Computer Science

We explore connections between the phenomenon of correlation decay and the location of Lee-Yang and Fisher zeros for various spin systems. In particular we show that, in many instances, proofs showing that weak spatial mixing on the Bethe lattice (infinite Δ -regular tree) implies strong spatial mixing on all graphs of maximum degree Δ can be lifted to the complex plane, establishing the absence of zeros of the associated partition function in a complex neighborhood of the region in parameter space corresponding to strong spatial mixing. This allows us to give unified proofs of several recent results of this kind, including the resolution by Peters and Regts of the Sokal conjecture for the partition function of the hard core lattice gas. It also allows us to prove new results on the location of Lee-Yang zeros of the anti-ferromagnetic Ising model. We show further that our methods extend to the case when weak spatial mixing on the Bethe lattice is not known to be equivalent to strong spatial mixing on all graphs. In particular, we show that results on strong spatial mixing in the anti-ferromagnetic Potts model can be lifted to the complex plane to give new zero-freeness results for the associated partition function. This extension allows us to give the first deterministic FPTAS for counting the number of q -colorings of a graph of maximum degree Δ provided only that q≥2Δ . This matches the natural bound for randomized algorithms obtained by a straightforward application of Markov chain Monte Carlo. We also give an improved version of this result for triangle-free graphs.

A conjecture of Hopkins (2018) posits that for certain high-dimensional hypothesis testing problems, no polynomial-time algorithm can outperform so-called "simple statistics", which are low-degree polynomials in the data. This conjecture formalizes the beliefs surrounding a line of recent work that seeks to understand statistical-versus-computational tradeoffs via the low-degree likelihood ratio. In this work, we refute the conjecture of Hopkins. However, our counterexample crucially exploits the specifics of the noise operator used in the conjecture, and we point out a simple way to modify the conjecture to rule out our counterexample. We also give an example illustrating that (even after the above modification), the symmetry assumption in the conjecture is necessary. These results do not undermine the low-degree framework for computational lower bounds, but rather aim to better understand what class of problems it is applicable to.

Counting problems in general and counting graph homomorphisms in particular have numerous applications in combinatorics, computer science, statistical physics, and elsewhere. One of the most well studied problems in this area is #GraphHom(H) --- the problem of finding the number of homomorphisms from a given graph G to the graph H. Not only the complexity of this basic problem is known, but also of its many variants for digraphs, more general relational structures, graphs with weights, and others. In this paper we consider a modification of #GraphHom(H), the #_p GraphHom(H) problem, p a prime number: Given a graph G, find the number of homomorphisms from G to H modulo p. In a series of papers Faben and Jerrum, and Goebel et al. determined the complexity of #_2 GraphHom(H) in the case H (or, in fact, a certain graph derived from H) is square-free, that is, does not contain a 4-cycle. Also, Goebel et al. found the complexity of #_p GraphHom(H) for an arbitrary prime p when H is a tree. Here we extend the above result to show that the #_p GraphHom(H) problem is #_p P-hard whenever the derived graph associated with H is square-free and is not a star, which completely classifies the complexity of #_p GraphHom(H) for square-free graphs H.

We study the problem of computing the parity of the number of homomorphisms from an input graph G to a fixed graph H . Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph H and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class ⊕P of parity problems. We verify their conjecture for all graphs H that exclude the complete graph on 4 vertices as a minor. Further, we rule out the existence of a subexponential-time algorithm for the ⊕P -complete cases, assuming the randomised Exponential Time Hypothesis. Our proofs introduce a novel method of deriving hardness from globally defined substructures of the fixed graph H . Using this, we subsume all prior progress towards resolving the conjecture (Faben and Jerrum [ToC'15]; Göbel, Goldberg and Richerby [ToCT'14,'16]). As special cases, our machinery also yields a proof of the conjecture for graphs with maximum degree at most 3 , as well as a full classification for the problem of counting list homomorphisms, modulo 2 .

Here we prove that counting maximum matchings in planar, bipartite graphs is #P-complete. This is somewhat surprising in the light that the number of perfect matchings in planar graphs can be computed in polynomial time. We also prove that counting non-necessarily perfect matchings in planar graphs is already #P-complete if the problem is restricted to bipartite graphs. So far hardness was proved only for general, non-necessarily bipartite graphs.

Given a graph property Φ , the problem #IndSub(Φ) asks, on input a graph G and a positive integer k , to compute the number of induced subgraphs of size k in G that satisfy Φ . The search for explicit criteria on Φ ensuring that #IndSub(Φ) is hard was initiated by Jerrum and Meeks [J. Comput. Syst. Sci. 15] and is part of the major line of research on counting small patterns in graphs. However, apart from an implicit result due to Curticapean, Dell and Marx [STOC 17] proving that a full classification into "easy" and "hard" properties is possible and some partial results on edge-monotone properties due to Meeks [Discret. Appl. Math. 16] and Dörfler et al. [MFCS 19], not much is known. In this work, we fully answer and explicitly classify the case of monotone, that is subgraph-closed, properties: We show that for any non-trivial monotone property Φ , the problem #IndSub(Φ) cannot be solved in time f(k)⋅|V(G) | o(k/ log 1/2 (k)) for any function f , unless the Exponential Time Hypothesis fails. By this, we establish that any significant improvement over the brute-force approach is unlikely; in the language of parameterized complexity, we also obtain a #W[1] -completeness result.

We consider the problem of counting the number of copies of a fixed graph H within an input graph G . This is one of the most well-studied algorithmic graph problems, with many theoretical and practical applications. We focus on solving this problem when the input G has bounded degeneracy. This is a rich family of graphs, containing all graphs without a fixed minor (e.g. planar graphs), as well as graphs generated by various random processes (e.g. preferential attachment graphs). We say that H is easy if there is a linear-time algorithm for counting the number of copies of H in an input G of bounded degeneracy. A seminal result of Chiba and Nishizeki from '85 states that every H on at most 4 vertices is easy. Bera, Pashanasangi, and Seshadhri recently extended this to all H on 5 vertices, and further proved that for every k>5 there is a k -vertex H which is not easy. They left open the natural problem of characterizing all easy graphs H . Bressan has recently introduced a framework for counting subgraphs in degenerate graphs, from which one can extract a sufficient condition for a graph H to be easy. Here we show that this sufficient condition is also necessary, thus fully answering the Bera--Pashanasangi--Seshadhri problem. We further resolve two closely related problems; namely characterizing the graphs that are easy with respect to counting induced copies, and with respect to counting homomorphisms. Our proofs rely on several novel approaches for proving hardness results in the context of subgraph-counting.

Counting homomorphisms from a graph H into another graph G is a fundamental problem of (parameterized) counting complexity theory. In this work, we study the case where \emph{both} graphs H and G stem from given classes of graphs: H∈H and G∈G . By this, we combine the structurally restricted version of this problem, with the language-restricted version. Our main result is a construction based on Kneser graphs that associates every problem P in #W[1] with two classes of graphs H and G such that the problem P is \emph{equivalent} to the problem #HOM(H→G) of counting homomorphisms from a graph in H to a graph in G . In view of Ladner's seminal work on the existence of NP -intermediate problems [J.ACM'75] and its adaptations to the parameterized setting, a classification of the class #W[1] in fixed-parameter tractable and #W[1] -complete cases is unlikely. Hence, obtaining a complete classification for the problem #HOM(H→G) seems unlikely. Further, our proofs easily adapt to W[1] . In search of complexity dichotomies, we hence turn to special graph classes. Those classes include line graphs, claw-free graphs, perfect graphs, and combinations thereof, and F -colorable graphs for fixed graphs F : If the class G is one of those classes and the class H is closed under taking minors, then we establish explicit criteria for the class H that partition the family of problems #HOM(H→G) into polynomial-time solvable and #W[1] -hard cases. In particular, we can drop the condition of H being minor-closed for F -colorable graphs.

We review and critique Boyu Sima's paper, "A solution of the P versus NP problem based on specific property of clique function," (arXiv:1911.00722) which claims to prove that P≠NP by way of removing the gap between the nonmonotone circuit complexity and the monotone circuit complexity of the clique function. We first describe Sima's argument, and then we describe where and why it fails. Finally, we present a simple example that clearly demonstrates the failure.

RNA cotranscriptional folding is the phenomenon in which an RNA transcript folds upon itself while being synthesized out of a gene. The oritatami system (OS) is a computation model of this phenomenon, which lets its sequence (transcript) of beads (abstract molecules) fold cotranscriptionally by the interactions between beads according to the binding ruleset. The OS is an useful computational model for predicting and simulating an RNA folding as well as constructing a biological structure. A fractal is an infinite pattern that is self-similar across different scales, and is an important structure in nature. Therefore, the fractal construction using self-assembly is one of the most important problems. We focus on the problem of generating an infinite fractal instead of a partial finite fractal, which is much more challenging. We use a cyclic OS, which has an infinite periodic transcript, to generate an infinite structure. We prove a negative result that it is impossible to make a Koch curve or a Minkowski curve, both of which are fractals, using a cyclic OS. We then establish sufficient conditions of infinite aperiodic curves that a cyclic OS cannot fold.