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Polynomial Time Randomised Approximation Schemes for the Tutte Polynomial of Dense Graphs

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Polynomial Time Randomised Approximation Schemes for the Tutte Polynomial of Dense Graphs Polynomial time randomised approximation schemes for the Tutte polynomial of dense graphs Noga Alon∗ Alan Friezey Dominic Welshz Abstract (iii) How many acyclic orientations has G? The Tutte-Gr¨othendieck polynomial T (G; x; y) of a graph Each of these is a special case of the general problem of G encodes numerous interesting combinatorial quanti- evaluating the Tutte polynomial of a graph (or matroid) ties associated with the graph. Its evaluation in various at a particular point of the (x; y)-plane | in other words points in the (x; y) plane give the number of spanning is a Tutte-Gr¨othendieck invariant. Other invariants in- forests of the graph, the number of its strongly connected clude: orientations, the number of its proper k-colorings, the (all terminal) reliability probability of the graph, and var- (iv) the chromatic and flow polynomials of a graph; ious other invariants the exact computation of each of which is well known to be #P -hard. Here we develop (v) the partition function of a Q-state Potts model; a general technique that supplies fully polynomial ran- domised approximation schemes for approximating the (vi) the Jones polynomial of an alternating link; value of T (G; x; y) for any dense graph G, that is, any graph on n vertices whose minimum degree is Ω(n), (vii) the weight enumerator of a linear code over GF (q). whenever x 1 and y 1, and in various additional ≥ ≥ points. This region includes evaluations of reliability and It has been shown in Vertigan and Welsh [19] that apart partition functions of the ferromagnetic Q-state Potts from a few special points and 2 special hyperbolae, the model. exact evaluation of any such invariant is #P -hard even for the very restricted class of planar bipartite graphs. However the question of which points have a fully poly- 1 Introduction nomial randomised approximation scheme (fpras) is wide open. A survey of what is currently known is given in [21], here we prove several new results concerning the Consider the following very simple counting problems existence of a fpras for dense graphs. More precisely, for associated with a graph G. 0 < α < 1, let α denote the set of graphs G = (V; E) with V = n andG δ(G) αn. A graph is α-dense if it is j j ≥ (i) What is the number of connected subgraphs of G? a member of α or, somewhat loosely, dense if we omit the α. G (ii) How many subgraphs of G are forests? Various counting and approximation problems are ∗Department of Mathematics, Raymond and Beverly Sackler known to be easier for graphs of sufficiently high den- Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel. sity than for general graphs. The number of perfect Email: [email protected]. Research supported in part by a matchings in bipartite graphs (which is not an evalua- USA-Israeli BSF grant and by the Fund for Basic Research ad- tion of the Tutte polynomial) is one such example. The ministered by the Israel Academy of Sciences. results of Broder [3] and of Jerrum and Sinclair [10] sup- yDepartment of Mathematics, Carnegie Mellon University, Pittsburgh PA15213, U.S.A., Supported by NSF grant CCR- ply a fpras for approximating that number for graphs 9225008 on two vertex classes of n vertices each which are (1/4)- zMerton College and Mathematical Institute, Oxford, UK. Re- dense. Dyer, Frieze and Jerrum [5] found an fpras for the search supported in part by ESPRIT RAND. number of Hamilton cycles in graphs which are α-dense 1 when α > 1=2. Annan [2] obtained a fpras for the num- First recall that if A E(G), the rank of A, r(A) is ber of forests of a dense graph (given by the value of the defined by ⊆ Tutte polynomial at (2; 1)). Edwards [6] showed that the number of proper k colorings of sufficiently dense graphs r(A) = V (G) k(A); (given by evaluating the Tutte polynomial at (1 k; 0)) j j − can be computed exactly in polynomial time, whereas− where k(A) is the number of connected components of it is clear that this number cannot be approximated in the graph G : A having vertex set V = V (G) and edge polynomial time for general graphs unless RP = NP . set A. Our main new result is a general technique that supplies It is now straightforward to prove: fully polynomial randomised approximation schemes for The Tutte polynomial T (G; x; y) can be expressed in the approximating the value of T (G; x; y) for any dense form graph G, whenever x 1 and y 1, and in various additional points. ≥ ≥ r(E) r(A) A r(A) T (G; x; y) = (x 1) − (y 1)j |− : The graph terminology used is standard. The complexity X − − A E theory and notation follows Garey and Johnson [8]. The ⊆ matroid terminology follows Oxley [13]. Further details of most of the concepts treated here can be found in These ideas can be extended to matroids - see for exam- Welsh [20]. ple [4] and [20]. 2 Tutte-Gr¨othendieck invariants 3 A catalogue of invariants First consider the following recursive definition of the We now collect together some of the naturally occurring function T (G; x; y) of a graph G, and two independent interpretations of the Tutte polynomial. Throughout G variables x; y. is a graph, M is a matroid and E will denote E(G);E(M) respectively. If G has no edges then T (G; x; y) = 1, otherwise for any e E(G); 2 (3.1) At (1,1) T counts the number of bases of M (span- ning trees in a connected graph). (2.1) T (G; x; y) = T (Ge0 ; x; y)+T (Ge00; x; y) if e is neither a loop nor an isthmus, where Ge0 denotes the deletion (3.2) At (2,1) T counts the number of independent sets of the edge e from G and Ge00 denotes the contraction of M, (forests in a graph). of e in G, (3.3) At (1,2) T counts the number of spanning sets of (2.2) T (G; x; y) = xT (Ge0 ; x; y) e an isthmus, M, that is sets which contain a base. (2.3) T (G; x; y) = yT (Ge00; x; y) e a loop. (3.4) At (2,0), T counts the number of acyclic orienta- tions of G. Stanley [17] also gives interpretations of From this, it is easy to show by induction that T is a T at (m; 0) for general positive integer m, in terms 2-variable polynomial in x; y, which we call the Tutte of acyclic orientations. polynomial of G. (3.5) Another interpretation at (2,0), and this for a dif- In other words, T may be calculated recursively by ferent class of matroids, was discovered by Zaslavsky choosing the edges in any order and repeatedly using [23]. This is in terms of counting the number of (2.1-3) to evaluate T . The remarkable fact is that T is different arrangements of sets of hyperplanes in n- well defined in the sense that the resulting polynomial is dimensional Euclidean space. independent of the order in which the edges are chosen. E d(B) (3.6) T (G; 1; 1) = ( 1)j j( 2) where B is the Alternatively, and this is often the easiest way to prove bicycle− space− of G,− see Read− and Rosenstiehl [15]. properties of T , we can show that T has the following When G is planar it also has interpretations in terms expansion. of the Arf invariant of the associated knot. 2 (3.7) The chromatic polynomial P (G; λ) is given by (3.15) Along the hyperbola xy = 1 when G is planar, T specialises to the Jones polynomial of the alternat- ing link or knot associated with G. This connection P (G; λ) = ( 1)r(E)λk(G)T (G; 1 λ, 0) − − was first discovered by Thistlethwaite [18]. where k(G) is the number of connected components. More details on these topics can be found in Welsh [20] (3.8) The flow polynomial F (G; λ) is given by and other more specialised interpretations can be found in the survey of Brylawski and Oxley [4]. E r(E) F (G; λ) = ( 1)j |− T (G; 0; 1 λ): − − (3.9) The (all terminal) reliability R(G; p) is given by 4 The ferromagnetic random E r(E) r(E) cluster model R(G; p) = qj |− p T (G; 1; 1=q) where q = 1 p. As we mentioned above, as Q varies between 0 < Q < , − T evaluates the partition function of the random cluster1 model. For integer Q this is the Q-state Potts model and In each of the above cases, the interesting quantity (on when Q = 2 it is the Ising model. When x 1 and y 1, the left hand side) is given (up to an easily determined we have the region corresponding to ferromagnetism≥ ≥. In term) by an evaluation of the Tutte polynomial. We shall the case Q = 2 we know from Jerrum and Sinclair [11] use the phrase \specialises to" to indicate this. Thus for that there is a fpras for all G. Here we obtain a similar example, along y = 0, T specialises to the chromatic result for general Q but only for the dense case. polynomial. For the remainder of this paper, except for Sections 8, we It turns out that the hyperbolae Hα defined by assume that we are dealing with the Tutte polynoomial of an α-dense graph G.
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