### On counting homomorphisms to directed acyclic graphsOn counting homomorphisms to directed acyclic graphs

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 Author Dyer, Martin ♦ Goldberg, Leslie Ann ♦ Paterson, Mike Source ACM Digital Library Content type Text Publisher Association for Computing Machinery (ACM) File Format PDF Copyright Year ©2007 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science Subject Keyword Counting ♦ Directed acyclic graphs ♦ Homomorphisms Abstract It is known that if P and NP are different then there is an infinite hierarchy of different complexity classes that lie strictly between them. Thus, if P ≠ NP, it is not possible to classify NP using any finite collection of complexity classes. This situation has led to attempts to identify smaller classes of problems within NP where $\textit{dichotomy}$ results may hold: every problem is either in P or is NP-complete. A similar situation exists for $\textit{counting}$ problems. If P ≠#P, there is an infinite hierarchy in between and it is important to identify subclasses of #P where dichotomy results hold. Graph homomorphism problems are a fertile setting in which to explore dichotomy theorems. Indeed, Feder and Vardi have shown that a dichotomy theorem for the problem of deciding whether there is a homomorphism to a fixed directed acyclic graph would resolve their long-standing dichotomy conjecture for all constraint satisfaction problems. In this article, we give a dichotomy theorem for the problem of counting homomorphisms to directed acyclic graphs. Let $\textit{H}$ be a fixed directed acyclic graph. The problem is, given an input digraph $\textit{G},$ determine how many homomorphisms there are from $\textit{G}$ to $\textit{H}.$ We give a graph-theoretic classification, showing that for some digraphs $\textit{H},$ the problem is in P and for the rest of the digraphs $\textit{H}$ the problem is #P-complete. An interesting feature of the dichotomy, which is absent from previously known dichotomy results, is that there is a rich supply of tractable graphs $\textit{H}$ with complex structure. ISSN 00045411 Age Range 18 to 22 years ♦ above 22 year Educational Use Research Education Level UG and PG Learning Resource Type Article Publisher Date 2007-12-01 Publisher Place New York e-ISSN 1557735X Journal Journal of the ACM (JACM) Volume Number 54 Issue Number 6

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Source: ACM Digital Library