use></g></g></g><use x=\\\"1679\\\" xlink:href=\\\"#MJMAIN-2B\\\" y=\\\"0\\\"></use><use x=\\\"2680\\\" xlink:href=\\\"#MJMATHI-3F5\\\" y=\\\"0\\\"></use><use x=\\\"3086\\\" xlink:href=\\\"#MJMAIN-29\\\" y=\\\"0\\\"></use></g></svg><span role=\\\"presentation\\\"><math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mo stretchy=\\\"false\\\">(</mo><mfrac><mn>93</mn><mn>95</mn></mfrac><mo>+</mo><mi>ϵ</mi><mo stretchy=\\\"false\\\">)</mo></math></span></span><script type=\\\"math/tex\\\">(\\\\frac{93}{95}+ \\\\epsilon )</script></span>-approximate MMS allocation does not always exist for two agents, while the MMS allocation problem without a knapsack constraint always has an MMS allocation for two agents. We propose a bag-filling based algorithm that can produce a <span><span style=\\\"\\\"></span><span data-mathml='&lt;math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"&gt;&lt;mfrac&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;' role=\\\"presentation\\\" style=\\\"font-size: 100%; display: inline-block; position: relative;\\\" tabindex=\\\"0\\\"><svg aria-hidden=\\\"true\\\" focusable=\\\"false\\\" height=\\\"2.914ex\\\" role=\\\"img\\\" style=\\\"vertical-align: -1.106ex;\\\" viewbox=\\\"0 -778.3 2042.9 1254.7\\\" width=\\\"4.745ex\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g fill=\\\"currentColor\\\" stroke=\\\"currentColor\\\" stroke-width=\\\"0\\\" transform=\\\"matrix(1 0 0 -1 0 0)\\\"><g transform=\\\"translate(120,0)\\\"><rect height=\\\"60\\\" stroke=\\\"none\\\" width=\\\"1802\\\" x=\\\"0\\\" y=\\\"220\\\"></rect><use transform=\\\"scale(0.707)\\\" x=\\\"974\\\" xlink:href=\\\"#MJMATHI-6E\\\" y=\\\"564\\\"></use><g transform=\\\"translate(60,-363)\\\"><use transform=\\\"scale(0.707)\\\" x=\\\"0\\\" xlink:href=\\\"#MJMAIN-33\\\" y=\\\"0\\\"></use><use transform=\\\"scale(0.707)\\\" x=\\\"500\\\" xlink:href=\\\"#MJMATHI-6E\\\" y=\\\"0\\\"></use><use transform=\\\"scale(0.707)\\\" x=\\\"1101\\\" xlink:href=\\\"#MJMAIN-2212\\\" y=\\\"0\\\"></use><use transform=\\\"scale(0.707)\\\" x=\\\"1879\\\" xlink:href=\\\"#MJMAIN-32\\\" y=\\\"0\\\"></use></g></g></g></svg><span role=\\\"presentation\\\"><math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mfrac><mi>n</mi><mrow><mn>3</mn><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span></span><script type=\\\"math/tex\\\">\\\\frac{n}{3n-2}</script></span>-approximate MMS allocation. When <span><span style=\\\"\\\"></span><span data-mathml='&lt;math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;' role=\\\"presentation\\\" style=\\\"font-size: 100%; display: inline-block; position: relative;\\\" tabindex=\\\"0\\\"><svg aria-hidden=\\\"true\\\" focusable=\\\"false\\\" height=\\\"1.912ex\\\" role=\\\"img\\\" style=\\\"vertical-align: -0.205ex;\\\" viewbox=\\\"0 -735.2 2435.1 823.4\\\" width=\\\"5.656ex\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g fill=\\\"currentColor\\\" stroke=\\\"currentColor\\\" stroke-width=\\\"0\\\" transform=\\\"matrix(1 0 0 -1 0 0)\\\"><use x=\\\"0\\\" xlink:href=\\\"#MJMATHI-6E\\\" y=\\\"0\\\"></use><use x=\\\"878\\\" xlink:href=\\\"#MJMAIN-3D\\\" y=\\\"0\\\"></use><use x=\\\"1934\\\" xlink:href=\\\"#MJMAIN-32\\\" y=\\\"0\\\"></use></g></svg><span role=\\\"presentation\\\"><math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>n</mi><mo>=</mo><mn>2</mn></math></span></span><script type=\\\"math/tex\\\">n=2</script></span> and <span><span style=\\\"\\\"></span><span data-mathml='&lt;math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;' role=\\\"presentation\\\" style=\\\"font-size: 100%; display: inline-block; position: relative;\\\" tabindex=\\\"0\\\"><svg aria-hidden=\\\"true\\\" focusable=\\\"false\\\" height=\\\"1.912ex\\\" role=\\\"img\\\" style=\\\"vertical-align: -0.205ex;\\\" viewbox=\\\"0 -735.2 2435.1 823.4\\\" width=\\\"5.656ex\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g fill=\\\"currentColor\\\" stroke=\\\"currentColor\\\" stroke-width=\\\"0\\\" transform=\\\"matrix(1 0 0 -1 0 0)\\\"><use x=\\\"0\\\" xlink:href=\\\"#MJMATHI-6E\\\" y=\\\"0\\\"></use><use x=\\\"878\\\" xlink:href=\\\"#MJMAIN-3D\\\" y=\\\"0\\\"></use><use x=\\\"1934\\\" xlink:href=\\\"#MJMAIN-33\\\" y=\\\"0\\\"></use></g></svg><span role=\\\"presentation\\\"><math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>n</mi><mo>=</mo><mn>3</mn></math></span></span><script type=\\\"math/tex\\\">n=3</script></span>, by more careful analysis, we improve the approximation ratios to <span><span style=\\\"\\\"></span><span data-mathml='&lt;math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"&gt;&lt;mfrac&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;' role=\\\"presentation\\\" style=\\\"font-size: 100%; display: inline-block; position: relative;\\\" tabindex=\\\"0\\\"><svg aria-hidden=\\\"true\\\" focusable=\\\"false\\\" height=\\\"3.209ex\\\" role=\\\"img\\\" style=\\\"vertical-align: -1.005ex;\\\" viewbox=\\\"0 -949.2 713.9 1381.8\\\" width=\\\"1.658ex\\\" xmlns:xlink=\\\"http://www.w3.org/1999/xlink\\\"><g fill=\\\"currentColor\\\" stroke=\\\"currentColor\\\" stroke-width=\\\"0\\\" transform=\\\"matrix(1 0 0 -1 0 0)\\\"><g transform=\\\"translate(120,0)\\\"><rect height=\\\"60\\\" stroke=\\\"none\\\" width=\\\"473\\\" x=\\\"0\\\" y=\\\"220\\\"></rect><use transform=\\\"scale(0.707)\\\" x=\\\"84\\\" xlink:href=\\\"#MJMAIN-32\\\" y=\\\"556\\\"></use><use transform=\\\"scale(0.707)\\\" x=\\\"84\\\" xlink:href=\\\"#MJMAIN-33\\\" y=\\\"-512\\\"></use></g></g></svg><span role=\\\"presentation\\\"><math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mfrac><mn>2</mn><mn>3</mn></mfrac></math></span></span><script type=\\\"math/tex\\\">\\\\frac{2}{3}</script></span> and <span><span style=\\\"\\\"></span><span data-mathml='&lt;math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"&gt;&lt;mfrac&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/math&gt;' role=\\\"presentation\\\" style=\\\"font-size: 100%; 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引用次数: 0

摘要

背包约束下的最大份额分配问题是将一组不可分割的商品分配给一组n个异构智能体,使分配商品的总成本不超过给定的预算,并且最大份额分配的近似比尽可能大。对于任意一个λ∈(0,1)\epsilon\in(0,1),我们证明了(9395+ λ)(\frac{93}{95} + \epsilon)-近似的MMS分配并不总是存在于两个智能体上,而没有背包约束的MMS分配问题总是有两个智能体的MMS分配。我们提出了一种基于袋填充的算法,可以产生n3n−2 \frac{n}{3n-2} -近似的MMS分配。当n=2n=2和n=3n=3时,通过更仔细的分析,我们将近似比分别提高到23 \frac{2}{3}和12 \frac{1}{2}。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Approximate maximin share allocation for indivisible goods under a knapsack constraint

The maximin share (MMS) allocation problem under a knapsack constraint is to allocate a set of indivisible goods to a set of n heterogeneous agents, such that the total cost of the allocated goods does not exceed the given budget, and the approximation ratio of the MMS allocation is as large as possible. For any ϵ(0,1), we prove that (9395+ϵ)-approximate MMS allocation does not always exist for two agents, while the MMS allocation problem without a knapsack constraint always has an MMS allocation for two agents. We propose a bag-filling based algorithm that can produce a n3n2-approximate MMS allocation. When n=2 and n=3, by more careful analysis, we improve the approximation ratios to 23 and 12, respectively.

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来源期刊
Journal of Combinatorial Optimization
Journal of Combinatorial Optimization 数学-计算机:跨学科应用
CiteScore
2.00
自引率
10.00%
发文量
83
审稿时长
6 months
期刊介绍: The objective of Journal of Combinatorial Optimization is to advance and promote the theory and applications of combinatorial optimization, which is an area of research at the intersection of applied mathematics, computer science, and operations research and which overlaps with many other areas such as computation complexity, computational biology, VLSI design, communication networks, and management science. It includes complexity analysis and algorithm design for combinatorial optimization problems, numerical experiments and problem discovery with applications in science and engineering. The Journal of Combinatorial Optimization publishes refereed papers dealing with all theoretical, computational and applied aspects of combinatorial optimization. It also publishes reviews of appropriate books and special issues of journals.
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