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Research on Algorithms in Discrete Convex Analysis

Research Project

Project/Area Number 15540118
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field General mathematics (including Probability theory/Statistical mathematics)
Research InstitutionKEIO UNIVERSITY (2004-2005)
Kyoto University (2003)

Principal Investigator

TAMURA Akihisa  Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (50217189)

Co-Investigator(Kenkyū-buntansha) FUJISHIGE Satoru  Keio Univ., RIMS, Professor, 数理解析研究所, 教授 (10092321)
MUROTA Kazuo  Univ.of Tokyo, Graduate School of Info.and Tech., Professor, 大学院・情報理工学系研究科, 教授 (50134466)
OHTA Katsuhiro  Keio Univ., Faculty of Sci.and Tech., Professor, 理工学部, 教授 (40213722)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2005: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2004: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2003: ¥1,200,000 (Direct Cost: ¥1,200,000)
Keywordsdiscrete convex analysis / algorithms / mathematical economics / game theory
Research Abstract

This research project had three aims on algorithms in discrete convex analysis : (1)development of a polynomial time algorithm for minimization of M_2 convex function ; (2)study on algorithms for minimizing continuous M-/L-convex functions ; and (3)study on relations between discrete convex analysis and convex function minimization problems.
Our group achieved the first aim. Murota who is one of our group together with Iwata and Moriguchi, developed a polynomial time algorithm for solving M-convex submodular flow problem which is equivalent to M_2 convex function minimization.
We are investigating the second aim.
On the third aim, we obtained several results and future works. These are divided into two subjects.
The first subject is an application of discrete convex analysis to two-sided matching market models. Fujishige and Tamura proposed several two-sided matching models by utilizing discrete convex analysis. These models contain many known models as special cases. They proved that their models always have pairwise-stable outcomes by modifying an algorithm for M_2-convex function minimization. Moreover, the proofs give polynomial time algorithms for finding pairwise-stable outcomes in the case where effective domains are contained in 0-1 hypercubes. Tamura together with Farooq investigated properties of M-convex functions in terms of mathematical economics.
The second subject is M-convex functions in jump systems. Murota gave a concept of M-convex functions in jump systems, an optimality criterion of these functions, a minimization algorithm.
Moreover, Tamura proposed a polynomial time algorithm for minimizing an M-convex function by using coordinatewise scaling technique which is a new idea in combinatorial optimization.

Report

(4 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (31 results)

All 2006 2005 2004 2003 Other

All Journal Article (28 results) Publications (3 results)

  • [Journal Article] A general two-sided matching market with discrete concave utility functions2006

    • Author(s)
      Fujishige, S.
    • Journal Title

      Discrete Applied Mathematics 154

      Pages: 950-970

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A steepest descent algorithm for M-convex functions on jump systems2006

    • Author(s)
      Murota, K.
    • Journal Title

      IEICE Trans. Fundanentals E89-A

      Pages: 1160-1165

    • NAID

      110007502826

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A general two-sided matching market with discrete concave utility functions2006

    • Author(s)
      Fujishige, S.
    • Journal Title

      Discrete Applied Mathematics vol.154

      Pages: 950-970

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A steepest descent algorithm for M-convex functions on jump systems2006

    • Author(s)
      Murota, K.
    • Journal Title

      IEICE Trans.Fundamentals vol.E89-A

      Pages: 1160-1165

    • NAID

      110007502826

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Coordinatewise domain scaling algorithm for M-convex function minimization2005

    • Author(s)
      Tamura, A.
    • Journal Title

      Mathematical Programming, Ser. A 102

      Pages: 339-354

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Discrete fixed point theorem reconsidered2005

    • Author(s)
      Iimura, T.
    • Journal Title

      Journal of Mathematical Economics 41

      Pages: 1030-1036

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A capacity scaling algorithm for M-convex submodular flow2005

    • Author(s)
      Iwata, S.
    • Journal Title

      Mathematical Programming, Ser. A 103

      Pages: 181-202

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Coordinatewise domain scaling algorithm for M-convex function minimization2005

    • Author(s)
      Tamura, A.
    • Journal Title

      Mathematical Programming, Ser.A vol.102

      Pages: 339-354

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Discrete fixed point theorem reconsidered2005

    • Author(s)
      Iimura, T.
    • Journal Title

      Journal of Mathematical Economics vol.41

      Pages: 1030-1036

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A capacity scaling algorithm for M-convex submodular flow2005

    • Author(s)
      Iwata, S.
    • Journal Title

      Mathematical Programming, Ser.A vol.103

      Pages: 181-202

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Discrete fixed point theorem reconsidered2005

    • Author(s)
      T.Iimura, K.Murota, A.Tamura
    • Journal Title

      Journal of Mathematical Economics 41

      Pages: 1030-1036

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A capacity scaling algorithm for M-convex submodular flow2005

    • Author(s)
      S.Iwata, S.Moriguchi, K.Murota
    • Journal Title

      Mathematical Programming 103・1

      Pages: 181-202

    • NAID

      110002812004

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Coordinatewise domain scaling algorithm for M-convex function minimization2005

    • Author(s)
      Tamura, A.
    • Journal Title

      Mathematical Programming 102・2

      Pages: 339-354

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Use of primal-dual technique in the network algorithm for two-way contingency tables2005

    • Author(s)
      T.Suzuki, S.Aoki, K.Murota
    • Journal Title

      Japan Journal of Industrial and Applied Mathematics 22・1

      Pages: 133-145

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Applications of discrete convex analysis to mathematical economics2004

    • Author(s)
      Tamura, A.
    • Journal Title

      Publications of RIMS 40

      Pages: 1015-1037

    • NAID

      110001050934

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A new characterization of M^#-convex set functions by substitutability2004

    • Author(s)
      Farooq, R.
    • Journal Title

      Journal of the Operations Research Society of Japan 47,1

      Pages: 18-24

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Applications of discrete convex analysis to mathematical economics2004

    • Author(s)
      Tamura, A.
    • Journal Title

      Publications of RIMS vol.40

      Pages: 1015-1037

    • NAID

      110001050934

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A new characterization of M^#-convex set functions by substitutability2004

    • Author(s)
      Farooq, R.
    • Journal Title

      Journal of the Operations Research Society of Japan vol.47

      Pages: 18-24

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A new characterization of M#-convex set functions by substitutability2004

    • Author(s)
      Farooq, R., Tamura, A.
    • Journal Title

      Journal of the Operations Research Society of Japan 47・1

      Pages: 18-24

    • NAID

      110001183589

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Applications of discrete convex analysis to mathematical economics2004

    • Author(s)
      Tamura, A.
    • Journal Title

      Publications of RIMS, Kyoto University 40・3

      Pages: 1015-1037

    • NAID

      110001050934

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A generalized Gale-Shapley Algorithm for a discrete-concave stable-marriage model2003

    • Author(s)
      Eguchi, A.
    • Journal Title

      Lecture Notes in Computer Science 2906

      Pages: 495-504

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A generalized Gale-Shapley Algorithm for a discrete-concave stable-marriage model2003

    • Author(s)
      Eguchi, A.
    • Journal Title

      Lecture Notes in Computer Science vol.2906

      Pages: 495-504

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A two-sided discrete-concave market with possibly bounded side payments : An approach by discrete convex analysis

    • Author(s)
      Fujishige, S.
    • Journal Title

      Mathematics of Operations Research 発表予定

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A two-sided discrete-concave market with possibly bounded side payments : An approach by discrete convex analysis

    • Author(s)
      Fujishige, S.
    • Journal Title

      Mathematics of Operations Research (to appear)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A steepest descent algorithm for M-convex functions on jump systems

    • Author(s)
      K.Murota, K.Tanaka
    • Journal Title

      IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences 発表予定(未定)

    • NAID

      110007502826

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A general two-sided matching market with discrete concave utility functions

    • Author(s)
      S.Fujishige, A.Tamura
    • Journal Title

      Discrete Applied Mathematics 発表予定(未定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A two-sided discrete-concave market with bounded side payments : An approach by discrete convex analysis

    • Author(s)
      S.Fujishige, A.Tamura
    • Journal Title

      Mathematics of Operations Research 発表予定(未定)

    • Related Report
      2005 Annual Research Report
  • [Journal Article] A capacity scaling algorithm for M-convex submodular flow

    • Author(s)
      S.Iwata, S.Moriguchi, K.Murota
    • Journal Title

      Mathematical Programming (発表予定)

    • NAID

      110002812004

    • Related Report
      2004 Annual Research Report
  • [Publications] Farooq, Rashid: "A new characterization of M^#-convex set functions by substitutability"Journal of the Operations Research Society of Japan. (発表予定). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Eguchi, A.: "A generalized Gale-Shapley algorithm for a discrete-concave stable-marriange model"Lecture Notes in Computer Science. 2906. 495-504 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Tamura, Akihisa: "Applications of discrete convex analysis to mathematical economics"Publications of RIMS. (発表予定). (2004)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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