The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VP_ ws and VNP. Mulmuley and Sohoni (SIAM J. Comput., 2001) suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of GL_ n^2 (C), which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem via multiplicity obstructions as proposed by Mulmuley and Sohoni.
A first attempt towards finding occurrence obstructions was by asymptotic considerations (moment polytopes): this was ruled out in @cite_4 , where it was proven that for all @math there exists a stretching factor @math such that @math . In @cite_5 it was shown that deciding positivity of Kronecker coefficients in general is NP-hard, but this proof fails for rectangular formats.
@cite_5 @cite_4
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Let P be a Poisson process of intensity one in a square Sn of area n. For a fixed integer k, join every point of P to its k nearest neighbours, creating an undirected random geometric graph Gn,k. We prove that there exists a critical constant ccrit such that for c ccrit, Gn,⌊clog n⌋ is connected with probability tending to 1 as n → ∞. This answers a question posed by the authors in [1]. Let P be a Poisson process of intensity one in a square Sn of area n. For a fixed integer k, we join every point of P to its k nearest neighbours, creating an undirected random geometric graph GSn,k = Gn,k in which every vertex has degree at least k. The connectivity of these graphs was studied by the present authors in [1]. It is not hard to see that Gn,k becomes connected around k = �(log n), and we proved in [1] that if k(n) ≤ 0.3043log n then the probability that Gn,k(n) is connected tends to zero as n → ∞, while if k(n) ≥ 0.5139log n then the probability that Gn,k(n) is connected tends to one as n → ∞. However, we were unable to prove the natural conjecture that there exists a critical constant ccrit such that for c ccrit, P(Gn,⌊clog n⌋ is connected) → 1 as n → ∞. In this paper we prove this conjecture. The problem of finding a d -simplex of maximum volume in an arbitrary d -dimensional V -polytope, for arbitrary d , was shown by [GKL] in 1995 to be NP-hard. They conjectured that the corresponding problem for H -polytopes is also NP-hard. This paper presents a unified way of proving the NP-hardness of both these problems. The approach also yields NP-hardness proofs for the problems of finding d -simplices of minimum volume containing d -dimensional V - or H -polytopes. The polytopes that play the key role in the hardness proofs are truncations of simplices. A construction is presented which associates a truncated simplex to a given directed graph, and the hardness results follow from the hardness of detecting whether a directed graph has a partition into directed triangles. We show that the problem of deciding positivity of Kronecker coefficients is NP-hard. Previously, this problem was conjectured to be in P, just as for the Littlewood–Richardson coefficients. Our result establishes in a formal way that Kronecker coefficients are more difficult than Littlewood–Richardson coefficients, unless P = NP. We consider the problem of computing the smallest possible distortion for embedding of a given n-point metric space into ℝ d , where d is fixed (and small). For d = 1, it was known that approximating the minimum distortion with a factor better than roughly n 1 12 is NP-hard. From this result we derive inapproximability with a factor roughly n 1 (22d-10) for every fixed d ≥ 2, by a conceptually very simple reduction. However, the proof of correctness involves a nontrivial result in geometric topology (whose current proof is based on ideas due to Jussi Vaisala), For d > 3, we obtain a stronger inapproximability result by a different reduction: assuming P≠NP, no polynomial-time algorithm can distinguish between spaces embeddable in ℝ d with constant distortion from spaces requiring distortion at least n c d , for a constant c > 0. The exponent c d has the correct order of magnitude, since every n-point metric space can be embedded in ℝ d with distortion O(n 2 d log 3 2 n) and such an embedding can be constructed in polynomial time by random projection. For d = 2, we give an example of a metric space that requires a large distortion for embedding in ℝ 2 , while all not too large subspaces of it embed almost isometrically.