Maximum Box Problem on Stochastic Points
Given a finite set of weighted points in R d (where there can be negative weights), the maximum box problem asks for an axis-aligned rectangle (i.e., box) such that the sum of the weights of the points that it contains is maximized. We consider that each point of the input has a probability of being...
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Veröffentlicht in: | Algorithmica 2021-12, Vol.83 (12), p.3741-3765 |
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creator | Caraballo, Luis E. Pérez-Lantero, Pablo Seara, Carlos Ventura, Inmaculada |
description | Given a finite set of weighted points in
R
d
(where there can be negative weights), the maximum box problem asks for an axis-aligned rectangle (i.e., box) such that the sum of the weights of the points that it contains is maximized. We consider that each point of the input has a probability of being present in the final random point set, and these events are mutually independent; then, the total weight of a maximum box is a random variable. We aim to compute both the probability that this variable is at least a given parameter, and its expectation. We show that even in
d
=
1
these computations are #P-hard, and give pseudo-polynomial time algorithms in the case where the weights are integers in a bounded interval. For
d
=
2
, we consider that each point is colored red or blue, where red points have weight
+
1
and blue points weight
-
∞
. The random variable is the maximum number of red points that can be covered with a box not containing any blue point. We prove that the above two computations are also #P-hard, and give a polynomial-time algorithm for computing the probability that there is a box containing exactly two red points, no blue point, and a given point of the plane. |
doi_str_mv | 10.1007/s00453-021-00882-z |
format | Article |
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R
d
(where there can be negative weights), the maximum box problem asks for an axis-aligned rectangle (i.e., box) such that the sum of the weights of the points that it contains is maximized. We consider that each point of the input has a probability of being present in the final random point set, and these events are mutually independent; then, the total weight of a maximum box is a random variable. We aim to compute both the probability that this variable is at least a given parameter, and its expectation. We show that even in
d
=
1
these computations are #P-hard, and give pseudo-polynomial time algorithms in the case where the weights are integers in a bounded interval. For
d
=
2
, we consider that each point is colored red or blue, where red points have weight
+
1
and blue points weight
-
∞
. The random variable is the maximum number of red points that can be covered with a box not containing any blue point. We prove that the above two computations are also #P-hard, and give a polynomial-time algorithm for computing the probability that there is a box containing exactly two red points, no blue point, and a given point of the plane.</description><identifier>ISSN: 0178-4617</identifier><identifier>EISSN: 1432-0541</identifier><identifier>DOI: 10.1007/s00453-021-00882-z</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algorithm Analysis and Problem Complexity ; Algorithms ; Computer Science ; Computer Systems Organization and Communication Networks ; Data Structures and Information Theory ; Graphs ; Mathematics of Computing ; Polynomials ; Random variables ; Theory of Computation</subject><ispartof>Algorithmica, 2021-12, Vol.83 (12), p.3741-3765</ispartof><rights>The Author(s) 2021</rights><rights>The Author(s) 2021. This work is published under http://creativecommons.org/licenses/by/4.0/ (the “License”). Notwithstanding the ProQuest Terms and Conditions, you may use this content in accordance with the terms of the License.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c314t-5379f73969eaa9f41b46130b921397cc5e808b04873377dac626f7db6486e1973</cites><orcidid>0000-0002-0095-1725</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00453-021-00882-z$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00453-021-00882-z$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Caraballo, Luis E.</creatorcontrib><creatorcontrib>Pérez-Lantero, Pablo</creatorcontrib><creatorcontrib>Seara, Carlos</creatorcontrib><creatorcontrib>Ventura, Inmaculada</creatorcontrib><title>Maximum Box Problem on Stochastic Points</title><title>Algorithmica</title><addtitle>Algorithmica</addtitle><description>Given a finite set of weighted points in
R
d
(where there can be negative weights), the maximum box problem asks for an axis-aligned rectangle (i.e., box) such that the sum of the weights of the points that it contains is maximized. We consider that each point of the input has a probability of being present in the final random point set, and these events are mutually independent; then, the total weight of a maximum box is a random variable. We aim to compute both the probability that this variable is at least a given parameter, and its expectation. We show that even in
d
=
1
these computations are #P-hard, and give pseudo-polynomial time algorithms in the case where the weights are integers in a bounded interval. For
d
=
2
, we consider that each point is colored red or blue, where red points have weight
+
1
and blue points weight
-
∞
. The random variable is the maximum number of red points that can be covered with a box not containing any blue point. We prove that the above two computations are also #P-hard, and give a polynomial-time algorithm for computing the probability that there is a box containing exactly two red points, no blue point, and a given point of the plane.</description><subject>Algorithm Analysis and Problem Complexity</subject><subject>Algorithms</subject><subject>Computer Science</subject><subject>Computer Systems Organization and Communication Networks</subject><subject>Data Structures and Information Theory</subject><subject>Graphs</subject><subject>Mathematics of Computing</subject><subject>Polynomials</subject><subject>Random variables</subject><subject>Theory of Computation</subject><issn>0178-4617</issn><issn>1432-0541</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><recordid>eNp9kEtLAzEYRYMoOFb_gKsBN26iX97JUosvqFhQ1yGTZnRKZ1KTKdT-ekdHcOfqbs69Fw5CpwQuCIC6zABcMAyUYACtKd7toYJwRjEITvZRAURpzCVRh-go5yUAocrIAp0_um3TbtryOm7LeYrVKrRl7MrnPvp3l_vGl_PYdH0-Rge1W-Vw8psT9Hp78zK9x7Onu4fp1Qx7RniPBVOmVsxIE5wzNSfVcMqgMpQwo7wXQYOugGvFmFIL5yWVtVpUkmsZiFFsgs7G3XWKH5uQe7uMm9QNl5YKo6XQQomBoiPlU8w5hdquU9O69GkJ2G8jdjRiByP2x4jdDSU2lvIAd28h_U3_0_oCaBxheQ</recordid><startdate>20211201</startdate><enddate>20211201</enddate><creator>Caraballo, Luis E.</creator><creator>Pérez-Lantero, Pablo</creator><creator>Seara, Carlos</creator><creator>Ventura, Inmaculada</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-0095-1725</orcidid></search><sort><creationdate>20211201</creationdate><title>Maximum Box Problem on Stochastic Points</title><author>Caraballo, Luis E. ; Pérez-Lantero, Pablo ; Seara, Carlos ; Ventura, Inmaculada</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c314t-5379f73969eaa9f41b46130b921397cc5e808b04873377dac626f7db6486e1973</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Algorithm Analysis and Problem Complexity</topic><topic>Algorithms</topic><topic>Computer Science</topic><topic>Computer Systems Organization and Communication Networks</topic><topic>Data Structures and Information Theory</topic><topic>Graphs</topic><topic>Mathematics of Computing</topic><topic>Polynomials</topic><topic>Random variables</topic><topic>Theory of Computation</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Caraballo, Luis E.</creatorcontrib><creatorcontrib>Pérez-Lantero, Pablo</creatorcontrib><creatorcontrib>Seara, Carlos</creatorcontrib><creatorcontrib>Ventura, Inmaculada</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><jtitle>Algorithmica</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Caraballo, Luis E.</au><au>Pérez-Lantero, Pablo</au><au>Seara, Carlos</au><au>Ventura, Inmaculada</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Maximum Box Problem on Stochastic Points</atitle><jtitle>Algorithmica</jtitle><stitle>Algorithmica</stitle><date>2021-12-01</date><risdate>2021</risdate><volume>83</volume><issue>12</issue><spage>3741</spage><epage>3765</epage><pages>3741-3765</pages><issn>0178-4617</issn><eissn>1432-0541</eissn><abstract>Given a finite set of weighted points in
R
d
(where there can be negative weights), the maximum box problem asks for an axis-aligned rectangle (i.e., box) such that the sum of the weights of the points that it contains is maximized. We consider that each point of the input has a probability of being present in the final random point set, and these events are mutually independent; then, the total weight of a maximum box is a random variable. We aim to compute both the probability that this variable is at least a given parameter, and its expectation. We show that even in
d
=
1
these computations are #P-hard, and give pseudo-polynomial time algorithms in the case where the weights are integers in a bounded interval. For
d
=
2
, we consider that each point is colored red or blue, where red points have weight
+
1
and blue points weight
-
∞
. The random variable is the maximum number of red points that can be covered with a box not containing any blue point. We prove that the above two computations are also #P-hard, and give a polynomial-time algorithm for computing the probability that there is a box containing exactly two red points, no blue point, and a given point of the plane.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s00453-021-00882-z</doi><tpages>25</tpages><orcidid>https://orcid.org/0000-0002-0095-1725</orcidid><oa>free_for_read</oa></addata></record> |
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subjects | Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Graphs Mathematics of Computing Polynomials Random variables Theory of Computation |
title | Maximum Box Problem on Stochastic Points |
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