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Documents authored by Mohapatra, Chandra Kanta

Document
DOI: 10.4230/LIPIcs.ESA.2026.129
Optimal Union Probability Interval Is NP-Hard

Authors: Petteri Kaski, Heikki Mannila, and Chandra Kanta Mohapatra

Published in: LIPIcs, Volume 388, 34th Annual European Symposium on Algorithms (ESA 2026)

Abstract
A problem dating back to Boole [Laws of Thought, Walton & Maberly, 1854] is what can be computed about the probability of a finite union of events when given as input the probabilities of intersections of some of the events. The modern geometric study of the problem can be traced back to Hailperin [Amer. Math. Monthly 2 (1965) 343-359] who phrased the problem in the language of linear programming and generalized it to logical formulas of the events other than disjunction, heralding a substantial body of work in probabilistic logic [Nilsson, Artif. Intell. 28 (1986) 71-87], including the probabilistic satisfiability problem of Georgakopoulos, Kavvadis, and Papadimitriou [J. Complexity 4 (1988) 1-11], as well as fundamental connections to the geometry of metrics via cut and correlation polytopes [Deza and Laurent, Geometry of Cuts and Metrics, Springer, 1997] and to the study of marginal polytopes in graphical models of machine learning [Wainwright and Jordan, Found. Trends Mach. Learn. 1 (2008) 1-305]. This paper (i) describes the pertinent geometry of Boole’s problem via coordinate projections of an elementary polytope arising essentially from Hailperin’s linear program on the atoms of a Venn diagram, and (ii) shows that computing the optimal interval for the union probability is NP-hard, resolving an apparent gap in the literature highlighted by Pitowsky [Math. Programming 50 (1991) 395-414] and Boros et al. [Math. Oper. Res. 39 (2014) 1311-1329 and 51 (2026) 134-148].
Cite as

Petteri Kaski, Heikki Mannila, and Chandra Kanta Mohapatra. Optimal Union Probability Interval Is NP-Hard. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 129:1-129:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kaski_et_al:LIPIcs.ESA.2026.129,
  author =	{Kaski, Petteri and Mannila, Heikki and Mohapatra, Chandra Kanta},
  title =	{{Optimal Union Probability Interval Is NP-Hard}},
  booktitle =	{34th Annual European Symposium on Algorithms (ESA 2026)},
  pages =	{129:1--129:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-445-1},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{388},
  editor =	{Bille, Philip and Pettie, Seth and Storandt, Sabine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2026.129},
  URN =		{urn:nbn:de:0030-drops-272650},
  doi =		{10.4230/LIPIcs.ESA.2026.129},
  annote =	{Keywords: combinatorial probability, computational complexity, correlation polytope, generalized Bonferroni inequalities, graphical models, inclusion-exclusion, intersection probability, linear programming, union polytope}
}
Document
DOI: 10.4230/LIPIcs.CCC.2020.14
Schur Polynomials Do Not Have Small Formulas If the Determinant Doesn't

Authors: Prasad Chaugule, Mrinal Kumar, Nutan Limaye, Chandra Kanta Mohapatra, Adrian She, and Srikanth Srinivasan

Published in: LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

Abstract
Schur Polynomials are families of symmetric polynomials that have been classically studied in Combinatorics and Algebra alike. They play a central role in the study of Symmetric functions, in Representation theory [Stanley, 1999], in Schubert calculus [Ledoux and Malham, 2010] as well as in Enumerative combinatorics [Gasharov, 1996; Stanley, 1984; Stanley, 1999]. In recent years, they have also shown up in various incarnations in Computer Science, e.g, Quantum computation [Hallgren et al., 2000; Ryan O'Donnell and John Wright, 2015] and Geometric complexity theory [Ikenmeyer and Panova, 2017]. However, unlike some other families of symmetric polynomials like the Elementary Symmetric polynomials, the Power Symmetric polynomials and the Complete Homogeneous Symmetric polynomials, the computational complexity of syntactically computing Schur polynomials has not been studied much. In particular, it is not known whether Schur polynomials can be computed efficiently by algebraic formulas. In this work, we address this question, and show that unless every polynomial with a small algebraic branching program (ABP) has a small algebraic formula, there are Schur polynomials that cannot be computed by algebraic formula of polynomial size. In other words, unless the algebraic complexity class VBP is equal to the complexity class VF, there exist Schur polynomials which do not have polynomial size algebraic formulas. As a consequence of our proof, we also show that computing the determinant of certain generalized Vandermonde matrices is essentially as hard as computing the general symbolic determinant. To the best of our knowledge, these are one of the first hardness results of this kind for families of polynomials which are not multilinear. A key ingredient of our proof is the study of composition of well behaved algebraically independent polynomials with a homogeneous polynomial, and might be of independent interest.
Cite as

Prasad Chaugule, Mrinal Kumar, Nutan Limaye, Chandra Kanta Mohapatra, Adrian She, and Srikanth Srinivasan. Schur Polynomials Do Not Have Small Formulas If the Determinant Doesn't. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 14:1-14:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chaugule_et_al:LIPIcs.CCC.2020.14,
  author =	{Chaugule, Prasad and Kumar, Mrinal and Limaye, Nutan and Mohapatra, Chandra Kanta and She, Adrian and Srinivasan, Srikanth},
  title =	{{Schur Polynomials Do Not Have Small Formulas If the Determinant Doesn't}},
  booktitle =	{35th Computational Complexity Conference (CCC 2020)},
  pages =	{14:1--14:27},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-156-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{169},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.14},
  URN =		{urn:nbn:de:0030-drops-125660},
  doi =		{10.4230/LIPIcs.CCC.2020.14},
  annote =	{Keywords: Schur polynomial, Jacobian, Algebraic independence, Generalized Vandermonde determinant, Taylor expansion, Formula complexity, Lower bound}
}

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