We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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Linear And Nonlinear Programming

Journal of the ACM

An Introduction to Probability Theory and its Applications, Volume I

We use the class of commuting quantum computations known as IQP (instantaneous quantum polynomial time) to strengthen the conjecture that quantum computers are hard to simulate classically We show that, if either of two plausible average-case hardness conjectures holds, then IQP computations are hard to simulate classically up to constant additive error One conjecture relates to the hardness of estimating the complex-temperature partition function for random instances of the Ising model; the other concerns approximating the number of zeroes of random low-degree polynomials We observe that both conjectures can be shown to be valid in the setting of worst-case complexity We arrive at these conjectures by deriving spin-based generalizations of the boson sampling problem that avoid the so-called permanent anticoncentration conjecture

Average-Case Complexity Versus Approximate Simulation of Commuting Quantum Computations.

Recent inapproximability results of Sly (2010), together with an approximation algorithm presented by Weitz (2006), establish a beautiful picture of the computational complexity of approximating the partition function of the hard-core model. Let λ c ( ) denote the critical activity for the hard-model on the infinite Δ-regular tree. Weitz presented an FPTAS for the partition function when λ 0 such that (unless RP = NP) there is no FPRAS for approximating the partition function on graphs of maximum degree Δ for activities λ satisfying λ c ( ) < λ < λ c ( ) + eΔ. We prove that a similar phenomenon holds for the antiferromagnetic Ising model. Sinclair, Srivastava and Thurley (2014) extended Weitz's approach to the antiferromagnetic Ising model, yielding an FPTAS for the partition function for all graphs of constant maximum degree Δ when the parameters of the model lie in the uniqueness region of the infinite Δ-regular tree. We prove the complementary result for the antiferromagnetic Ising model without external field, namely, that unless RP = NP, for all Δ ⩾ 3, there is no FPRAS for approximating the partition function on graphs of maximum degree Δ when the inverse temperature lies in the non-uniqueness region of the infinite tree . Our proof works by relating certain second moment calculations for random Δ-regular bipartite graphs to the tree recursions used to establish the critical points on the infinite tree.

Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

Recent inapproximability results of Sly (2010), together with an approximation algorithm presented by Weitz (2006) establish a beautiful picture for the computational complexity of approximating the partition function of the hard-core model. Let $\lambda_c(T_\Delta)$ denote the critical activity for the hard-model on the infinite $\Delta$-regular tree. Weitz presented an FPTAS for the partition function when $\lambda 0$ such that (unless RP=NP) there is no FPRAS for approximating the partition function on graphs of maximum degree $\Delta$ for activities $\lambda$ satisfying $\lambda_c(T_\Delta)<\lambda<\lambda_c(T_\Delta)+\epsilon_\Delta$. 
We prove that a similar phenomenon holds for the antiferromagnetic Ising model. Recent results of Li et al. and Sinclair et al. extend Weitz's approach to any 2-spin model, which includes the antiferromagnetic Ising model, to yield an FPTAS for the partition function for all graphs of constant maximum degree $\Delta$ when the parameters of the model lie in the uniqueness regime of the infinite tree $T_\Delta$. We prove the complementary result that for the antiferrogmanetic Ising model without external field that, unless RP=NP, for all $\Delta\geq 3$, there is no FPRAS for approximating the partition function on graphs of maximum degree $\Delta$ when the inverse temperature lies in the non-uniqueness regime of the infinite tree $T_\Delta$. Our results extend to a region of the parameter space for general 2-spin models. Our proof works by relating certain second moment calculations for random $\Delta$-regular bipartite graphs to the tree recursions used to establish the critical points on the infinite tree.

Recent results establish for the hard-core model (and more generally for 2-spin antiferromagnetic systems) that the computational complexity of approximating the partition function on graphs of maximum degree $\Delta$ undergoes a phase transition that coincides with the uniqueness/non-uniqueness phase transition on the infinite $\Delta$-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs, so-called #BIS-hardness. We improve this hardness result by establishing it for bipartite graphs of maximum degree $\Delta$. To this end, we first present a detailed picture for the phase diagram for the infinite $\Delta$-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and order...

/pdf/ferromagnetic-potts-model-refined-bis-hardness-and-related-2ppqtyrgvd.pdf

Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results

A remarkable connection has been established for antiferromagnetic 2-spin systems, including the Ising and hard-core models, showing that the computational complexity of approximating the partition function for graphs with maximum degree Δ undergoes a phase transition that coincides with the statistical physics uniqueness/nonuniqueness phase transition on the infinite Δ-regular tree. Despite this clear picture for 2-spin systems, there is little known for multispin systems. We present the first analog of this in approximability results for multispin systems.The main difficulty in previous inapproximability results was analyzing the behavior of the model on random Δ-regular bipartite graphs, which served as the gadget in the reduction. To this end, one needs to understand the moments of the partition function. Our key contribution is connecting: (i) induced matrix norms, (ii) maxima of the expectation of the partition function, and (iii) attractive fixed points of the associated tree recursions (belief propagation). The view through matrix norms allows a simple and generic analysis of the second moment for any spin system on random Δ-regular bipartite graphs. This yields concentration results for any spin system in which one can analyze the maxima of the first moment. The connection to fixed points of the tree recursions enables an analysis of the maxima of the first moment for specific models of interest.For k-colorings we prove that for even k, in a tree nonuniqueness region (which corresponds to k

Inapproximability for Antiferromagnetic Spin Systems in the Tree Nonuniqueness Region

We study the computational complexity of approximately counting the number of independent sets of a graph with maximum degree Delta. More generally, for an input graph G=(V,E) and an activity lambda>0, we are interested in the quantity Z_G(lambda) defined as the sum over independent sets I weighted as w(I) = lambda^|I|. In statistical physics, Z_G(lambda) is the partition function for the hard-core model, which is an idealized model of a gas where the particles have non-negibile size. 
Recently, an interesting phase transition was shown to occur for the complexity of approximating the partition function. Weitz showed an FPAS for the partition function for any graph of maximum degree Delta when Delta is constant and lambda 0. We remove the upper bound in the assumptions of Sly's result for Delta not equal to 4 and 5, that is, we show that there does not exist efficient randomized approximation algorithms for all lambda>lambda_c(Tree_Delta) for Delta=3 and Delta>= 6. Sly's inapproximability result uses a clever reduction, combined with a second-moment analysis of Mossel, Weitz and Wormald which prove torpid mixing of the Glauber dynamics for sampling from the associated Gibbs distribution on almost every regular graph of degree Delta for the same range of lambda as in Sly's result. We extend Sly's result by improving upon the technical work of Mossel et al., via a more detailed analysis of independent sets in random regular graphs.

Andreas Galanis

Papers

Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

Inapproximability of the Partition Function for the Antiferromagnetic Ising and Hard-Core Models

Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results

Inapproximability for Antiferromagnetic Spin Systems in the Tree Nonuniqueness Region

Improved Inapproximability Results for Counting Independent Sets in the Hard-Core Model