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Decreasing minimization on M-convex sets: background and structures

Abstract

The present work is the first member of a pair of papers concerning decreasingly-minimal (dec-min) elements of a set of integral vectors, where a vector is dec-min if its largest component is as small as possible, within this, the next largest component is as small as possible, and so on. This discrete notion, along with its fractional counterpart, showed up earlier in the literature under various names. The domain we consider is an M-convex set, that is, the set of integral elements of an integral base-polyhedron. A fundamental difference between the fractional and the discrete case is that a base-polyhedron has always a unique dec-min element, while the set of dec-min elements of an M-convex set admits a rich structure, described here with the help of a ‘canonical chain’. As a consequence, we prove that this set arises from a matroid by translating the characteristic vectors of its bases with an integral vector. By relying on these characterizations, we prove that an element is dec-min if and only if the square-sum of its components is minimum, a property resulting in a new type of min-max theorems. The characterizations also give rise, as shown in the companion paper, to a strongly polynomial algorithm, and to several applications in the areas of resource allocation, network flow, matroid, and graph orientation problems, which actually provided a major motivation to the present investigations. In particular, we prove a conjecture on graph orientation.

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References

  1. 1.

    Ahuja, R.K., Hochbaum, D.S., Orlin, J.B.: A cut-based algorithm for the nonlinear dual of the minimum cost network flow problem. Algorithmica 39, 189–208 (2004)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Arnold, B.C., Sarabia, J.M.: Majorization and the Lorenz Order with Applications in Applied Mathematics and Economics, Springer International Publishing, Cham (2018), (1st edn., 1987)

  3. 3.

    Bokal, D., Brešar, B., Jerebic, J.: A generalization of Hungarian method and Hall’s theorem with applications in wireless sensor networks. Discrete Appl. Math. 160, 460–470 (2012)

  4. 4.

    Borradaile, G., Iglesias, J., Migler, T., Ochoa, A., Wilfong, G., Zhang, L.: Egalitarian graph orientations. J. Graph Algorithms Appl. 21, 687–708 (2017)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Borradaile, G., Migler, T., Wilfong, G.: Density decompositions of networks. J. Graph Algorithms Appl. 23, 625–651 (2019)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Edmonds, J.: Submodular functions, matroids and certain polyhedra. In: Guy, R., Hanani, H., Sauer, N., Schönheim, J. (eds.) Combinatorial Structures and Their Applications, pp. 69–87. Gordon and Breach, New York (1970)

    Google Scholar 

  7. 7.

    Federgruen, A., Groenevelt, H.: The greedy procedure for resource allocation problems: necessary and sufficient conditions for optimality. Oper. Res. 34, 909–918 (1986)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Frank, A.: On the orientation of graphs. J. Combin. Theory Ser. B 28, 251–261 (1980)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Frank, A.: Connections in Combinatorial Optimization. Oxford University Press, Oxford (2011)

    MATH  Google Scholar 

  10. 10.

    Frank, A., Murota, K.: Discrete decreasing minimization, Part II: Views from discrete convex analysis. arXiv: 1808.08477 (2018)

  11. 11.

    Frank, A., Murota, K.: Decreasing minimization on M-convex sets: Algorithms and applications. Mathematical Programming, published online (October 15, 2021). https://doi.org/10.1007/s10107-021-01711-5

  12. 12.

    Frank, A., Murota, K.: A discrete convex min-max formula for box-TDI polyhedra. Mathematics of Operations Research, to appear. arXiv:2007.03507

  13. 13.

    Frank, A., Murota, K.: Fair integral flows. Submitted for publication. arXiv: 1907.02673v3 (2020)

  14. 14.

    Frank, A., Murota, K.: Fair integral submodular flows. Submitted for publication. arXiv: 2012.07325 (2020)

  15. 15.

    Fujishige, S.: Lexicographically optimal base of a polymatroid with respect to a weight vector. Math. Oper. Res. 5, 186–196 (1980)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Fujishige, S.: Submodular Functions and Optimization, 2nd edn. Annals of Discrete Mathematics 58. Elsevier, Amsterdam (2005)

  17. 17.

    Fujishige, S.: Theory of principal partitions revisited. In: Cook, W., Lovász, L., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, pp. 127–162. Springer, Berlin (2009)

    Chapter  Google Scholar 

  18. 18.

    Ghodsi, A., Zaharia, M., Shenker, S., Stoica, I.: Choosy: Max-min fair sharing for datacenter jobs with constraints. In: EuroSys ’13 Proceedings of the 8th ACM European Conference on Computer Systems, pp. 365–378, ACM New York, NY (2013)

  19. 19.

    Groenevelt, H.: Two algorithms for maximizing a separable concave function over a polymatroid feasible region. Eur. J. Oper. Res. 54, 227–236 (1991)

    Article  Google Scholar 

  20. 20.

    Harada, Y., Ono, H., Sadakane, K., Yamashita, M.: Optimal balanced semi-matchings for weighted bipartite graphs. IPSJ Digital Courier 3, 693–702 (2007)

    Article  Google Scholar 

  21. 21.

    Harvey, N.J.A., Ladner, R.E., Lovász, L., Tamir, T.: Semi-matchings for bipartite graphs and load balancing. J. Algorithms 59, 53–78 (2006)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Hochbaum, D.S.: Solving integer programs over monotone inequalities in three variables: a framework for half integrality and good approximations. Eur. J. Oper. Res. 140, 291–321 (2002)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Hochbaum, D.S.: Complexity and algorithms for nonlinear optimization problems. Ann. Oper. Res. 153, 257–296 (2007)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Hochbaum, D.S., Hong, S.-P.: About strongly polynomial time algorithms for quadratic optimization over submodular constraints. Math. Program. 69, 269–309 (1995)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Ibaraki, T., Katoh, N.: Resource Allocation Problems: Algorithmic Approaches. MIT Press, Boston (1988)

    MATH  Google Scholar 

  26. 26.

    Katoh, N., Ibaraki, T.: Resource allocation problems. In: Du, D.-Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, vol. 2, pp. 159–260. Kluwer Academic Publishers, Boston (1998)

    Google Scholar 

  27. 27.

    Katoh, N., Shioura, A., Ibaraki, T.: Resource allocation problems. In: Pardalos, P.M., Du, D.-Z., Graham, R.L. (eds.) Handbook of Combinatorial Optimization, vol. 5, 2nd edn., pp. 2897–2988. Springer, Berlin (2013)

    Chapter  Google Scholar 

  28. 28.

    Katrenič, J., Semanišin, G.: Maximum semi-matching problem in bipartite graphs. Discussiones Mathematicae Graph Theory 33, 559–569 (2013)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Levin, A., Onn, S.: Shifted matroid optimization. Oper. Res. Lett. 44, 535–539 (2016)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Lovász, L.: Submodular functions and convexity. In: Bachem, A., Grötschel, M., Korte, B. (eds.) Mathematical Programming-The State of the Art, pp. 235–257. Springer, Berlin (1983)

    Chapter  Google Scholar 

  31. 31.

    Marshall, A.W., Olkin, I., Arnold, B.C.: Inequalities: Theory of Majorization and Iis Applications, 2nd edn. Springer, New York (2011), (1st edn., 1979)

  32. 32.

    Maruyama, F.: A unified study on problems in information theory via polymatroids. Graduation Thesis, University of Tokyo, Japan (1978) (in Japanese)

  33. 33.

    Megiddo, N.: Optimal flows in networks with multiple sources and sinks. Math. Program. 7, 97–107 (1974)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Megiddo, N.: A good algorithm for lexicographically optimal flows in multi-terminal networks. Bull. Am. Math. Soc. 83, 407–409 (1977)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Murota, K.: Discrete convex analysis. Math. Program. 83, 313–371 (1998)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Murota, K.: Discrete Convex Analysis. Society for Industrial and Applied Mathematics, Philadelphia (2003)

    Book  Google Scholar 

  37. 37.

    Nagano, K.: On convex minimization over base polytopes. In: Fischetti, M., Williamson, D.P. (eds.): Integer Programming and Combinatorial Optimization. Lecture Notes in Computer Science, vol. 4513, pp. 252–266 (2007)

  38. 38.

    Tamir, A.: Least majorized elements and generalized polymatroids. Math. Oper. Res. 20, 583–589 (1995)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

We are grateful to the six authors of the paper by Borradaile et al. [4] because that work triggered the present research (and this is so even if we realized later that there had been several related works). We thank S. Fujishige and S. Iwata for discussion about the history of convex minimization over base-polyhedra. We also thank A. Jüttner and T. Maehara for illuminating the essence of the Newton–Dinkelbach algorithm. J. Tapolcai kindly drew our attention to engineering applications in resource allocation. Z. Király played a similar role by finding an article which pointed to a work of Levin and Onn on decreasingly minimal optimization in matroid theory. We are also grateful to M. Kovács for drawing our attention to some important papers in the literature concerning fair resource allocation problems. Special thanks are due to T. Migler for her continuous availability to answer our questions concerning the paper [4] and the work by Borradaile, Migler, and Wilfong [5], which paper was also a prime driving force in our investigations. We are grateful to B. Shepherd and K. Bérczi for their advice that led to restructuring our presentation appropriately. We are also grateful to an anonymous referee of the paper whose strategic suggestions were particularly important to shape the final form of our work. This research was supported through the program “Research in Pairs” by the Mathematisches Forschungsinstitut Oberwolfach in 2019. The two weeks we could spend at Oberwolfach provided an exceptional opportunity to conduct particularly intensive research. The research was partially supported by the National Research, Development and Innovation Fund of Hungary (FK_18) – No. NKFI-128673, and by CREST, JST, Grant Number JPMJCR14D2, Japan, and JSPS KAKENHI Grant Numbers JP26280004, JP20K11697.

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Correspondence to Kazuo Murota.

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The research was partially supported by the National Research, Development and Innovation Fund of Hungary (FK_18) – No. NKFI-128673. The research was supported by CREST, JST, Grant Number JPMJCR14D2, Japan, and JSPS KAKENHI Grant Numbers JP26280004, JP20K11697.

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Frank, A., Murota, K. Decreasing minimization on M-convex sets: background and structures. Math. Program. (2021). https://doi.org/10.1007/s10107-021-01722-2

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Keywords

  • Submodular optimization
  • Matroid
  • Base-polyhedron
  • M-convex set
  • Lexicographic minimization

Mathematics Subject Classification

  • 90C27
  • 05C
  • 68R10