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{"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='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>&#x03F5;</mi><mo>&#x2208;</mo><mo stretchy=\"false\">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math>' 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='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mo stretchy=\"false\">(</mo><mfrac><mn>93</mn><mn>95</mn></mfrac><mo>+</mo><mi>&#x03F5;</mi><mo stretchy=\"false\">)</mo></math>' 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='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mi>n</mi><mrow><mn>3</mn><mi>n</mi><mo>&#x2212;</mo><mn>2</mn></mrow></mfrac></math>' 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='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi><mo>=</mo><mn>2</mn></math>' 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='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>n</mi><mo>=</mo><mn>3</mn></math>' 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='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mn>2</mn><mn>3</mn></mfrac></math>' 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='<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mfrac><mn>1</mn><mn>2</mn></mfrac></math>' 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='<math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>&#x03F5;</mi><mo>&#x2208;</mo><mo stretchy=\\\"false\\\">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy=\\\"false\\\">)</mo></math>' 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='<math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mo stretchy=\\\"false\\\">(</mo><mfrac><mn>93</mn><mn>95</mn></mfrac><mo>+</mo><mi>&#x03F5;</mi><mo stretchy=\\\"false\\\">)</mo></math>' 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='<math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mfrac><mi>n</mi><mrow><mn>3</mn><mi>n</mi><mo>&#x2212;</mo><mn>2</mn></mrow></mfrac></math>' 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='<math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>n</mi><mo>=</mo><mn>2</mn></math>' 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='<math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mi>n</mi><mo>=</mo><mn>3</mn></math>' 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='<math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mfrac><mn>2</mn><mn>3</mn></mfrac></math>' 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='<math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\"><mfrac><mn>1</mn><mn>2</mn></mfrac></math>' 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\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Optimization\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10878-025-01331-1\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10878-025-01331-1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
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