The Computational Complexity of Counting

The complexity theory of counting contrasts intriguingly with that of existence or optimization. 1. Counting versus existence The branch of theoretical computer science known as computational complexity is concerned with quantifying the computational resources required to achieve specified computational goals. Classically, the goal is often to decide the existence of a certain combinatorial structure, for example, whether a given graph G contains a Hamilton cycle. Alternatively, the goal might be to find an occurrence of the structure that is optimal with respect to a certain measure; in the context of the structure "Hamilton cycle," the notorious Travelling Salesman Problem may be cited as an example. Less well studied, and somewhat less well understood, are counting problems, such as determining how many Hamilton cycles a graph G contains. In some areas, such as statistical physics, counting problems arise directly; in many others they appear in the guise of discrete approximations to continuous problems involving multivariate integration. This article aims to sketch the complexity theory of counting, highlighting the ways in which it diverges from that of existence or optimization. Let E be an alphabet, possibly the binary alphabet, in which the objects of interest (e.g., graphs and Hamilton cycles) may be encoded. A witness-testing predicate for some combinatorial structure S is a predicate ip : ZJ* x XJ* —> {0,1}, where the truth of ij)(x, y) is to be interpreted as "?/ is an occurrence of structure S within instance x." Specializing to the structure "Hamilton cycle," %b(x,y) would be true precisely if the words x and y encode (respectively) a graph G and a subgraph H of G, and H is a Hamilton cycle in G. The existence predicate Q(x) for the structure S may be expressed as 4>(x) **3yeZ* [\y\ = p(\x\) A(*,y)], (1) *The author is a Nuffield Foundation Science Research Fellow, and is supported in part by grant G R / F 90363 of the UK Science and Engineering Research Council, and by Esprit Working Group No. 7097, "RAND." Proceedings of t he In terna t iona l Congress of Mathemat ic ians , Zürich, Switzerland 1994 © Birkhäuser Verlag, Basel, Switzerland 1995