Linearly Representable Submodular Functions: An Algebraic Algorithm for Minimization

Authors Rohit Gurjar, Rajat Rathi

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Rohit Gurjar
  • Indian Institute of Technology Bombay, India
Rajat Rathi
  • Indian Institute of Technology Bombay, India


We would like to thank Siddharth Barman for helpful comments.

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Rohit Gurjar and Rajat Rathi. Linearly Representable Submodular Functions: An Algebraic Algorithm for Minimization. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 61:1-61:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


A set function f : 2^E → ℝ on the subsets of a set E is called submodular if it satisfies a natural diminishing returns property: for any S ⊆ E and x,y ∉ S, we have f(S ∪ {x,y}) - f(S ∪ {y}) ≤ f(S ∪ {x}) - f(S). Submodular minimization problem asks for finding the minimum value a given submodular function takes. We give an algebraic algorithm for this problem for a special class of submodular functions that are "linearly representable". It is known that every submodular function f can be decomposed into a sum of two monotone submodular functions, i.e., there exist two non-decreasing submodular functions f₁,f₂ such that f(S) = f₁(S) + f₂(E ⧵ S) for each S ⊆ E. Our class consists of those submodular functions f, for which each of f₁ and f₂ is a sum of k rank functions on families of subspaces of 𝔽ⁿ, for some field 𝔽. Our algebraic algorithm for this class of functions can be parallelized, and thus, puts the problem of finding the minimizing set in the complexity class randomized NC. Further, we derandomize our algorithm so that it needs only O(log²(kn|E|)) many random bits. We also give reductions from two combinatorial optimization problems to linearly representable submodular minimization, and thus, get such parallel algorithms for these problems. These problems are (i) covering a directed graph by k a-arborescences and (ii) packing k branchings with given root sets in a directed graph.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Theory of computation → Parallel algorithms
  • Theory of computation → Pseudorandomness and derandomization
  • Submodular Minimization
  • Parallel Algorithms
  • Derandomization
  • Algebraic Algorithms


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  1. Sepehr Assadi, Yu Chen, and Sanjeev Khanna. Polynomial pass lower bounds for graph streaming algorithms. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, page 265–276, New York, NY, USA, 2019. Association for Computing Machinery. URL:
  2. Eric Balkanski and Yaron Singer. Minimizing a submodular function from samples. In Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS’17, page 814–822, Red Hook, NY, USA, 2017. Curran Associates Inc. Google Scholar
  3. Eric Balkanski and Yaron Singer. A lower bound for parallel submodular minimization. In Proceedings of the 52nd Annual ACM Symposium on Theory of Computing (STOC) June 22–26, 2020 in Chicago, IL, 2020. URL:
  4. Allan Borodin, Joachim von zur Gathen, and John Hopcroft. Fast parallel matrix and GCD computations. Information and Control, 52(3):241-256, 1982. Google Scholar
  5. G. Cornuéjols, G.L. Nemhauser, and L.A. Wolsey. The uncapacitated facility location problem. In In: Discrete Location Theory (P. Mirchandani, R. Francis, eds.), pages 119-171. Wiley, New York, 1990. Google Scholar
  6. William H. Cunningham. On submodular function minimization. Combinatorica, 5(3):185-192, 1985. URL:
  7. G. Davis, S. Mallat, and M. Avellaneda. Adaptive greedy approximations. Constructive Approximation, 13(1):57-98, 1997. URL:
  8. Richard A. Demillo and Richard J. Lipton. A probabilistic remark on algebraic program testing. Information Processing Letters, 7(4):193-195, 1978. Google Scholar
  9. Uriel Feige, Vahab S. Mirrokni, and Jan Vondrák. Maximizing non-monotone submodular functions. SIAM J. Comput., 40(4):1133-1153, 2011. URL:
  10. Stephen A. Fenner, Rohit Gurjar, and Thomas Thierauf. Bipartite perfect matching is in quasi-nc. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, pages 754-763, 2016. Google Scholar
  11. Martin Grötschel, László Lovász, and Alexander Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1(2):169-197, 1981. URL:
  12. Martin Grötschel, László Lovász, and Alexander Schrijver. Geometric Algorithms and Combinatorial Optimization, volume 2 of Algorithms and Combinatorics. Springer, 1988. URL:
  13. Jiong Guo, Rolf Niedermeier, and Sebastian Wernicke. Parameterized complexity of vertex cover variants. Theory Comput. Syst., 41:501-520, October 2007. URL:
  14. Rohit Gurjar and Thomas Thierauf. Linear matroid intersection is in quasi-NC. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 821-830, 2017. URL:
  15. Satoru Iwata, Lisa Fleischer, and Satoru Fujishige. A combinatorial strongly polynomial algorithm for minimizing submodular functions. J. ACM, 48(4):761-777, 2001. URL:
  16. Richard M. Karp. Reducibility among combinatorial problems. In Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, pages 85-103, 1972. URL:
  17. Richard M. Karp, Eli Upfal, and Avi Wigderson. Constructing a perfect matching is in random NC. Combinatorica, 6(1):35-48, 1986. Google Scholar
  18. E. L. Lawler. Cutsets and partitions of hypergraphs. Networks, 3(3):275-285, 1973. URL:
  19. Yin Tat Lee, Aaron Sidford, and Sam Chiu-Wai Wong. A faster cutting plane method and its implications for combinatorial and convex optimization. In Proceedings of the 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), FOCS ’15, page 1049–1065, USA, 2015. IEEE Computer Society. URL:
  20. L. Lovász. Submodular functions and convexity. In Achim Bachem, Bernhard Korte, and Martin Grötschel, editors, Mathematical Programming The State of the Art: Bonn 1982, pages 235-257. Springer Berlin Heidelberg, Berlin, Heidelberg, 1983. URL:
  21. László Lovász. On determinants, matchings, and random algorithms. In Fundamentals of Computation Theory, pages 565-574, 1979. Google Scholar
  22. Ketan Mulmuley. A fast parallel algorithm to compute the rank of a matrix over an arbitrary field. Combinatorica, 7(1):101-104, 1987. Google Scholar
  23. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7:105-113, 1987. URL:
  24. H. Narayanan, Huzur Saran, and Vijay V. Vazirani. Randomized parallel algorithms for matroid union and intersection, with applications to arboresences and edge-disjoint spanning trees. SIAM J. Comput., 23(2):387-397, 1994. URL:
  25. Øystein Ore. Über höhere Kongruenzen. Norsk Mat. Forenings Skrifter Ser. I, 7(15):27, 1922. Google Scholar
  26. James G. Oxley. Matroid Theory (Oxford Graduate Texts in Mathematics). Oxford University Press, Inc., New York, NY, USA, 2006. Google Scholar
  27. M. J. Piff and D. J. A. Welsh. On the number of combinatorial geometries. Bulletin of the London Mathematical Society, 3(1):55-56, 1971. URL:
  28. Alexander Schrijver. A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J. Comb. Theory, Ser. B, 80(2):346-355, 2000. URL:
  29. Alexander Schrijver. Combinatorial optimization : polyhedra and efficiency. Vol. B. , Matroids, trees, stable sets. chapters 39-69. Algorithms and combinatorics. Springer-Verlag, Berlin, Heidelberg, New York, N.Y., et al., 2003. URL:
  30. Jacob T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM, 27(4):701-717, October 1980. Google Scholar
  31. Zoya Svitkina and Lisa Fleischer. Submodular approximation: Sampling-based algorithms and lower bounds. SIAM J. Comput., 40(6):1715–1737, December 2011. URL:
  32. K. Vidyasankar. Covering the edge set of a directed graph with trees. Discrete Mathematics, 24(1):79-85, 1978. URL:
  33. Richard Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposium on Symbolic and Algebraic Computation (EUROSAM), pages 216-226. Springer-Verlag, 1979. Google Scholar
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