Approximation algorithms for maximum independent set of a unit disk graph
We propose a 2-approximation algorithm for the maximum independent set problem for a unit disk graph. The time and space complexities are O(n3) and O(n2), respectively. For a penny graph, our proposed 2-approximation algorithm works in O(nlogn) time using O(n) space. We also propose a polynomial-ti...
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| Veröffentlicht in: | Information processing letters 2015-03, Vol.115 (3), p.439-446 |
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| creator | Das, Gautam K. De, Minati Kolay, Sudeshna Nandy, Subhas C. Sur-Kolay, Susmita |
| description | We propose a 2-approximation algorithm for the maximum independent set problem for a unit disk graph. The time and space complexities are O(n3) and O(n2), respectively. For a penny graph, our proposed 2-approximation algorithm works in O(nlogn) time using O(n) space. We also propose a polynomial-time approximation scheme (PTAS) for the maximum independent set problem for a unit disk graph. Given an integer k>1, it produces a solution of size 1(1+1k)2|OPT| in O(k4nσklogk+nlogn) time and O(n+klogk) space, where OPT is the subset of disks in an optimal solution and σk≤7k3+2. For a penny graph, the proposed PTAS produces a solution of size 1(1+1k)|OPT| in O(22σknk+nlogn) time using O(2σk+n) space.
•A 2-factor approximation for computing the maximum independent set of unit disk graph is proposed. It runs in O(n3) time and O(n2) space.•A similar technique works for penny graph in O(nlogn) time and produces a 2-approximation result.•Efficient PTAS are proposed for computing the maximum independent set of unit disk graph and penny graph. |
| doi_str_mv | 10.1016/j.ipl.2014.11.002 |
| format | Article |
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•A 2-factor approximation for computing the maximum independent set of unit disk graph is proposed. It runs in O(n3) time and O(n2) space.•A similar technique works for penny graph in O(nlogn) time and produces a 2-approximation result.•Efficient PTAS are proposed for computing the maximum independent set of unit disk graph and penny graph.</description><identifier>ISSN: 0020-0190</identifier><identifier>EISSN: 1872-6119</identifier><identifier>DOI: 10.1016/j.ipl.2014.11.002</identifier><identifier>CODEN: IFPLAT</identifier><language>eng</language><publisher>Amsterdam: Elsevier B.V</publisher><subject>Algorithms ; Approximation ; Approximation algorithms ; Complexity ; Computational geometry ; Computer science ; Data processing ; Disks ; Graph algorithms ; Graphs ; Integer programming ; Integers ; Mathematical analysis ; Mathematical problems ; Maximum independent set ; Optimization ; Optimization algorithms ; PTAS ; Studies ; Theorems ; Unit disk graph</subject><ispartof>Information processing letters, 2015-03, Vol.115 (3), p.439-446</ispartof><rights>2014 Elsevier B.V.</rights><rights>Copyright Elsevier Sequoia S.A. Mar 2015</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c358t-edebd324e789f991b1e663f24eddccff898fa41ab5ad38d8e3e8dfd896d13b543</citedby><cites>FETCH-LOGICAL-c358t-edebd324e789f991b1e663f24eddccff898fa41ab5ad38d8e3e8dfd896d13b543</cites><orcidid>0000-0002-1859-8800 ; 0000-0003-0330-2891</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://dx.doi.org/10.1016/j.ipl.2014.11.002$$EHTML$$P50$$Gelsevier$$H</linktohtml><link.rule.ids>314,780,784,3550,27924,27925,45995</link.rule.ids></links><search><creatorcontrib>Das, Gautam K.</creatorcontrib><creatorcontrib>De, Minati</creatorcontrib><creatorcontrib>Kolay, Sudeshna</creatorcontrib><creatorcontrib>Nandy, Subhas C.</creatorcontrib><creatorcontrib>Sur-Kolay, Susmita</creatorcontrib><title>Approximation algorithms for maximum independent set of a unit disk graph</title><title>Information processing letters</title><description>We propose a 2-approximation algorithm for the maximum independent set problem for a unit disk graph. The time and space complexities are O(n3) and O(n2), respectively. For a penny graph, our proposed 2-approximation algorithm works in O(nlogn) time using O(n) space. We also propose a polynomial-time approximation scheme (PTAS) for the maximum independent set problem for a unit disk graph. Given an integer k>1, it produces a solution of size 1(1+1k)2|OPT| in O(k4nσklogk+nlogn) time and O(n+klogk) space, where OPT is the subset of disks in an optimal solution and σk≤7k3+2. For a penny graph, the proposed PTAS produces a solution of size 1(1+1k)|OPT| in O(22σknk+nlogn) time using O(2σk+n) space.
•A 2-factor approximation for computing the maximum independent set of unit disk graph is proposed. It runs in O(n3) time and O(n2) space.•A similar technique works for penny graph in O(nlogn) time and produces a 2-approximation result.•Efficient PTAS are proposed for computing the maximum independent set of unit disk graph and penny graph.</description><subject>Algorithms</subject><subject>Approximation</subject><subject>Approximation algorithms</subject><subject>Complexity</subject><subject>Computational geometry</subject><subject>Computer science</subject><subject>Data processing</subject><subject>Disks</subject><subject>Graph algorithms</subject><subject>Graphs</subject><subject>Integer programming</subject><subject>Integers</subject><subject>Mathematical analysis</subject><subject>Mathematical problems</subject><subject>Maximum independent set</subject><subject>Optimization</subject><subject>Optimization algorithms</subject><subject>PTAS</subject><subject>Studies</subject><subject>Theorems</subject><subject>Unit disk graph</subject><issn>0020-0190</issn><issn>1872-6119</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2015</creationdate><recordtype>article</recordtype><recordid>eNp9kD1PwzAQhi0EEqXwA9gssbAk-OLUdcRUVXxUqsQCs-XG59YliYOdIPj3uCoTA4tPunte6-4h5BpYDgzE3T53fZMXDMocIGesOCETkPMiEwDVKZmkDssYVOycXMS4Z4yJks8nZLXo--C_XKsH5zuqm60Pbti1kVofaKvTZGyp6wz2mJ5uoBEH6i3VdOzcQI2L73QbdL-7JGdWNxGvfuuUvD0-vC6fs_XL02q5WGc1n8khQ4Mbw4sS57KyVQUbQCG4TQ1j6tpaWUmrS9CbmTZcGokcpbFGVsIA38xKPiW3x3_T3h8jxkG1LtbYNLpDP0YFQlRScikgoTd_0L0fQ5e2SxQXpZRM8ETBkaqDjzGgVX1IPsK3AqYOctVeJbnqIFcBqKQyZe6PGUyXfjoMKtYOuxqNC1gPynj3T_oH7KuCwA</recordid><startdate>20150301</startdate><enddate>20150301</enddate><creator>Das, Gautam K.</creator><creator>De, Minati</creator><creator>Kolay, Sudeshna</creator><creator>Nandy, Subhas C.</creator><creator>Sur-Kolay, Susmita</creator><general>Elsevier B.V</general><general>Elsevier Sequoia S.A</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><orcidid>https://orcid.org/0000-0002-1859-8800</orcidid><orcidid>https://orcid.org/0000-0003-0330-2891</orcidid></search><sort><creationdate>20150301</creationdate><title>Approximation algorithms for maximum independent set of a unit disk graph</title><author>Das, Gautam K. ; De, Minati ; Kolay, Sudeshna ; Nandy, Subhas C. ; Sur-Kolay, Susmita</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c358t-edebd324e789f991b1e663f24eddccff898fa41ab5ad38d8e3e8dfd896d13b543</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2015</creationdate><topic>Algorithms</topic><topic>Approximation</topic><topic>Approximation algorithms</topic><topic>Complexity</topic><topic>Computational geometry</topic><topic>Computer science</topic><topic>Data processing</topic><topic>Disks</topic><topic>Graph algorithms</topic><topic>Graphs</topic><topic>Integer programming</topic><topic>Integers</topic><topic>Mathematical analysis</topic><topic>Mathematical problems</topic><topic>Maximum independent set</topic><topic>Optimization</topic><topic>Optimization algorithms</topic><topic>PTAS</topic><topic>Studies</topic><topic>Theorems</topic><topic>Unit disk graph</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Das, Gautam K.</creatorcontrib><creatorcontrib>De, Minati</creatorcontrib><creatorcontrib>Kolay, Sudeshna</creatorcontrib><creatorcontrib>Nandy, Subhas C.</creatorcontrib><creatorcontrib>Sur-Kolay, Susmita</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Information processing letters</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Das, Gautam K.</au><au>De, Minati</au><au>Kolay, Sudeshna</au><au>Nandy, Subhas C.</au><au>Sur-Kolay, Susmita</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Approximation algorithms for maximum independent set of a unit disk graph</atitle><jtitle>Information processing letters</jtitle><date>2015-03-01</date><risdate>2015</risdate><volume>115</volume><issue>3</issue><spage>439</spage><epage>446</epage><pages>439-446</pages><issn>0020-0190</issn><eissn>1872-6119</eissn><coden>IFPLAT</coden><abstract>We propose a 2-approximation algorithm for the maximum independent set problem for a unit disk graph. The time and space complexities are O(n3) and O(n2), respectively. For a penny graph, our proposed 2-approximation algorithm works in O(nlogn) time using O(n) space. We also propose a polynomial-time approximation scheme (PTAS) for the maximum independent set problem for a unit disk graph. Given an integer k>1, it produces a solution of size 1(1+1k)2|OPT| in O(k4nσklogk+nlogn) time and O(n+klogk) space, where OPT is the subset of disks in an optimal solution and σk≤7k3+2. For a penny graph, the proposed PTAS produces a solution of size 1(1+1k)|OPT| in O(22σknk+nlogn) time using O(2σk+n) space.
•A 2-factor approximation for computing the maximum independent set of unit disk graph is proposed. It runs in O(n3) time and O(n2) space.•A similar technique works for penny graph in O(nlogn) time and produces a 2-approximation result.•Efficient PTAS are proposed for computing the maximum independent set of unit disk graph and penny graph.</abstract><cop>Amsterdam</cop><pub>Elsevier B.V</pub><doi>10.1016/j.ipl.2014.11.002</doi><tpages>8</tpages><orcidid>https://orcid.org/0000-0002-1859-8800</orcidid><orcidid>https://orcid.org/0000-0003-0330-2891</orcidid></addata></record> |
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| subjects | Algorithms Approximation Approximation algorithms Complexity Computational geometry Computer science Data processing Disks Graph algorithms Graphs Integer programming Integers Mathematical analysis Mathematical problems Maximum independent set Optimization Optimization algorithms PTAS Studies Theorems Unit disk graph |
| title | Approximation algorithms for maximum independent set of a unit disk graph |
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