Computational complexity of counting coincidences

Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in R3 have the same number of domino tilings? There are two versions of the problem, with 2×1×1 and 2×2×1 boxes. We prove that in both cases the coincidenc...

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Vydané v:Theoretical computer science Ročník 1015; s. 114776
Hlavní autori: Chan, Swee Hong, Pak, Igor
Médium: Journal Article
Jazyk:English
Vydavateľské údaje: Elsevier B.V 01.11.2024
Predmet:
Domino tiling
Graph matching
Kronecker coefficient
Linear extensions of posets
Matroid bases
Order ideals of posets
P-completeness
Permanent
Standard Young tableau
Standard Young tableau
Domino tiling
Permanent
Order ideals of posets
Kronecker coefficient
Matroid bases
Linear extensions of posets
P-completeness
Graph matching
ISSN:0304-3975
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Popis
Shrnutí:Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in R3 have the same number of domino tilings? There are two versions of the problem, with 2×1×1 and 2×2×1 boxes. We prove that in both cases the coincidence problem is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. While the conclusions are the same, the proofs are notably different and generalize in different directions. We proceed to explore the coincidence problem for counting independent sets and matchings in graphs, matroid bases, order ideals and linear extensions in posets, permutation patterns, and the Kronecker coefficients. We also make a number of conjectures for counting other combinatorial objects such as plane triangulations, contingency tables, standard Young tableaux, reduced factorizations and the Littlewood–Richardson coefficients.
ISSN:0304-3975
DOI:10.1016/j.tcs.2024.114776