Computational Complexity in Algebraic Combinatorics

Question

1. What is the computational complexity of computing the number of standard Young tableaux?

2. Is KostkaPos P?

3. Is ComputeLR #P-complete when input is in unary?

4. Is KroneckerPos NP-hard in unary?

Accepted Answer

The computational complexity of computing the number of standard Young tableaux depends on the input encoding. If the input is encoded in unary, the problem is in FP, as the HLF formula allows the computation in O(n) time. However, if the input is in binary, the input size is O(log 2(|I|)), and the number of SYTs of shape l can be computed in polynomial time. For binary input, the answer is too large to be encoded in exponential space. The next natural problem is to compute the number of SSYTs of shape l and given type a, but no product or determinantal formula is known for this problem.

Accepted Answer

KostkaPos is a computational problem that checks if Kostka numbers K l,a are positive. Given l and a in either binary or unary, it verifies the inequalities in O(l(l)) time. The problem is not trivial, but it can be solved in polynomial time. The computational problem ComputeKostka, which computes the value of K l,a, is #P-complete when the input is in binary. However, when the input is in unary, it is conjectured to be #P-complete. The problem's complexity depends on the input encoding, which is usually part of the problem specification.

Accepted Answer

Yes, ComputeLR is conjectured to be #P-complete when the input is in unary. This is based on the Littlewood-Richardson rule and its polytopal equivalent formulation. The hard part is to show that computing them is still hard, meaning that an #P-complete problem like 3SAT would reduce to ComputeLR. However, this has not been proven for Kronecker and plethysm coefficients due to the lack of any positive combinatorial formula.

Accepted Answer

Yes, KroneckerPos is NP-hard when the input l, u, n is in unary. This means that the problem cannot be solved in polynomial time. The proof utilizes the fact that g(l, u, n) represents the number of pyramids with marginals l, u, n, which is NP-complete. However, the problem is not in NP as there are no polynomially verifiable witnesses to show that g(l, u, n) > 0 when it occurs. The problem becomes even more challenging when the input is in binary, but it is not considered in this context. Additionally, Mulmuley conjectured that computing the Kronecker coefficients would be in #P, similar to the Littlewood-Richardson coefficients.

Accepted Answer

ComputeKron is not in #P under reasonable complexity theoretic assumptions such as PH not collapsing. If the above is proven, that would make any solution to Open Problem 5.1 as unlikely as the polynomial hierarchy collapsing. Any reasonable combinatorial interpretation as counting certain objects would show that the problem is in #P, as the objects would likely be verifiable in polynomial time. The author's experience with Kronecker coefficients suggests that some particular families would be as hard as the general problem. Conjecture 5.17 (Panova) states that ComputeKron is in #P when l(l) = 2 if and only if ComputeKron is in #P in the general case. Similarly, ComputeKron for u = n = (n d ) and l nd is in #P if and only if the general problem is in #P. When the partitions have fixed lengths, ComputeKron is in FP even when the input is in binary, as shown in [CDW12, PP17b]. From the asymptotic analysis and symmetric function identities in [PP23], it follows that g(l, u, n) can be computed in time O(n 4k 3) when k is fixed and (l, u, n) are partitions with diagonals at most k. This does not guarantee the efficiency of the algorithm, but it can be easily derived. However, when the lengths of the partitions are bounded by k, efficient algorithms run in time O((log n) k 3 log k). It is not expected that a similar efficient algorithm exists in the more general case of fixed diagonals.

Accepted Answer

Conjecture 6.5 was the first conjecture to be disproved, stating that there exist l, s.t.g(l, n d , n d ) = 0 and g l,d,n,m > 0 for some n > poly(m). This conjecture was significant as it led to further research and the development of Theorem 3, which states that if g(l, n d , n d ) = 0, then g l,d,n,m = 0. This theorem was crucial in understanding the behavior of g l,d,n,m and its relationship with l and n. The work of Kadish and Landsberg provided a characterization of g l,d,n,m, leading to Proposition 6.6, which established conditions for g l,d,n,m to be greater than zero. Overall, Conjecture 6.5 played a pivotal role in advancing the understanding of g l,d,n,m and its properties.

Accepted Answer

Structure constants, such as Kronecker and plethysm coefficients, are crucial in Geometric Complexity Theory for separating algebraic complexity classes and polynomials. Understanding the multiplicities of irreducible components in the coordinate rings of orbit closures is essential. Simply determining if multiplicities are 0 or not is insufficient. A deeper understanding of these multiplicities and their potential size is required. Finding their combinatorial interpretation is an open problem in Algebraic Combinatorics and Representation Theory. Computational Complexity theory can provide insights into these fundamental problems, especially for negative answers. Additionally, exploring the asymptotic properties and effective bounds of structure constants, despite the lack of exact formulas, is a significant open problem in the field.

Computational Complexity in Algebraic Combinatorics

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Most frequently asked questions

1. What is the computational complexity of computing the number of standard Young tableaux?

2. Is KostkaPos P?

3. Is ComputeLR #P-complete when input is in unary?

4. Is KroneckerPos NP-hard in unary?

5. Is ComputeKron in #P under PH not collapsing?

6. What is the significance of Conjecture 6.5 in relation to g(l, n d , n d)?

7. What role do structure constants play in Geometric Complexity Theory?

Citations

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Computational complexity of counting coincidences

Computational Complexity of Counting Coincidences

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