On a quadratic programming problem involving distances in trees

Let T be a tree and let D be the distance matrix of the tree. The problem of finding the maximum of x ′ D x subject to x being a nonnegative vector with sum one occurs in many different contexts. These include some classical work on the transfinite diameter of a finite metric space, equilibrium poin...

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Vydané v:	Annals of operations research Ročník 243; číslo 1-2; s. 365 - 373
Hlavní autori:	Bapat, R. B., Neogy, S. K.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.08.2016 Springer Springer Nature B.V
Predmet:	Business and Management Combinatorics Equilibrium Game theory Games Graphs Mathematical analysis Mathematical research Metric space Operations research Operations Research/Decision Theory Polynomials Quadratic programming Texts Theory of Computation Trees Trees (Graph theory) Vectors (mathematics) 90C20 Quadratic programming problem Finite metric space 94C15 Tree Symmetric bimatrix game Lemke’s algorithm 05C05 Distance matrix Polynomial time
ISSN:	0254-5330, 1572-9338
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Abstract	Let T be a tree and let D be the distance matrix of the tree. The problem of finding the maximum of x ′ D x subject to x being a nonnegative vector with sum one occurs in many different contexts. These include some classical work on the transfinite diameter of a finite metric space, equilibrium points of symmetric bimatrix games and maximizing weighted average distance in graphs. We show that the problem can be converted into a strictly convex quadratic programming problem and hence it can be solved in polynomial time.
AbstractList	Let T be a tree and let D be the distance matrix of the tree. The problem of finding the maximum of x ′ D x subject to x being a nonnegative vector with sum one occurs in many different contexts. These include some classical work on the transfinite diameter of a finite metric space, equilibrium points of symmetric bimatrix games and maximizing weighted average distance in graphs. We show that the problem can be converted into a strictly convex quadratic programming problem and hence it can be solved in polynomial time. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image).Let ... be a tree and let ... be the distance matrix of the tree. The problem of finding the maximum of ... subject to ... being a nonnegative vector with sum one occurs in many different contexts. These include some classical work on the transfinite diameter of a finite metric space, equilibrium points of symmetric bimatrix games and maximizing weighted average distance in graphs. We show that the problem can be converted into a strictly convex quadratic programming problem and hence it can be solved in polynomial time. Let be a tree and let be the distance matrix of the tree. The problem of finding the maximum of subject to being a nonnegative vector with sum one occurs in many different contexts. These include some classical work on the transfinite diameter of a finite metric space, equilibrium points of symmetric bimatrix games and maximizing weighted average distance in graphs. We show that the problem can be converted into a strictly convex quadratic programming problem and hence it can be solved in polynomial time.
Audience	Academic
Author	Neogy, S. K. Bapat, R. B.
Author_xml	– sequence: 1 givenname: R. B. surname: Bapat fullname: Bapat, R. B. email: rbb@isid.ac.in organization: Theoretical Statistics and Mathematics Unit, Indian Statistical Institute – sequence: 2 givenname: S. K. surname: Neogy fullname: Neogy, S. K. organization: Statistical Quality Control and Operations Research Unit, Indian Statistical Institute
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References	TardosEA strongly polynomial algorithm to solve combinatorial linear programsOperations Research198634225025610.1287/opre.34.2.250 Bapat, R. B. (2010). Graphs and matrices. London/New Delhi: Springer/Hindustan Book Agency. WestDIntroduction to graph theory20012Englewood Cliffs, NJPrentice-Hall KojimaMMizunoSYoshiseAA polynomial-time algorithm for a class of linear complementarity problemsMathematical Programming19894412610.1007/BF01587074 GranotFSkorin-KapovJTowards a strongly polynomial algorithm for strictly convex quadratic programs: An extension of Tardos’ algorithmMathematical Programming19904622523610.1007/BF01585740 Chung, F. R. K. (1986). Diameters of communication networks. Mathematics of information processing (Louisville, Ky., 1984). In Proceedings of the symposia in applied mathematics (Vol. 34, pp. 1–18). Providence, RI: American Mathematical Society. LemkeCEBimatrix equilibrium points and mathematical programmingManagement Science19651168168910.1287/mnsc.11.7.681 BapatRBRaghavanTESNonnegative matrices and applications, encyclopedia of mathematics and its applications1997CambridgeCambridge University Press10.1017/CBO9780511529979 MurtyKGKabadiSNSome NP-complete problems in quadratic and nonlinear programmingMathematical Programming19873911712910.1007/BF02592948 HjorthPLisoněkPMarkvorsenSThomassencFinite metric spaces of strictly negative typeLinear Algebra and its Applications199827025527310.1016/S0024-3795(97)00242-5 ParsonsTDKuhnHWApplications of principal pivotingProceedings of the Princeton symposium on mathematical programming1970Princeton, NJPrinceton University Press567581 DankelmannPAverage distance in weighted graphsDiscrete Mathematics2012312122010.1016/j.disc.2011.02.010 BomzeIMRegularity versus degeneracy in dynamics, games and optimization: A unified approach to different aspectsSIAM Review200244339441410.1137/S00361445003756 CottleRWPangJSStoneREThe linear complementarity problem1992New YorkAcademic Press BapatRBSivasubramanianSIdentities for minors of the Laplacian, resistance and distance matricesLinear Algebra and its Applications20114351479148910.1016/j.laa.2011.03.028 RB Bapat (1743_CR2) 1997 P Dankelmann (1743_CR7) 2012; 312 TD Parsons (1743_CR13) 1970 RW Cottle (1743_CR5) 1992 1743_CR6 CE Lemke (1743_CR11) 1965; 11 1743_CR1 P Hjorth (1743_CR9) 1998; 270 M Kojima (1743_CR10) 1989; 44 RB Bapat (1743_CR3) 2011; 435 F Granot (1743_CR8) 1990; 46 D West (1743_CR15) 2001 E Tardos (1743_CR14) 1986; 34 IM Bomze (1743_CR4) 2002; 44 KG Murty (1743_CR12) 1987; 39
References_xml	– reference: Bapat, R. B. (2010). Graphs and matrices. London/New Delhi: Springer/Hindustan Book Agency. – reference: LemkeCEBimatrix equilibrium points and mathematical programmingManagement Science19651168168910.1287/mnsc.11.7.681 – reference: MurtyKGKabadiSNSome NP-complete problems in quadratic and nonlinear programmingMathematical Programming19873911712910.1007/BF02592948 – reference: DankelmannPAverage distance in weighted graphsDiscrete Mathematics2012312122010.1016/j.disc.2011.02.010 – reference: BapatRBRaghavanTESNonnegative matrices and applications, encyclopedia of mathematics and its applications1997CambridgeCambridge University Press10.1017/CBO9780511529979 – reference: GranotFSkorin-KapovJTowards a strongly polynomial algorithm for strictly convex quadratic programs: An extension of Tardos’ algorithmMathematical Programming19904622523610.1007/BF01585740 – reference: CottleRWPangJSStoneREThe linear complementarity problem1992New YorkAcademic Press – reference: ParsonsTDKuhnHWApplications of principal pivotingProceedings of the Princeton symposium on mathematical programming1970Princeton, NJPrinceton University Press567581 – reference: KojimaMMizunoSYoshiseAA polynomial-time algorithm for a class of linear complementarity problemsMathematical Programming19894412610.1007/BF01587074 – reference: HjorthPLisoněkPMarkvorsenSThomassencFinite metric spaces of strictly negative typeLinear Algebra and its Applications199827025527310.1016/S0024-3795(97)00242-5 – reference: TardosEA strongly polynomial algorithm to solve combinatorial linear programsOperations Research198634225025610.1287/opre.34.2.250 – reference: BomzeIMRegularity versus degeneracy in dynamics, games and optimization: A unified approach to different aspectsSIAM Review200244339441410.1137/S00361445003756 – reference: Chung, F. R. K. (1986). Diameters of communication networks. Mathematics of information processing (Louisville, Ky., 1984). In Proceedings of the symposia in applied mathematics (Vol. 34, pp. 1–18). Providence, RI: American Mathematical Society. – reference: WestDIntroduction to graph theory20012Englewood Cliffs, NJPrentice-Hall – reference: BapatRBSivasubramanianSIdentities for minors of the Laplacian, resistance and distance matricesLinear Algebra and its Applications20114351479148910.1016/j.laa.2011.03.028 – volume: 11 start-page: 681 year: 1965 ident: 1743_CR11 publication-title: Management Science doi: 10.1287/mnsc.11.7.681 – volume: 44 start-page: 394 issue: 3 year: 2002 ident: 1743_CR4 publication-title: SIAM Review doi: 10.1137/S00361445003756 – volume: 270 start-page: 255 year: 1998 ident: 1743_CR9 publication-title: Linear Algebra and its Applications doi: 10.1016/S0024-3795(97)00242-5 – volume-title: The linear complementarity problem year: 1992 ident: 1743_CR5 – volume: 46 start-page: 225 year: 1990 ident: 1743_CR8 publication-title: Mathematical Programming doi: 10.1007/BF01585740 – ident: 1743_CR1 doi: 10.1007/978-1-84882-981-7 – ident: 1743_CR6 doi: 10.1090/psapm/034/846852 – volume: 44 start-page: 1 year: 1989 ident: 1743_CR10 publication-title: Mathematical Programming doi: 10.1007/BF01587074 – volume: 34 start-page: 250 issue: 2 year: 1986 ident: 1743_CR14 publication-title: Operations Research doi: 10.1287/opre.34.2.250 – volume: 39 start-page: 117 year: 1987 ident: 1743_CR12 publication-title: Mathematical Programming doi: 10.1007/BF02592948 – volume: 435 start-page: 1479 year: 2011 ident: 1743_CR3 publication-title: Linear Algebra and its Applications doi: 10.1016/j.laa.2011.03.028 – start-page: 567 volume-title: Proceedings of the Princeton symposium on mathematical programming year: 1970 ident: 1743_CR13 – volume-title: Introduction to graph theory year: 2001 ident: 1743_CR15 – volume-title: Nonnegative matrices and applications, encyclopedia of mathematics and its applications year: 1997 ident: 1743_CR2 doi: 10.1017/CBO9780511529979 – volume: 312 start-page: 1220 year: 2012 ident: 1743_CR7 publication-title: Discrete Mathematics doi: 10.1016/j.disc.2011.02.010
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SubjectTerms	Business and Management Combinatorics Equilibrium Game theory Games Graphs Mathematical analysis Mathematical research Metric space Operations research Operations Research/Decision Theory Polynomials Quadratic programming Texts Theory of Computation Trees Trees (Graph theory) Vectors (mathematics)
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