Spin glasses are fundamental probability distributions at the core of statistical physics, the theory of average-case computational complexity, and modern high-dimensional statistical inference. In the mean-field setting, we design deterministic quasipolynomial-time algorithms for estimating the partition function to arbitrarily high accuracy for all inverse temperatures in the second moment regime. In particular, for the Sherrington--Kirkpatrick model, our algorithms succeed for the entire replica-symmetric phase. To achieve this, we study the locations of the zeros of the partition function. Notably, our methods are conceptually simple, and apply equally well to the spherical case and the case of Ising spins.

Abstract Answering a question of Gamarnik and Smedira [15], we give a polynomial time algorithm that approximately computes the volume of a truncation of a relaxation of the independent set polytope, improving on their quasi-polynomial time algorithm. Our algorithm is obtained by viewing the volume as an evaluation of a graph polynomial and we approximate this evaluation using Barvinok’s interpolation method.

Abstract We show that for any non-real algebraic number q, such that $$|q-1|>1$$ | q - 1 | > 1 or $$\Re(q)>\frac{3}{2}$$ ℜ ( q ) > 3 2 it is #P-hard to compute a multiplicative (resp. additive) approximation to the absolute value (resp. argument) of the chromatic polynomial evaluated at q on planar graphs. This implies #P-hardness for all non-real algebraic q on the family of all graphs. We, moreover, prove several hardness results for q, such that $$|q-1|\leq 1$$ | q - 1 | ≤ 1 . Our hardness results are obtained by showing that a polynomial time algorithm for approximately computing the chromatic polynomial of a planar graph at non-real algebraic q (satisfying some properties) leads to a polynomial time algorithm for exactly computing it, which is known to be hard by a result of Vertigan. Many of our results extend in fact to the more general partition function of the random cluster model, a well-known reparametrization of the Tutte polynomial.

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 and λ is a non-real on the unit circle. This result contrasts with known approximation algorithms when , 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. The full version (with all proofs) is available on arXiv at arxiv.org/abs/2006.14828.

We show how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture. The variant, which we call Count Unique Games, is a promise problem in which the "yes" case guarantees a certain number of highly satisfiable assignments to the Unique Games instance. In the standard Unique Games problem, the "yes" case only guarantees at least one such assignment. We exhibit efficient algorithms for Count Unique Games based on approximating a suitable partition function for the Unique Games instance via (i) a zero-free region and polynomial interpolation, and (ii) the cluster expansion. We also show that a modest improvement to the parameters for which we give results would be strong negative evidence for the truth of the Unique Games Conjecture.

In this paper we show that for any graph H of order m and any graph G of order n and maximum degree $$\Delta $$ one can compute the number of subsets S of V(G) that induces a graph isomorphic to H in time $$O(c^m \cdot n )$$ for some constant $$c = c(\Delta ) >0$$ . This is essentially best possible (in the sense that there is no $$c^{o(m)}poly(n)$$ -time algorithm under the exponential time hypothesis).

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 [ Found. Comput. Math., 16 (2016), pp. 329--342; Theory Comput., 11 (2015), pp. 339--355; Computing the Partition Function of a Polynomial on the Boolean Cube, 2015; Discrete Anal., 2 (2017), 34pp], which provides a new algorithmic approach besides the existing Markov chain Monte Carlo method and the correlation decay method for these types of problems.

We consider the variation of Ramsey numbers introduced by Erdös and Pach [J. Graph Theory, 7 (1983), pp. 137--147], where instead of seeking complete or independent sets we only seek a $t$-homogeneous set, a vertex subset that induces a subgraph of minimum degree at least $t$ or the complement of such a graph. For any $\nu > 0$ and positive integer $k$, we show that any graph $G$ or its complement contains as an induced subgraph some graph $H$ on $\ell \ge k$ vertices with minimum degree at least $\frac12(\ell-1) + \nu$ provided that $G$ has at least $k^{\Omega(\nu^2)}$ vertices. We also show this to be the best possible in a sense. This may be viewed as correction to a result claimed in [P. Erdös and J. Pach, J. Graph Theory, 7 (1983), pp. 137--147]. For the above result, we permit $H$ to have order at least $k$. In the harder problem, where we insist that $H$ have exactly $k$ vertices, we do not obtain sharp results, although we show a way to translate results of one form of the problem to the other.