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Establishment of Discrete Convexity Paradigm

Research Project

Project/Area Number 15360043
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Engineering fundamentals
Research InstitutionThe University of Tokyo

Principal Investigator

MUROTA Kazuo  The University of Tokyo, Graduate School of Information Sciences and Technology, Professor, 大学院・情報理工学系研究科, 教授 (50134466)

Co-Investigator(Kenkyū-buntansha) TAMURA Akihisa  Keio University, Science and Technology, Assistant Professor, 理工学部, 教授 (50217189)
IWATA Satoru  The University of Tokyo, Graduate School of Information Sciences and Technology, Assistant Professor, 大学院・情報理工学系研究科, 助教授 (00263161)
TSUCHIMURA Nobuyuki  The University of Tokyo, Graduate School of Information Sciences and Technology, Research Associate, 大学院・情報理工学系研究科, 助手 (20345119)
塩浦 昭義  東北大学, 大学院・情報科学研究科, 助教授 (10296882)
松浦 史郎  東京大学, 大学院・情報理工学系研究科, 学術研究支援員 (00332619)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥10,200,000 (Direct Cost: ¥10,200,000)
Fiscal Year 2005: ¥3,100,000 (Direct Cost: ¥3,100,000)
Fiscal Year 2004: ¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2003: ¥3,800,000 (Direct Cost: ¥3,800,000)
Keywordsconvex analysis / convex function / convex set / matroid / discrete optimization / mathematical programming / nonlinear programming / network flow / 双対性 / 劣モジュラ関数 / アルゴリズム / 離散凸関数 / 組合せ最適化
Research Abstract

The main aim of this research project is to establish a novel paradigm of discrete convexity by exploiting applications of the theory of discrete convex analysis to fundamental problems in economics, system engineering, operations research, optimization, algorithm science, and others. In this research project, we obtained a number of results described below. All of the results are presented at refereed international conference or accepted for publications in international scientific journals.
(1)For optimization problems related to discrete convexity (in particular, for the submodular flow problem with M-convex cost function), we developed a number of efficient algorithms using the capacity scaling technique. The obtained algorithm has the best known complexity bound.
(2)We pointed out that the concept of multimodular functions, which had been conceived independently of discrete convex analysis, is essentially equivalent to the concept of L-convex functions. On the basis of this observation we derived some properties including discrete separation.
(3)We revealed a close relationship between M-convex functions and tree metrics. Specifically, we proved that the Hessian matrix of a quadratic M-convex function is essentially the same as the negative of the tree metric matrix.
(4)We established a discrete fixed point theorem for functions defined on integrally convex sets. The theorem finds applications in mathematical economics and game theory.
(5)We provided new characterizations of M-convex functions in terms of gross substitution and single improvement property known in mathematical economics.
(6)As applications of discrete convex analysis to mathematical economics, we developed a theory for economic equilibrium models with indivisible goods. We also developed an algorithm for computing a competitive equilibrium using the framework of M-convex submodular flow problem.

Report

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

    (63 results)

All 2006 2005 2004 2003 Other

All Journal Article (54 results) Book (2 results) Publications (7 results)

  • [Journal Article] Bisubmodular function minimization2006

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

      SIAM Journal on Discrete Mathematics 19

      Pages: 1065-1073

    • NAID

      110002812420

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] M-convex functions on jump systems : A general framework for minsquare graph factor problem2006

    • Author(s)
      K.Murota
    • Journal Title

      SIAM Journal on Discrete mathematics 20

      Pages: 213-226

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Matroid Matching via mexed skew-symmetric matrices2005

    • Author(s)
      J.F.Geelen, S.Iwata
    • Journal Title

      Combinatorica 25

      Pages: 187-215

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Discrete Hessian matrix for L-convex functions2005

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

      IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E88-A

      Pages: 1104-1108

    • NAID

      110003213419

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Note on multimodularity and L-convexity2005

    • Author(s)
      K.Murota
    • Journal Title

      Mathematics of Operations Research 30

      Pages: 658-661

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] M-convex functions on jump systems : A general framework for minsquare graph factor problem2005

    • Author(s)
      K.Murota
    • Journal Title

      SIAM Journal on Discrete Mathematics 20

      Pages: 213-226

    • 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)
      S.Iwata, S.Moriguchi, K.Murota
    • Journal Title

      Mathematical Programming 103

      Pages: 181-202

    • NAID

      110002812004

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [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

      Pages: 133-145

    • 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)
      A.Tamura
    • Journal Title

      Mathematical Programming 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] A strongly polynomial cut canceling algorithm for minimum cost submodular flow2005

    • Author(s)
      S.Iwata, S.T.McCormick, M.Shigeno
    • Journal Title

      SIAM Journal on Discrete Mathematics 19

      Pages: 304-320

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Computational results for Gaussian moat problem2005

    • Author(s)
      N.Tsuchimura
    • Journal Title

      IEICE Trans. Fundamentals E88-A

      Pages: 1267-1273

    • NAID

      110002811996

    • 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)
      T.Iimura, K.Murota, A.Tamura
    • 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 Annual Research Report 2005 Final Research Report Summary
  • [Journal Article] Substitutes and complements in network flows viewed as discrete convexity2005

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

      Discrete optimization 2

      Pages: 256-268

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A note on the equivalence between substitutability and M^# -convexity2005

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

      Pacific Journal of optimization 1

      Pages: 243-252

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Matroid matching via mixed skew-symmetric matrices2005

    • Author(s)
      J.F.Geelen, S.Iwata
    • Journal Title

      Combinatorica 25

      Pages: 187-215

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Computational results for Gaussian moat problem2005

    • Author(s)
      N.Tsuchimura
    • Journal Title

      IEICE Trans.Fundamentals E88-A

      Pages: 1267-1273

    • NAID

      110002811996

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Substitutes and complements in network flows viewed as discrete convexity2005

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

      Discrete Optimization 2

      Pages: 256-268

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A note on the equivalence between substitutability and M〓-convexity2005

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

      Pacific Journal of Optimization 1

      Pages: 243-252

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] SVM kernel by electric network2005

    • Author(s)
      H.Hirai, K.Murota, M.Rikitoku
    • Journal Title

      Pacific Journal of Optimization 1

      Pages: 509-526

    • 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 (掲載予定)

    • NAID

      110002812004

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Note on multimodularity and L-convexity2005

    • Author(s)
      K.Murota
    • Journal Title

      Mathematics of Operations Research (掲載予定)

    • Related Report
      2004 Annual Research Report
  • [Journal Article] A constrained independent set problem for matroids2004

    • Author(s)
      T.Fleiner, A.Frank, S.Iwata
    • Journal Title

      Operations Research Letters 32

      Pages: 23-26

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Conjugacy relationship between M-convex and L-convex functions in continuous variables2004

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

      Mathematical Programming 101

      Pages: 415-433

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Optimality criteria for a class of nonlinear integer programs2004

    • Author(s)
      K.Murota, H.Saito, R.Weismantel
    • Journal Title

      Operations Research Letters 32

      Pages: 468-472

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Fundamental properties of M-convex and L-convex functions in continuous variables2004

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

      IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences E87-A

      Pages: 1042-1052

    • NAID

      110003213000

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Quadratic M-convex and L-convex functions2004

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

      Advances in Applied Mathematics 33

      Pages: 318-341

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Proximity theorems of discrete convex functions2004

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

      Mathematical Programming 99

      Pages: 539-562

    • NAID

      110003478708

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] M-convex functions and tree metrics2004

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

      Japan Journal of Industrial and Applied Mathematics 21

      Pages: 391-403

    • NAID

      10018379869

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] The MA-ordering max-flow algorithm is not strongly polynomial for directed networks2004

    • Author(s)
      A.Shioura
    • Journal Title

      Operations Research Letters 32

      Pages: 31-35

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A network flow approach to cost allocation for rooted trees2004

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

      Networks 44

      Pages: 297-301

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Fast scaling algorithms for M-convex function minimization with application to the resource allocation problem2004

    • Author(s)
      A.Shioura
    • Journal Title

      Discrete Applied Mathematics 134

      Pages: 303-316

    • NAID

      110003178709

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On Hochbaum's scaling algorithm for the general resource allocation problem2004

    • Author(s)
      S.Moriguchi, A.Shioura
    • Journal Title

      Mathematics of Operations Research 29

      Pages: 394-397

    • 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)
      R.Farooq, A.Tamura
    • Journal Title

      Journal ofthe operations Research Society of Japan 47

      Pages: 18-24

    • NAID

      110001183589

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Fast scaling algorithms for M-convex function Minimization with application to the resource allocation problem2004

    • Author(s)
      A.Shioura
    • Journal Title

      Discrete Applied Mathematics 134

      Pages: 303-316

    • NAID

      110003178709

    • 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)
      R.Farooq, A.Tamura
    • Journal Title

      Journal of the Operations Research Society of Japan 47

      Pages: 18-24

    • NAID

      110001183589

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

    • Author(s)
      A.Tamura
    • Journal Title

      Publications of Research Institute for Mathematical Sciences 40

      Pages: 1015-1037

    • NAID

      110001050934

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Numerical indefinite integration by double exponential sinc method2004

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

      Mathematics of Computation 74

      Pages: 655-679

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Quasi M-convex and L-convex Functions---Quasi-convexity in discrete optimization2003

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

      Discrete Applied Mathematics 131/132

      Pages: 467-494

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The linear delta-matroid parity problem2003

    • Author(s)
      J.Geelen, S.Iwata, K.Mruota
    • Journal Title

      J. Combinatorial Theory, B 88

      Pages: 377-398

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On convoloution of L-convex functions2003

    • Author(s)
      A.Tamura
    • Journal Title

      Optimization Methods and Software 18

      Pages: 231-245

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Capacity scaling algorithm for scalable M-convex submodular flow problems2003

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

      Optimization Methods and Software 18

      Pages: 207-218

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Conjugate scaling algorithm for Fenchel-type duality in discrete convex optimization2003

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

      SIAM Journal on Optimization 13

      Pages: 204-211

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A push-relabel framework for submodular function minimization and applications to parametric optimization2003

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

      Discrete Applied Mathematics 133

      Pages: 311-322

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Computing the maximun degree of minors in matrix pencils via combinatorial relaxation2003

    • Author(s)
      S.Iwata
    • Journal Title

      Algorithmica 36

      Pages: 331-341

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Fast cycle canceling algorithms for minimum cost submodular flow2003

    • Author(s)
      S.Iwata, S.T.McCormick, M.Shigeno
    • Journal Title

      Combinatorica 23

      Pages: 503-525

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On steepest descent algorithms for discrete convex functions2003

    • Author(s)
      K.Murota
    • Journal Title

      SIAM Journal on Optimization 14

      Pages: 699-707

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] A faster scaling algorithm for minimizing submodular functions2003

    • Author(s)
      S.Iwata
    • Journal Title

      SIAM Journal on Computing 32

      Pages: 833-840

    • NAID

      110002812483

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] New characterizations of M-convex functions and their applications to economic equilibrium models2003

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

      Discrete Applied Mathematics 131

      Pages: 495-512

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Application of M-convex submodular flow problem to mathematical economics2003

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

      Japan Journal of Industrial and Applied Mathematics 20

      Pages: 257-277

    • NAID

      10018380082

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Quasi M-convex and L-convex functions ・・・ Quasi-convexity in discrete optimization2003

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

      Discrete Applied Mathematics 131/132

      Pages: 467-494

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] The linear delta-matroid parity problem2003

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

      J.Combinatorial Theory, B 88

      Pages: 377-398

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On convolution of L-convex functions2003

    • Author(s)
      A.Tamura
    • Journal Title

      Optimization Methods and Software 18

      Pages: 231-245

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Capacity scaling algorithm for scalable M-convex submod ular flow problems2003

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

      Optimization Methods and Software 18

      Pages: 207-218

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Computing the maximum degree of minors in matrix pencils via combinatorial relaxation2003

    • Author(s)
      S.Iwata
    • Journal Title

      Algorithmica 36

      Pages: 331-341

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Book] Discrete Convex Analysis2003

    • Author(s)
      K.Murota
    • Total Pages
      389
    • Publisher
      Soceity for Industrial and Applied Mathematics
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Book] Society for Industrial and Applied Mathematics2003

    • Author(s)
      K.Murota
    • Total Pages
      389
    • Publisher
      Discrete Convex Analysis
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Publications] K.Murota, A.Shioura: "Conjugacy Relationship between M-convex and L-convex Functions in Continuous Variables"Mathematical Programming. (印刷中). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Iwata: "A Faster Scaling Algorithm for Minimizing Submodular Functions"SIAM Journal on Computing. 32. 833-840 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] S.Iwata, M.Shigeno: "Conjugate Scaling Algorithm for Fenchel-type Duality in Discrete Convex Optimization"SIAM Journal on Optimization. 13. 204-211 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Murota: "On Steepest Descent Algorithms for Discrete Convex Functions"SIAM Journal on Optimization. 14. 699-707 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Murota, H.Saito, R.Weismantel: "Optimality Criteria for a Class of Nonlinear Integer Programs"Operation Research Letters. (印刷中).

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Murota, A.Tamura: "New characterizations of M-convex Functions and Their Applications to Economic Equilibrium Models with Indivisibilities"Discrete Applied Mathematics. 131. 495-512 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] K.Murota: "Discrete Convex Analysis"Society for Industrial and Applied Mathematics. 389 (2003)

    • Related Report
      2003 Annual Research Report

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

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