The Complexity of Computing the Sign of the Tutte Polynomial

We study the complexity of computing the sign of the Tutte polynomial of a graph. As there are only three possible outcomes (positive, negative, and zero), this seems at first sight more like a decision problem than a counting problem. Surprisingly, however, there are large regions of the parameter space for which computing the sign of the Tutte polynomial is actually #P-hard. As a trivial consequence, approximating the polynomial is also #P-hard in this case. Thus, approximately evaluating the Tutte polynomial in these regions is as hard as exactly counting the satisfying assignments to a CNF Boolean formula. For most other points in the parameter space, we show that computing the sign of the polynomial is in FP, whereas approximating the polynomial can be done in polynomial time with an NP oracle. As a special case, we completely resolve the complexity of computing the sign of the chromatic polynomial - this is easily computable at q=2 and when q is less than or equal to 32/27, and is NP-hard to compute for all other values of the parameter q.Comment: minor updates. This is the final version (to appear in SICOMP

Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials

In this paper we show a new way of constructing deterministic polynomial-time approximation algorithms for computing complex-valued evaluations of a large class of graph polynomials on bounded degree graphs. In particular, our approach works for the Tutte polynomial and independence polynomial, as well as partition functions of complex-valued spin and edge-coloring models. More specifically, we define a large class of graph polynomials $\mathcal C$ and show that if $p\in \cal C$ and there is a disk $D$ centered at zero in the complex plane such that $p(G)$ does not vanish on $D$ for all bounded degree graphs $G$, then for each $z$ in the interior of $D$ there exists a deterministic polynomial-time approximation algorithm for evaluating $p(G)$ at $z$. This gives an explicit connection between absence of zeros of graph polynomials and the existence of efficient approximation algorithms, allowing us to show new relationships between well-known conjectures. Our work builds on a recent line of work initiated by. Barvinok, which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.Comment: 27 pages; some changes have been made based on referee comments. In particular a tiny error in Proposition 4.4 has been fixed. The introduction and concluding remarks have also been rewritten to incorporate the most recent developments. Accepted for publication in SIAM Journal on Computatio

The complexity of approximating the matching polynomial in the complex plane

We study the problem of approximating the value of the matching polynomial on graphs with edge parameter $\gamma$, where $\gamma$ takes arbitrary values in the complex plane. When $\gamma$ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of $\gamma$, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree $\Delta$ as long as $\gamma$ is not a negative real number less than or equal to $-1/(4(\Delta-1))$. Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all $\Delta\geq 3$ and all real $\gamma$ less than $-1/(4(\Delta-1))$, the problem of approximating the value of the matching polynomial on graphs of maximum degree $\Delta$ with edge parameter $\gamma$ is #P-hard. We then explore whether the maximum degree parameter can be replaced by the connective constant. Sinclair et al. showed that for positive real $\gamma$ it is possible to approximate the value of the matching polynomial using a correlation decay algorithm on graphs with bounded connective constant (and potentially unbounded maximum degree). We first show that this result does not extend in general in the complex plane; in particular, the problem is #P-hard on graphs with bounded connective constant for a dense set of $\gamma$ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value $\gamma$ that does not lie on the negative real axis. Our analysis accounts for complex values of $\gamma$ using geodesic distances in the complex plane in the metric defined by an appropriate density function

The complexity of approximating the complex-valued Potts model

We study the complexity of approximating the partition function of the $q$-state Potts model and the closely related Tutte polynomial for complex values of the underlying parameters. Apart from the classical connections with quantum computing and phase transitions in statistical physics, recent work in approximate counting has shown that the behaviour in the complex plane, and more precisely the location of zeros, is strongly connected with the complexity of the approximation problem, even for positive real-valued parameters. Previous work in the complex plane by Goldberg and Guo focused on $q=2$, which corresponds to the case of the Ising model; for $q>2$, the behaviour in the complex plane is not as well understood and most work applies only to the real-valued Tutte plane. Our main result is a complete classification of the complexity of the approximation problems for all non-real values of the parameters, by establishing \#P-hardness results that apply even when restricted to planar graphs. Our techniques apply to all $q\geq 2$ and further complement/refine previous results both for the Ising model and the Tutte plane, answering in particular a question raised by Bordewich, Freedman, Lov\'{a}sz and Welsh in the context of quantum computations.Comment: 58 pages. Changes on version 2: minor change

Lee-yang zeros and the complexity of the ferromagnetic ising model on bounded-degree graphs

We study the computational complexity of approximating the partition function of the ferromagnetic Ising model in the Lee-Yang circle of zeros given by |λ| = 1, where λ is the external field of the model. Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics, but have also been key in the recent deterministic approximation scheme for all |λ| ≠ 1 by Liu, Sinclair, and Srivastava. Here, we focus on the unresolved complexity picture on the unit circle, and on the tantalising question of what happens in the circular arc around λ = 1, where on one hand the classical algorithm of Jerrum and Sinclair gives a randomised approximation scheme on the real axis suggesting tractability, and on the other hand the presence of Lee-Yang zeros alludes to computational hardness. Our main result establishes a sharp computational transition at the point λ = 1; in fact, our techniques apply more generally to the whole unit circle |λ| = 1. We show #P-hardness for approximating the partition function on graphs of maximum degree Δ when b, the edge-interaction parameter, is in the interval [EQUATION] and λ is a non-real on the unit circle. This result contrasts with known approximation algorithms when |λ| ≠ 1 or [EQUATION], and shows that the Lee-Yang circle of zeros is computationally intractable, even on bounded-degree graphs. Our inapproximability result is based on constructing rooted tree gadgets via a detailed understanding of the underlying dynamical systems, which are further parameterised by the degree of the root. The ferromagnetic Ising model has radically different behaviour than previously considered anti-ferromagnetic models, and showing our #P-hardness results in the whole Lee-Yang circle requires a new high-level strategy to construct the gadgets. To this end, we devise an elaborate inductive procedure to construct the required gadgets by taking into account the dependence between the degree of the root of the tree and the magnitude of the derivative at the fixpoint of the corresponding dynamical system