On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients

Narayanan, Hariharan

doi:10.1007/s10801-006-0008-5

Cited by 85 publications

(70 citation statements)

References 12 publications

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“…10: end if 11: end for 12: if not foundpath then 13: print( f + π( pq)) with a shortest q ∈ C (R p f ) and return. 14: end if implies f -extendability.…”

Section: Algorithm 3 Findneighmentioning

confidence: 99%

Small Littlewood–Richardson coefficients

Ikenmeyer

2016

J Algebr Comb

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We develop structural insights into the Littlewood-Richardson graph, whose number of vertices equals the Littlewood-Richardson coefficient c ν λ,μ for given partitions λ, μ, and ν. This graph was first introduced in Bürgisser and Ikenmeyer (SIAM J Discrete Math 27(4):1639-1681, 2013), where its connectedness was proved. Our insights are useful for the design of algorithms for computing the LittlewoodRichardson coefficient: We design an algorithm for the exact computation of c ν λ,μ with running time O (c ν λ,μ ) 2 · poly(n) , where λ, μ, and ν are partitions of length at most n. Moreover, we introduce an algorithm for deciding whether c ν λ,μ ≥ t whose running time is O t 2 · poly(n) . Even the existence of a polynomial-time algorithm for deciding whether c ν λ,μ ≥ 2 is a nontrivial new result on its own. Our insights also lead to the proof of a conjecture by King et al. (Symmetry in physics. American Mathematical Society, Providence, 2004), stating that c ν λ,μ = 2 implies c Mν Mλ,Mμ = M + 1 for all M ∈ N. Here, the stretching of partitions is defined componentwise.

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“…10: end if 11: end for 12: if not foundpath then 13: print( f + π( pq)) with a shortest q ∈ C (R p f ) and return. 14: end if implies f -extendability.…”

Section: Algorithm 3 Findneighmentioning

confidence: 99%

Small Littlewood–Richardson coefficients

Ikenmeyer

2016

J Algebr Comb

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“…Kronecker coefficients are #P-hard to compute as they are generalizations of the well-known LittlewoodRichardson coefficients [Nar06]. But the positivity of Littlewood-Richardson coefficients can be decided in polynomial time [DLM06,MNS12], even by a combinatorial max-flow algorithm [BI13].…”

Section: (A) Complexity Lower Bounds Via Representation Theorymentioning

confidence: 99%

Rectangular Kronecker Coefficients and Plethysms in Geometric Complexity Theory

Ikenmeyer

Panova

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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We prove that in the geometric complexity theory program the vanishing of rectangular Kronecker coefficients cannot be used to prove superpolynomial determinantal complexity lower bounds for the permanent polynomial.Moreover, we prove the positivity of rectangular Kronecker coefficients for a large class of partitions where the side lengths of the rectangle are at least quadratic in the length of the partition. We also compare rectangular Kronecker coefficients with their corresponding plethysm coefficients, which leads to a new lower bound for rectangular Kronecker coefficients. Moreover, we prove that the saturation of the rectangular Kronecker semigroup is trivial, we show that the rectangular Kronecker positivity stretching factor is 2 for a long first row, and we completely classify the positivity of rectangular limit Kronecker coefficients that were introduced by Manivel in 2011.

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“…It is natural to ask the complexity of deciding when two partitions are comparable. Since the positivity of Kostka number is equivalent to the comparability of partitions, by Proposition 1 of [15] we know that given λ and µ, whether or not λ and µ are comparable can be answered in polynomial time.…”

Section: Final Remarks and Problemsmentioning

confidence: 99%

A Note on Murthy's Conjecture

Li¹

2007

Demonstratio Mathematica

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Abstract. In this paper, we consider a conjecture made by Murthy to the effect that a CQ f"l Qo matrix is positive semidefinite (PSD) and show that the conjecture is true for n x n matrices of rank 1 or 3 x 3 matrices that are not in Q. We also consider the class of Pi-matrices which is a subclass of Qo and obtain the following: for A 6 Pi , if A S CQ and A £ Q, then A is PSD; if A 6 C{ and A € Q, then A is PSD for n < 3 and A isn't PSD for n > 3. IntroductionThe linear complementarity problem (LCP) with data A G R nxn and q € R n involves finding a vector z E R n such thatLCP has numerous applications, both in theory and practice, treated by vast literature (see [l]). A number of matrix classes have been defined in connection with LCP. We shall briefly introduce the concepts and notation required for presentation of the results of this paper.For most of other notion, we shall follow that used in [1]. For any positive integer n, write n = {1,2, • • •, n}, and for any subset a of n, write a = n\a. Consider A £ R nxn .If a C n such that detA aa ^ 0, then the matrix M defined byis known as the principal pivotal transform (PPT) of A with respect to a This work is supported by YSF of Guangdong University of Technology (062058)

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On the complexity of computing Kostka numbers and Littlewood-Richardson coefficients

Cited by 85 publications

References 12 publications

Small Littlewood–Richardson coefficients

Small Littlewood–Richardson coefficients

Rectangular Kronecker Coefficients and Plethysms in Geometric Complexity Theory

A Note on Murthy's Conjecture

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