arXiv subjects

Leslie Ann Goldberg

Publications and source records attributed to Leslie Ann Goldberg.

At least 37 records · Page 2Linked to original sources

arXiv2023

Sampling from the random cluster model on random regular graphs at all temperatures via Glauber dynamics

We consider the performance of Glauber dynamics for the random cluster model with real parameter $q>1$ and temperature $β>0$. Recent work by Helmuth, Jenssen and Perkins detailed the ordered/disordered transition of the model on random $Δ$-regular graphs for all sufficiently large $q$ and obtained an efficient sampling algorithm for all temperatures $β$ using cluster expansion methods. Despite this major progress, the performance of natural Markov chains, including Glauber dynamics, is not yet well understood on the random regular graph, partly because of the non-local nature of the model (especially at low temperatures) and partly because of severe bottleneck phenomena that emerge in a window around the ordered/disordered transition. Nevertheless, it is widely conjectured that the bottleneck phenomena that impede mixing from worst-case starting configurations can be avoided by initialising the chain more judiciously. Our main result establishes this conjecture for all sufficiently large $q$ (with respect to $Δ$). Specifically, we consider the mixing time of Glauber dynamics initialised from the two extreme configurations, the all-in and all-out, and obtain a pair of fast mixing bounds which cover all temperatures $β$, including in particular the bottleneck window. Our result is inspired by the recent approach of Gheissari and Sinclair for the Ising model who obtained a similar-flavoured mixing-time bound on the random regular graph for sufficiently low temperatures. To cover all temperatures in the RC model, we refine appropriately the structural results of Helmuth, Jenssen and Perkins about the ordered/disordered transition and show spatial mixing properties "within the phase", which are then related to the evolution of the chain.

math.PR↗

arXiv2023

Graph Ranking and the Cost of Sybil Defense

Ranking functions such as PageRank assign numeric values (ranks) to nodes of graphs, most notably the web graph. Node rankings are an integral part of Internet search algorithms, since they can be used to order the results of queries. However, these ranking functions are famously subject to attacks by spammers, who modify the web graph in order to give their own pages more rank. We characterize the interplay between rankers and spammers as a game. We define the two critical features of this game, spam resistance and distortion, based on how spammers spam and how rankers protect against spam. We observe that all the ranking functions that are well-studied in the literature, including the original formulation of PageRank, have poor spam resistance, poor distortion, or both. Finally, we study Min-PPR, the form of PageRank used at Google itself, but which has received no (theoretical or empirical) treatment in the literature. We prove that Min-PPR has low distortion and high spam resistance. A secondary benefit is that Min-PPR comes with an explicit cost function on nodes that shows how important they are to the spammer; thus a ranker can focus their spam-detection capacity on these vulnerable nodes. Both Min-PPR and its associated cost function are straightforward to compute.

cs.DS↗

arXiv2023

Parameterised Approximation of the Fixation Probability of the Dominant Mutation in the Multi-Type Moran Process

The multi-type Moran process is an evolutionary process on a connected graph $G$ in which each vertex has one of $k$ types and, in each step, a vertex $v$ is chosen to reproduce its type to one of its neighbours. The probability of a vertex $v$ being chosen for reproduction is proportional to the fitness of the type of $v$. So far, the literature was almost solely concerned with the $2$-type Moran process in which each vertex is either healthy (type $0$) or a mutant (type $1$), and the main problem of interest has been the (approximate) computation of the so-called fixation probability, i.e., the probability that eventually all vertices are mutants. In this work we initiate the study of approximating fixation probabilities in the multi-type Moran process on general graphs. Our main result is an FPTRAS (fixed-parameter tractable randomised approximation scheme) for computing the fixation probability of the dominant mutation; the parameter is the number of types and their fitnesses. In the course of our studies we also provide novel upper bounds on the expected absorption time, i.e., the time that it takes the multi-type Moran process to reach a state in which each vertex has the same type.

cs.DS↗

arXiv2023

Metastability of the Potts ferromagnet on random regular graphs

We study the performance of Markov chains for the $q$-state ferromagnetic Potts model on random regular graphs. It is conjectured that their performance is dictated by metastability phenomena, i.e., the presence of "phases" (clusters) in the sample space where Markov chains with local update rules, such as the Glauber dynamics, are bound to take exponential time to escape. The phases that are believed to drive these metastability phenomena in the case of the Potts model emerge as local, rather than global, maxima of the so-called Bethe functional, and previous approaches of analysing these phases based on optimisation arguments fall short of the task. Our first contribution is to detail the emergence of the metastable phases for the $q$-state Potts model on the $d$-regular random graph for all integers $q,d\geq 3$, and establish that for an interval of temperatures, which is delineated by the uniqueness and a broadcasting threshold on the $d$-regular tree, the two phases coexist. The proofs are based on a conceptual connection between spatial properties and the structure of the Potts distribution on the random regular graph, rather than complicated moment calculations. Based on this new structural understanding of the model, we obtain various algorithmic consequences. We first complement recent fast mixing results for Glauber dynamics by Blanca and Gheissari below the uniqueness threshold, showing an exponential lower bound on the mixing time above the uniqueness threshold. Then, we obtain tight results even for the non-local Swendsen-Wang chain, where we establish slow mixing/metastability for the whole interval of temperatures where the chain is conjectured to mix slowly on the random regular graph. The key is to bound the conductance of the chains using a random graph "planting" argument combined with delicate bounds on random-graph percolation.

math.PR↗

arXiv2022

Implementations and the independent set polynomial below the Shearer threshold

The independent set polynomial is important in many areas. For every integer $Δ\geq 2$, the Shearer threshold is the value $λ^*(Δ)=(Δ-1)^{Δ-1}/Δ^Δ$ . It is known that for $λ< - λ^*(Δ)$, there are graphs~$G$ with maximum degree~$Δ$ whose independent set polynomial, evaluated at~$λ$, is at most~$0$. Also, there are no such graphs for any $λ> -λ^*(Δ)$. This paper is motivated by the computational problem of approximating the independent set polynomial when $λ< - λ^*(Δ)$. The key issue in complexity bounds for this problem is "implementation". Informally, an implementation of a real number $λ'$ is a graph whose hard-core partition function, evaluated at~$λ$, simulates a vertex-weight of~$λ'$ in the sense that $λ'$ is the ratio between the contribution to the partition function from independent sets containing a certain vertex and the contribution from independent sets that do not contain that vertex. Implementations are the cornerstone of intractability results for the problem of approximately evaluating the independent set polynomial. Our main result is that, for any $λ< - λ^*(Δ)$, it is possible to implement a set of values that is dense over the reals. The result is tight in the sense that it is not possible to implement a set of values that is dense over the reals for any $λ> λ^*(Δ)$. Our result has already been used in a paper with \bezakova{} (STOC 2018) to show that it is \#P-hard to approximate the evaluation of the independent set polynomial on graphs of degree at most~$Δ$ at any value $λ<-λ^*(Δ)$. In the appendix, we give an additional incomparable inapproximability result (strengthening the inapproximability bound to an exponential factor, but weakening the hardness to NP-hardness).

cs.CC↗

arXiv2022

The complexity of approximating the complex-valued Ising model on bounded degree graphs

We study the complexity of approximating the partition function $Z_{\mathrm{Ising}}(G; β)$ of the Ising model in terms of the relation between the edge interaction $β$ and a parameter $Δ$ which is an upper bound on the maximum degree of the input graph $G$. Following recent trends in both statistical physics and algorithmic research, we allow the edge interaction $β$ to be any complex number. Many recent partition function results focus on complex parameters, both because of physical relevance and because of the key role of the complex case in delineating the tractability/intractability phase transition of the approximation problem. In this work we establish both new tractability results and new intractability results. Our tractability results show that $Z_{\mathrm{Ising}}(-; β)$ has an FPTAS when $\lvert β- 1 \rvert / \lvert β+ 1 \rvert < \tan(π/ (4 Δ- 4))$. The core of the proof is showing that there are no inputs~$G$ that make the partition function $0$ when $β$ is in this range. Our result significantly extends the known zero-free region of the Ising model (and hence the known approximation results). Our intractability results show that it is $\mathrm{\#P}$-hard to multiplicatively approximate the norm and to additively approximate the argument of $Z_{\mathrm{Ising}}(-; β)$ when $β\in \mathbb{C}$ is an algebraic number such that $β\not \in \mathbb{R} \cup \{i,-i\}$ and $\lvert β- 1\rvert / \lvert β+ 1 \rvert > 1 / \sqrt{Δ- 1}$. These are the first results to show intractability of approximating $Z_{\mathrm{Ising}}(-, β)$ on bounded degree graphs with complex $β$. Moreover, we demonstrate situations in which zeros of the partition function imply hardness of approximation in the Ising model.

cs.CC↗

arXiv2022

Fast mixing via polymers for random graphs with unbounded degree

The polymer model framework is a classical tool from statistical mechanics that has recently been used to obtain approximation algorithms for spin systems on classes of bounded-degree graphs; examples include the ferromagnetic Potts model on expanders and on the grid. One of the key ingredients in the analysis of polymer models is controlling the growth rate of the number of polymers, which has been typically achieved so far by invoking the bounded-degree assumption. Nevertheless, this assumption is often restrictive and obstructs the applicability of the method to more general graphs. For example, sparse random graphs typically have bounded average degree and good expansion properties, but they include vertices with unbounded degree, and therefore are excluded from the current polymer-model framework. We develop a less restrictive framework for polymer models that relaxes the standard bounded-degree assumption, by reworking the relevant polymer models from the edge perspective. The edge perspective allows us to bound the growth rate of the number of polymers in terms of the total degree of polymers, which in turn can be related more easily to the expansion properties of the underlying graph. To apply our methods, we consider random graphs with unbounded degrees from a fixed degree sequence (with minimum degree at least 3) and obtain approximation algorithms for the ferromagnetic Potts model, which is a standard benchmark for polymer models. Our techniques also extend to more general spin systems.

cs.DS↗

arXiv2021

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, Lovász and Welsh in the context of quantum computations.

cs.CC↗

arXiv2021

Faster Exponential-time Algorithms for Approximately Counting Independent Sets

Counting the independent sets of a graph is a classical #P-complete problem, even in the bipartite case. We give an exponential-time approximation scheme for this problem which is faster than the best known algorithm for the exact problem. The running time of our algorithm on general graphs with error tolerance $\varepsilon$ is at most $O(2^{0.2680n})$ times a polynomial in $1/\varepsilon$. On bipartite graphs, the exponential term in the running time is improved to $O(2^{0.2372n})$. Our methods combine techniques from exact exponential algorithms with techniques from approximate counting. Along the way we generalise (to the multivariate case) the FPTAS of Sinclair, Srivastava, Štefankovič and Yin for approximating the hard-core partition function on graphs with bounded connective constant. Also, we obtain an FPTAS for counting independent sets on graphs with no vertices with degree at least 6 whose neighbours' degrees sum to 27 or more. By a result of Sly, there is no FPTAS that applies to all graphs with maximum degree 6 unless $\mbox{P}=\mbox{NP}$.

cs.DS↗

arXiv2021

Counting Homomorphisms to $K_4$-minor-free Graphs, modulo 2

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 $\oplus\mathrm{P}$ 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 $\oplus\mathrm{P}$-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 $2$.

cs.CC↗

arXiv2021

Counting solutions to random CNF formulas

We give the first efficient algorithm to approximately count the number of solutions in the random $k$-SAT model when the density of the formula scales exponentially with $k$. The best previous counting algorithm for the permissive version of the model was due to Montanari and Shah and was based on the correlation decay method, which works up to densities $(1+o_k(1))\frac{2\log k}{k}$, the Gibbs uniqueness threshold for the model. Instead, our algorithm harnesses a recent technique by Moitra to work for random formulas. The main challenge in our setting is to account for the presence of high-degree variables whose marginal distributions are hard to control and which cause significant correlations within the formula.

cs.DS↗

arXiv2021

Fast algorithms for general spin systems on bipartite expanders

A spin system is a framework in which the vertices of a graph are assigned spins from a finite set. The interactions between neighbouring spins give rise to weights, so a spin assignment can also be viewed as a weighted graph homomorphism. The problem of approximating the partition function (the aggregate weight of spin assignments) or of sampling from the resulting probability distribution is typically intractable for general graphs. In this work, we consider arbitrary spin systems on bipartite expander $Δ$-regular graphs, including the canonical class of bipartite random $Δ$-regular graphs. We develop fast approximate sampling and counting algorithms for general spin systems whenever the degree and the spectral gap of the graph are sufficiently large. Our approach generalises the techniques of Jenseen et al. and Chen et al. by showing that typical configurations on bipartite expanders correspond to "bicliques" of the spin system; then, using suitable polymer models, we show how to sample such configurations and approximate the partition function in $\tilde{O}(n^2)$ time, where $n$ is the size of the graph.

cs.DS↗

arXiv2021

Fast algorithms at low temperatures via Markov chains

We define a discrete-time Markov chain for abstract polymer models and show that under sufficient decay of the polymer weights, this chain mixes rapidly. We apply this Markov chain to polymer models derived from the hard-core and ferromagnetic Potts models on bounded-degree (bipartite) expander graphs. In this setting, Jenssen, Keevash and Perkins (2019) recently gave an FPTAS and an efficient sampling algorithm at sufficiently high fugacity and low temperature respectively. Their method is based on using the cluster expansion to obtain a complex zero-free region for the partition function of a polymer model, and then approximating this partition function using the polynomial interpolation method of Barvinok. Our approach via the polymer model Markov chain circumvents the zero-free analysis and the generalization to complex parameters, and leads to a sampling algorithm with a fast running time of $O(n \log n)$ for the Potts model and $O(n^2 \log n)$ for the hard-core model, in contrast to typical running times of $n^{O(\log Δ)}$ for algorithms based on Barvinok's polynomial interpolation method on graphs of maximum degree $Δ$. We finally combine our results for the hard-core and ferromagnetic Potts models with standard Markov chain comparison tools to obtain polynomial mixing time for the usual spin Glauber dynamics restricted to even and odd or `red' dominant portions of the respective state spaces.

cs.DS↗

arXiv2021

The Complexity of Approximately Counting Retractions to Square-Free Graphs

A retraction is a homomorphism from a graph $G$ to an induced subgraph $H$ of $G$ that is the identity on $H$. In a long line of research, retractions have been studied under various algorithmic settings. Recently, the problem of approximately counting retractions was considered. We give a complete trichotomy for the complexity of approximately counting retractions to all square-free graphs (graphs that do not contain a cycle of length $4$). It turns out there is a rich and interesting class of graphs for which this problem is complete in the class $\#\mathrm{BIS}$. As retractions generalise homomorphisms, our easiness results extend to the important problem of approximately counting homomorphisms. By giving new $\#\mathrm{BIS}$-easiness results we now settle the complexity of approximately counting homomorphisms for a whole class of non-trivial graphs which were previously unresolved.

cs.CC↗

arXiv2021

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 $γ$, where $γ$ takes arbitrary values in the complex plane. When $γ$ is a positive real, Jerrum and Sinclair showed that the problem admits an FPRAS on general graphs. For general complex values of $γ$, Patel and Regts, building on methods developed by Barvinok, showed that the problem admits an FPTAS on graphs of maximum degree $Δ$ as long as $γ$ is not a negative real number less than or equal to $-1/(4(Δ-1))$. Our first main result completes the picture for the approximability of the matching polynomial on bounded degree graphs. We show that for all $Δ\geq 3$ and all real $γ$ less than $-1/(4(Δ-1))$, the problem of approximating the value of the matching polynomial on graphs of maximum degree $Δ$ with edge parameter $γ$ 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 $γ$ 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 $γ$ values on the negative real axis. Nevertheless, we show that the result does extend for any complex value $γ$ that does not lie on the negative real axis. Our analysis accounts for complex values of $γ$ using geodesic distances in the complex plane in the metric defined by an appropriate density function.

cs.DM↗

arXiv2020

Approximating partition functions of bounded-degree Boolean counting Constraint Satisfaction Problems

We study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language $Γ$ and a degree bound $Δ$, we study the complexity of #CSP$_Δ(Γ)$, which is the problem of counting satisfying assignments to CSP instances with constraints from $Γ$ and whose variables can appear at most $Δ$ times. Our main result shows that: (i) if every function in $Γ$ is affine, then #CSP$_Δ(Γ)$ is in FP for all $Δ$, (ii) otherwise, if every function in $Γ$ is in a class called IM$_2$, then for all sufficiently large $Δ$, #CSP$_Δ(Γ)$ is equivalent under approximation-preserving (AP) reductions to the counting problem #BIS (the problem of counting independent sets in bipartite graphs) (iii) otherwise, for all sufficiently large $Δ$, it is NP-hard to approximate the number of satisfying assignments of an instance of #CSP$_Δ(Γ)$, even within an exponential factor. Our result extends previous results, which apply only in the so-called "conservative" case.

cs.DS↗

arXiv2020

The Complexity of Approximately Counting Retractions

Let $G$ be a graph that contains an induced subgraph $H$. A retraction from $G$ to $H$ is a homomorphism from $G$ to $H$ that is the identity function on $H$. Retractions are very well-studied: Given $H$, the complexity of deciding whether there is a retraction from an input graph $G$ to $H$ is completely classified, in the sense that it is known for which $H$ this problem is tractable (assuming $\mathrm{P}\neq \mathrm{NP}$). Similarly, the complexity of (exactly) counting retractions from $G$ to $H$ is classified (assuming $\mathrm{FP}\neq \#\mathrm{P}$). However, almost nothing is known about approximately counting retractions. Our first contribution is to give a complete trichotomy for approximately counting retractions to graphs of girth at least $5$. Our second contribution is to locate the retraction counting problem for each $H$ in the complexity landscape of related approximate counting problems. Interestingly, our results are in contrast to the situation in the exact counting context. We show that the problem of approximately counting retractions is separated both from the problem of approximately counting homomorphisms and from the problem of approximately counting list homomorphisms --- whereas for exact counting all three of these problems are interreducible. We also show that the number of retractions is at least as hard to approximate as both the number of surjective homomorphisms and the number of compactions. In contrast, exactly counting compactions is the hardest of all of these exact counting problems.

cs.CC↗

arXiv2020

Random Walks on Small World Networks

We study the mixing time of random walks on small-world networks modelled as follows: starting with the 2-dimensional periodic grid, each pair of vertices $\{u,v\}$ with distance $d>1$ is added as a "long-range" edge with probability proportional to $d^{-r}$, where $r\geq 0$ is a parameter of the model. Kleinberg studied a close variant of this network model and proved that the (decentralised) routing time is $O((\log n)^2)$ when $r=2$ and $n^{Ω(1)}$ when $r\neq 2$. Here, we prove that the random walk also undergoes a phase transition at $r=2$, but in this case the phase transition is of a different form. We establish that the mixing time is $Θ(\log n)$ for $r<2$, $O((\log n)^4)$ for $r=2$ and $n^{Ω(1)}$ for $r>2$.

cs.DM↗