{"title":"Computational complexity of counting coincidences","authors":"","doi":"10.1016/j.tcs.2024.114776","DOIUrl":null,"url":null,"abstract":"<div><p>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 <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> have the same number of domino tilings? There are two versions of the problem, with <span><math><mn>2</mn><mo>×</mo><mn>1</mn><mo>×</mo><mn>1</mn></math></span> and <span><math><mn>2</mn><mo>×</mo><mn>2</mn><mo>×</mo><mn>1</mn></math></span> 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.</p><p>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.</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0304397524003931/pdfft?md5=7678638db921f3cd70363b4dc17d17d4&pid=1-s2.0-S0304397524003931-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397524003931","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
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 have the same number of domino tilings? There are two versions of the problem, with and 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.
期刊介绍:
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.