Computing the Tutte Polynomial in Vertex-Exponential Time Andreas Bjorklund¨ 1 Thore Husfeldt1;2 Petteri Kaski3 Mikko Koivisto3 Abstract Here, G is a graph with vertex set V and edge set E; by c(F) we denote the number of connected components in the The deletion–contraction algorithm is perhaps the most graph with vertex set V and edge set F. Later, Oxley and popular method for computing a host of fundamental graph Welsh [35] showed in their celebrated Recipe Theorem that, invariants such as the chromatic, flow, and reliability poly- in a very strong sense, the Tutte polynomial TG is indeed the nomials in graph theory, the Jones polynomial of an alter- most general graph invariant that can be computed using nating link in knot theory, and the partition functions of the deletion–contraction. models of Ising, Potts, and Fortuin–Kasteleyn in statistical Since the 1980s it has become clear that this construction physics. Prior to this work, deletion–contraction was also has deep connections to many fields outside of computer the fastest known general-purpose algorithm for these in- science and algebraic graph theory. It appears in various variants, running in time roughly proportional to the num- guises and specialisations in enumerative combinatorics, ber of spanning trees in the input graph. statistical physics, knot theory, and network theory. It Here, we give a substantially faster algorithm that com- subsumes the chromatic, flow, and reliability polynomials, putes the Tutte polynomial—and hence, all the aforemen- the Jones polynomial of an alternating link, and, perhaps tioned invariants and more—of an arbitrary graph in time most importantly, the models of Ising, Potts, and Fortuin– within a polynomial factor of the number of connected ver- Kasteleyn, which appear in tens of thousands of research tex sets. The algorithm actually evaluates a multivariate papers. A number of surveys written for various audiences generalization of the Tutte polynomial by making use of present and explain these specialisations [38, 42, 43, 44]. an identity due to Fortuin and Kasteleyn. We also provide Computing the Tutte polynomial has been a very fruitful a polynomial-space variant of the algorithm and give an topic in theoretical computer science, resulting in seminal analogous result for Chung and Graham’s cover polyno- work on the computational complexity of counting, several mial. algorithmic breakthroughs both classical and quantum, and whole research programmes devoted to the existence and 1 Introduction nonexistence of approximation algorithms. Its specialisa- tion to graph colouring has been one of the main bench- Tutte’s motivation for studying what he called the “dichro- marks of progress in exact algorithms. matic polynomial” was algorithmic. By his own enter- The deletion–contraction algorithm computes TG for a taining account [40], he was intrigued by the variety of connected G in time within a polynomial factor of τ(G), graph invariants that could be computed with the deletion– the number of spanning trees of the graph, and no es- contraction algorithm, and “playing” with it he discovered sentially faster algorithm was known. In this paper we a bivariate polynomial that we can define as show that the Tutte polynomial—and hence, by virtue of the Recipe Theorem, every graph invariant admitting a X c(F)−c(E) c(F)+jF|−|Vj TG(x; y) = (x − 1) (y − 1) : (1) deletion–contraction recursion—can be computed in time F⊆E within a polynomial factor of σ(G), the number of ver- tex subsets that induce a connected subgraph. Especially, 1Lund University, Department of Computer Science, P.O.Box 118, SE- the algorithm runs in time expO(n), that is, in “vertex- 22100 Lund, Sweden. 2 exponential” time, while τ(G) typically is exp !(n) and can IT University of Copenhagen, Rued Langgaards Vej 7, DK-2300 n−2 Copenhagen S, Denmark. be as large as n [12]. 3Helsinki Institute for Information Technology HIIT, Department of Computer Science, University of Helsinki, P.O.Box 68, FI-00014 University of Helsinki, Finland. 1.1 Result and consequences 4This research was supported in part by the Academy of Finland, Grants 117499 (P.K.) and 109101 (M.K.) and by the Swedish Research By “computing the Tutte polynomial” we mean computing i j Council, project “Exact Algorithms” (A.B., T.H.). the coefficients ti j of the monomials x y in TG(x; y) for a 1 ) n ) n q See x2.3 for references. The only points that are known to ( 2 y ∗ ( ∗ P O admit algorithms with better bounds than our main result O : 1 ‘Jones’ H − ; − ; 1 are the “colouring” points ( 2 0) and ( 3 0), the “Ising” ) n 3 = ‘Potts’: ( hyperbola H , for which a faster algorithm in observed in y ‘Ising’: ∗ 2 ) x : 2 O : : H x ; 2.3, and of course the points in P. w’ 4 ; 3 ‘flo For bounded-degree graphs G, the deletion–contraction 0 = q = algorithm itself runs in vertex-exponential time because ( q x ‘reliability’: H τ(G) = exp O(n) . Theorem 1 still gives a better bound 1 n = because it is known that σ(G) = O((2 − ) ) for bounded x degree [7, Lemma 6], while τ(G) grows faster than 2:3n al- ) ) n n 2 conn. ready for 3-regular graphs (see x2.4). The precise bound is spann. 9464 6262 : : subgraphs as follows: 1 1 ( ( O O P t 1 (G): P forests: Corollary 2 The Tutte polynomial of an n-vertex graph P O ∗ (2 n) with maximum vertex degree ∆ can be computed in time ∗ n ∆+1 1=(∆+1) O (ξ ), where ξ∆ = (2 − 1) . 4-colourings: 3-colourings: bipartitions: empty: ∆ x −3 −2 −1 0 1 2 ac ycl. orientations y = 0 ‘chromatic’: O∗(2n) The question about solving deletion–contraction based algorithmic problems in vertex-exponential time makes −1 dim. Eulerian: sense in directed graphs as well. Here, the most success- of bicycle P space: ful attempt to define an analogue of the Tutte polynomial P is Chung and Graham’s cover polynomial, which satisfies directed analogues to the deletion–contraction operations Figure 1. Tutte plane with prior complexities. [13]. Let D be a digraph, possibly with parallel edges and graph G given as input. Of course, the coefficients also en- loops. Denote by cD(i; j) the number of ways of disjointly able the efficient evaluation of TG(x; y) at any given point covering all the vertices of D with i directed paths and j (x; y). Our main result is as follows. directed cycles. The cover polynomial of D is defined as X i j Theorem 1 The Tutte polynomial of an n-vertex graph G CD(x; y) = cD(i; j)x y ; can be computed i; j (a) in time and space O∗(σ(G)); where xi = x(x − 1) ··· (x − i + 1) and x0 = 1. It is known that C (x; y) is #P-complete to evaluate except at a handful (b) in time O∗(3n) and polynomial space; and D of points (x; y) [9]. (c) in time O∗(3n−s2s) and space O∗(2s) for any integer s, In analogy to Theorem 1, we can show that CD can be 0 ≤ s ≤ n. computed in vertex-exponential time: Especially, the Tutte polynomial can be evaluated every- Theorem 3 The cover polynomial of an n-vertex directed where in vertex-exponential time. In some sense, this is graph can be computed both surprising and optimal, a claim that we solidify under (a) in time and space O∗ n ; and the Exponential Time Hypothesis in x2.5. (2 ) Prescribing the point of evaluation (x; y) leads to the (b) in time O∗(3n) and polynomial space. more restricted task of computing the value TG(x; y) for a given G. In this setting vertex-exponential (or faster) algo- rithms were known before for a number of points (x; y); see 1.2 Overview of techniques Figure 1. The hyperbolas Hq are defined by (x − 1)(y − 1) = q, only their positive branches are drawn. The points (0; 0), The Tutte polynomial is, in essence, a sum over connected (−1; 0), (0; −1), (−1; −1), (1; 1) and the hyperbola H1 are in spanning subgraphs. Managing this connectedness property P, all other points are #P-complete. Those points and lines introduces a computational challenge not present with its where algorithms with complexity expO(n) were previ- specialisations, e.g., with the chromatic polynomial. Nei- ously known (sometimes only in exponential space), are ther the dynamic programming algorithm across vertex sub- labelled with their running time; the hyperbolas Hq were sets by Lawler [33] nor the recent inclusion–exclusion al- known to be vertex-exponential only for positive integer q. gorithm [8], which apply for counting k-colourings, seems 2 to work directly for the Tutte polynomial. Perhaps surpris- To simplify running time bounds, we use the notation O∗ ingly, they do work for the cover polynomial, even though to suppress a polynomial factor (always in n), so O∗( f (r)) the application is quite involved; the details are in x5 and means O( f (r)nk) for some constant k. We also assume m is can be seen as an attempt to explain just how far these con- bounded by a polynomial in n and remark that this assump- cepts get us. tion is implicit already in Theorem 1. (Without this assump- For the Tutte polynomial, we take a detour via the Potts tion, all the time bounds require an additional multiplicative model. The idea is to evaluate the partition function of term polynomial in m.) For a set of vertices U ⊆ V(G), we the q-state Potts model at suitable points using inclusion– write G[U] for the subgraph induced by U in G.