Andreas Galanis
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Andreas Galanis has a total of 26 co-authors. Here are the top co-authors by number of shared publications:
- Daniel Štefankovič
Computer Science Department University of Rochester 14627-0226 Rochester NY
9 shared publications - Leslie Ann Goldberg
Department of Computer Science, University of Oxford, Oxford, UK
7 shared publications - Eric Vigoda
School of Computer Science Georgia Institute of Technology 30332 Atlanta GA
4 shared publications - Amin Coja-Oghlan
TU Dortmund University
2 shared publications Eric Vigoda
2 shared publications- Eric Vigoda
Department of Computer Science University of California Santa Barbara
2 shared publications
Latest Publications
- Open Access
- 1 January 2026
- Published by Institute of Mathematical Statistics in Electronic Journal of Probability
- Vol. 31 (none) , 1-40
Low-temperature sampling on sparse random graphs
We consider sampling in the so-called low-temperature regime, which is typically characterised by non-local behaviour and strong global correlations. Canonical examples include sampling independent sets on bipartite graphs and sampling from the ferromagnetic q-state Potts model. Low-temperature sampling is computationally intractable for general graphs, but recent advances based on the polymer method have made significant progress for graph families that exhibit certain expansion properties that reinforce the correlations, including for example expanders, lattices and dense graphs. One of the most natural graph classes that has so far escaped this algorithmic framework is the class of sparse Erdős-Rényi random graphs whose expansion only manifests for sufficiently large subsets of vertices; small sets of vertices on the other hand have vanishing expansion which makes them behave independently from the bulk of the graph and therefore weakens the correlations. At a more technical level, the expansion of small sets is crucial for establishing the Kotecky-Priess condition which underpins the applicability of the framework. Our main contribution is to develop the polymer method in the low-temperature regime for sparse random graphs. As our running example, we use the Potts and random-cluster models on G(n,d∕n) for d=Θ(1), where we show a polynomial-time sampling algorithm for all sufficiently large q and d, at all temperatures. Our approach applies more generally for models that are monotone. Key to our result is a simple polymer definition that blends easily with the connectivity properties of the graph and allows us to show that polymers have size at most O(logn).
- 1 January 2026
- Published by Institute of Mathematical Statistics in Electronic Journal of Probability
- Vol. 31 (none) , 1-40
Open AccessLow-temperature sampling on sparse random graphs
We consider sampling in the so-called low-temperature regime, which is typically characterised by non-local behaviour and strong global correlations. Canonical examples include sampling independent sets on bipartite graphs and sampling from the ferromagnetic q-state Potts model. Low-temperature sampling is computationally intractable for general graphs, but recent advances based on the polymer method have made significant progress for graph families that exhibit certain expansion properties that reinforce the correlations, including for example expanders, lattices and dense graphs. One of the most natural graph classes that has so far escaped this algorithmic framework is the class of sparse Erdős-Rényi random graphs whose expansion only manifests for sufficiently large subsets of vertices; small sets of vertices on the other hand have vanishing expansion which makes them behave independently from the bulk of the graph and therefore weakens the correlations. At a more technical level, the expansion of small sets is crucial for establishing the Kotecky-Priess condition which underpins the applicability of the framework. Our main contribution is to develop the polymer method in the low-temperature regime for sparse random graphs. As our running example, we use the Potts and random-cluster models on G(n,d∕n) for d=Θ(1), where we show a polynomial-time sampling algorithm for all sufficiently large q and d, at all temperatures. Our approach applies more generally for models that are monotone. Key to our result is a simple polymer definition that blends easily with the connectivity properties of the graph and allows us to show that polymers have size at most O(logn).
- 9 December 2024
- Published by Cambridge University Press (CUP) in Combinatorics, Probability and Computing
- Vol. 34 (3) , 359-391
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 $q\gt 1$ and temperature $\beta \gt 0$ . Recent work by Helmuth, Jenssen, and Perkins detailed the ordered/disordered transition of the model on random $\Delta$ -regular graphs for all sufficiently large $q$ and obtained an efficient sampling algorithm for all temperatures $\beta$ 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 $\Delta$ ). 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 $\beta$ , 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.- 9 December 2024
- Published by Cambridge University Press (CUP) in Combinatorics, Probability and Computing
- Vol. 34 (3) , 359-391
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 $q\gt 1$ and temperature $\beta \gt 0$ . Recent work by Helmuth, Jenssen, and Perkins detailed the ordered/disordered transition of the model on random $\Delta$ -regular graphs for all sufficiently large $q$ and obtained an efficient sampling algorithm for all temperatures $\beta$ 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 $\Delta$ ). 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 $\beta$ , 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.- Open Access
On Sampling from Ising Models with Spectral Constraints
We consider the problem of sampling from the Ising model when the underlying interaction matrix has eigenvalues lying within an interval of length $γ$. Recent work in this setting has shown various algorithmic results that apply roughly when $γ< 1$, notably with nearly-linear running times based on the classical Glauber dynamics. However, the optimality of the range of $γ$ was not clear since previous inapproximability results developed for the antiferromagnetic case (where the matrix has entries $\leq 0$) apply only for $γ>2$. To this end, Kunisky (SODA'24) recently provided evidence that the problem becomes hard already when $γ>1$ based on the low-degree hardness for an inference problem on random matrices. Based on this, he conjectured that sampling from the Ising model in the same range of $γ$ is NP-hard. Here we confirm this conjecture, complementing in particular the known algorithmic results by showing NP-hardness results for approximately counting and sampling when $γ>1$, with strong inapproximability guarantees; we also obtain a more refined hardness result for matrices where only a constant number of entries per row are allowed to be non-zero. The main observation in our reductions is that, for $γ>1$, Glauber dynamics mixes slowly when the interactions are all positive (ferromagnetic) for the complete and random regular graphs, due to a bimodality in the underlying distribution. While ferromagnetic interactions typically preclude NP-hardness results, here we work around this by introducing in an appropriate way mild antiferromagnetism, keeping the spectrum roughly within the same range. This allows us to exploit the bimodality of the aforementioned graphs and show the target NP-hardness by adapting suitably previous inapproximability techniques developed for antiferromagnetic systems.Open AccessOn Sampling from Ising Models with Spectral Constraints
We consider the problem of sampling from the Ising model when the underlying interaction matrix has eigenvalues lying within an interval of length $γ$. Recent work in this setting has shown various algorithmic results that apply roughly when $γ< 1$, notably with nearly-linear running times based on the classical Glauber dynamics. However, the optimality of the range of $γ$ was not clear since previous inapproximability results developed for the antiferromagnetic case (where the matrix has entries $\leq 0$) apply only for $γ>2$. To this end, Kunisky (SODA'24) recently provided evidence that the problem becomes hard already when $γ>1$ based on the low-degree hardness for an inference problem on random matrices. Based on this, he conjectured that sampling from the Ising model in the same range of $γ$ is NP-hard. Here we confirm this conjecture, complementing in particular the known algorithmic results by showing NP-hardness results for approximately counting and sampling when $γ>1$, with strong inapproximability guarantees; we also obtain a more refined hardness result for matrices where only a constant number of entries per row are allowed to be non-zero. The main observation in our reductions is that, for $γ>1$, Glauber dynamics mixes slowly when the interactions are all positive (ferromagnetic) for the complete and random regular graphs, due to a bimodality in the underlying distribution. While ferromagnetic interactions typically preclude NP-hardness results, here we work around this by introducing in an appropriate way mild antiferromagnetism, keeping the spectrum roughly within the same range. This allows us to exploit the bimodality of the aforementioned graphs and show the target NP-hardness by adapting suitably previous inapproximability techniques developed for antiferromagnetic systems. - Open Access
- 22 January 2024
- Published by Association for Computing Machinery (ACM) in ACM Transactions on Algorithms
- Vol. 20 (1) , 1-26
Fast Sampling via Spectral Independence Beyond Bounded-degree Graphs
Spectral independence is a recently developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal O(n log n) sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using L p -norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, and Yin, FOCS’13). The non-linearity of L p -norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the L p -analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph G (n, d/n) , where the previously known algorithms run in time n O (log d ) or applied only to large d . We refine these algorithmic bounds significantly, and develop fast nearly linear algorithms based on Glauber dynamics that apply to all constant d , throughout the uniqueness regime.
- 22 January 2024
- Published by Association for Computing Machinery (ACM) in ACM Transactions on Algorithms
- Vol. 20 (1) , 1-26
Open AccessFast Sampling via Spectral Independence Beyond Bounded-degree Graphs
Spectral independence is a recently developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal O(n log n) sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using L p -norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, and Yin, FOCS’13). The non-linearity of L p -norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the L p -analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph G (n, d/n) , where the previously known algorithms run in time n O (log d ) or applied only to large d . We refine these algorithmic bounds significantly, and develop fast nearly linear algorithms based on Glauber dynamics that apply to all constant d , throughout the uniqueness regime.
- Open Access
Learning Hard-Constrained Models with One Sample
We consider the problem of estimating the parameters of a Markov Random Field with hard-constraints using a single sample. As our main running examples, we use the $k$-SAT and the proper coloring models, as well as general $H$-coloring models; for all of these we obtain both positive and negative results. In contrast to the soft-constrained case, we show in particular that single-sample estimation is not always possible, and that the existence of an estimator is related to the existence of non-satisfiable instances. Our algorithms are based on the pseudo-likelihood estimator. We show variance bounds for this estimator using coupling techniques inspired, in the case of $k$-SAT, by Moitra's sampling algorithm (JACM, 2019); our positive results for colorings build on this new coupling approach. For $q$-colorings on graphs with maximum degree $d$, we give a linear-time estimator when $q>d+1$, whereas the problem is non-identifiable when $q\leq d+1$. For general $H$-colorings, we show that standard conditions that guarantee sampling, such as Dobrushin's condition, are insufficient for one-sample learning; on the positive side, we provide a general condition that is sufficient to guarantee linear-time learning and obtain applications for proper colorings and permissive models. For the $k$-SAT model on formulas with maximum degree $d$, we provide a linear-time estimator when $k\gtrsim 6.45\log d$, whereas the problem becomes non-identifiable when $k\lesssim \log d$.Open AccessLearning Hard-Constrained Models with One Sample
We consider the problem of estimating the parameters of a Markov Random Field with hard-constraints using a single sample. As our main running examples, we use the $k$-SAT and the proper coloring models, as well as general $H$-coloring models; for all of these we obtain both positive and negative results. In contrast to the soft-constrained case, we show in particular that single-sample estimation is not always possible, and that the existence of an estimator is related to the existence of non-satisfiable instances. Our algorithms are based on the pseudo-likelihood estimator. We show variance bounds for this estimator using coupling techniques inspired, in the case of $k$-SAT, by Moitra's sampling algorithm (JACM, 2019); our positive results for colorings build on this new coupling approach. For $q$-colorings on graphs with maximum degree $d$, we give a linear-time estimator when $q>d+1$, whereas the problem is non-identifiable when $q\leq d+1$. For general $H$-colorings, we show that standard conditions that guarantee sampling, such as Dobrushin's condition, are insufficient for one-sample learning; on the positive side, we provide a general condition that is sufficient to guarantee linear-time learning and obtain applications for proper colorings and permissive models. For the $k$-SAT model on formulas with maximum degree $d$, we provide a linear-time estimator when $k\gtrsim 6.45\log d$, whereas the problem becomes non-identifiable when $k\lesssim \log d$. - Open Access
- 9 February 2023
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 401 (1) , 185-225
Metastability of the Potts Ferromagnet on Random Regular Graphs
We study the performance of Markov chains for theq-state ferromagnetic Potts model on random regular graphs. While the cases of the grid and the complete graph are by now well-understood, the case of random regular graphs has resisted a detailed analysis and, in fact, even analysing the properties of the Potts distribution has remained elusive. It is conjectured that the performance of Markov chains 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, and therefore cause slow mixing. 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 two relevant phases for theq-state Potts model on thed-regular random graph for all integers $$q,d\ge 3$$ , and establish that for an interval of temperatures, delineated by the uniqueness and a broadcasting threshold on thed-regular tree, the two phases coexist (as possible metastable states). 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. This significantly refines earlier results by Helmuth, Jenssen, and Perkins who had established phase coexistence for a small interval around the so-called ordered-disordered threshold (via different arguments) that applied for largeqand $$d\ge 5$$ . Based on our new structural understanding of the model, our second contribution is to obtain metastability results for two classical Markov chains for the Potts model. We first complement recent fast mixing results for Glauber dynamics by Blanca and Gheissari below the uniqueness threshold, by showing an exponential lower bound on the mixing time above the uniqueness threshold. Then, we obtain tight results even for the non-local and more elaborate 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.- 9 February 2023
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 401 (1) , 185-225
Open AccessMetastability of the Potts Ferromagnet on Random Regular Graphs
We study the performance of Markov chains for theq-state ferromagnetic Potts model on random regular graphs. While the cases of the grid and the complete graph are by now well-understood, the case of random regular graphs has resisted a detailed analysis and, in fact, even analysing the properties of the Potts distribution has remained elusive. It is conjectured that the performance of Markov chains 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, and therefore cause slow mixing. 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 two relevant phases for theq-state Potts model on thed-regular random graph for all integers $$q,d\ge 3$$ , and establish that for an interval of temperatures, delineated by the uniqueness and a broadcasting threshold on thed-regular tree, the two phases coexist (as possible metastable states). 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. This significantly refines earlier results by Helmuth, Jenssen, and Perkins who had established phase coexistence for a small interval around the so-called ordered-disordered threshold (via different arguments) that applied for largeqand $$d\ge 5$$ . Based on our new structural understanding of the model, our second contribution is to obtain metastability results for two classical Markov chains for the Potts model. We first complement recent fast mixing results for Glauber dynamics by Blanca and Gheissari below the uniqueness threshold, by showing an exponential lower bound on the mixing time above the uniqueness threshold. Then, we obtain tight results even for the non-local and more elaborate 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. - Open Access
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.Open AccessMetastability 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. - Open Access
- 31 December 2022
- Published by Association for Computing Machinery (ACM) in ACM Transactions on Computation Theory
- Vol. 14 (3-4) , 1-33
Inapproximability of Counting Hypergraph Colourings
Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to the Lovász Local Lemma. Nevertheless, the boundaries of these methods for CSPs with non-Boolean domain are not well-understood. Our goal in this article is to fill in this gap and obtain strong inapproximability results by studying the prototypical problem in this class of CSPs, hypergraph colourings. More precisely, we focus on the problem of approximately counting q -colourings on K -uniform hypergraphs with bounded degree Δ. An efficient algorithm exists if \({{\Delta \lesssim \frac{q^{K/3-1}}{4^KK^2}}}\) [Jain et al. 25 ; He et al. 23 ]. Somewhat surprisingly however, a hardness bound is not known even for the easier problem of finding colourings. For the counting problem, the situation is even less clear and there is no evidence of the right constant controlling the growth of the exponent in terms of K . To this end, we first establish that for general q computational hardness for finding a colouring on simple/linear hypergraphs occurs at Δ ≳ Kq K , almost matching the algorithm from the Lovász Local Lemma. Our second and main contribution is to obtain a far more refined bound for the counting problem that goes well beyond the hardness of finding a colouring and which we conjecture is asymptotically tight (up to constant factors). We show in particular that for all even q ≥ 4 it is NP -hard to approximate the number of colourings when Δ ≳ q K/2 . Our approach is based on considering an auxiliary weighted binary CSP model on graphs, which is obtained by “halving” the K -ary hypergraph constraints. This allows us to utilise reduction techniques available for the graph case, which hinge upon understanding the behaviour of random regular bipartite graphs that serve as gadgets in the reduction. The major challenge in our setting is to analyse the induced matrix norm of the interaction matrix of the new CSP which captures the most likely solutions of the system. In contrast to previous analyses in the literature, the auxiliary CSP demonstrates both symmetry and asymmetry, making the analysis of the optimisation problem severely more complicated and demanding the combination of delicate perturbation arguments and careful asymptotic estimates.- 31 December 2022
- Published by Association for Computing Machinery (ACM) in ACM Transactions on Computation Theory
- Vol. 14 (3-4) , 1-33
Open AccessInapproximability of Counting Hypergraph Colourings
Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to the Lovász Local Lemma. Nevertheless, the boundaries of these methods for CSPs with non-Boolean domain are not well-understood. Our goal in this article is to fill in this gap and obtain strong inapproximability results by studying the prototypical problem in this class of CSPs, hypergraph colourings. More precisely, we focus on the problem of approximately counting q -colourings on K -uniform hypergraphs with bounded degree Δ. An efficient algorithm exists if \({{\Delta \lesssim \frac{q^{K/3-1}}{4^KK^2}}}\) [Jain et al. 25 ; He et al. 23 ]. Somewhat surprisingly however, a hardness bound is not known even for the easier problem of finding colourings. For the counting problem, the situation is even less clear and there is no evidence of the right constant controlling the growth of the exponent in terms of K . To this end, we first establish that for general q computational hardness for finding a colouring on simple/linear hypergraphs occurs at Δ ≳ Kq K , almost matching the algorithm from the Lovász Local Lemma. Our second and main contribution is to obtain a far more refined bound for the counting problem that goes well beyond the hardness of finding a colouring and which we conjecture is asymptotically tight (up to constant factors). We show in particular that for all even q ≥ 4 it is NP -hard to approximate the number of colourings when Δ ≳ q K/2 . Our approach is based on considering an auxiliary weighted binary CSP model on graphs, which is obtained by “halving” the K -ary hypergraph constraints. This allows us to utilise reduction techniques available for the graph case, which hinge upon understanding the behaviour of random regular bipartite graphs that serve as gadgets in the reduction. The major challenge in our setting is to analyse the induced matrix norm of the interaction matrix of the new CSP which captures the most likely solutions of the system. In contrast to previous analyses in the literature, the auxiliary CSP demonstrates both symmetry and asymmetry, making the analysis of the optimisation problem severely more complicated and demanding the combination of delicate perturbation arguments and careful asymptotic estimates. - Open Access
Inapproximability of counting hypergraph colourings
Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to the Lovasz Local Lemma. Nevertheless, the boundaries of these methods for CSPs with non-Boolean domain are not well-understood. Our goal in this paper is to fill in this gap and obtain strong inapproximability results by studying the prototypical problem in this class of CSPs, hypergraph colourings. More precisely, we focus on the problem of approximately counting $q$-colourings on $K$-uniform hypergraphs with bounded degree $\Delta$. An efficient algorithm exists if $\Delta\lesssim \frac{q^{K/3-1}}{4^KK^2}$ (Jain, Pham, and Vuong, 2021; He, Sun, and Wu, 2021). Somewhat surprisingly however, a hardness bound is not known even for the easier problem of finding colourings. For the counting problem, the situation is even less clear and there is no evidence of the right constant controlling the growth of the exponent in terms of $K$. To this end, we first establish that for general $q$ computational hardness for finding a colouring on simple/linear hypergraphs occurs at $\Delta\gtrsim Kq^K$, almost matching the algorithm from the Lovasz Local Lemma. Our second and main contribution is to obtain a far more refined bound for the counting problem that goes well beyond the hardness of finding a colouring and which we conjecture that is asymptotically tight (up to constant factors). We show in particular that for all even $q\geq 4$ it is NP-hard to approximate the number of colourings when $\Delta\gtrsim q^{K/2}$.Open AccessInapproximability of counting hypergraph colourings
Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to the Lovasz Local Lemma. Nevertheless, the boundaries of these methods for CSPs with non-Boolean domain are not well-understood. Our goal in this paper is to fill in this gap and obtain strong inapproximability results by studying the prototypical problem in this class of CSPs, hypergraph colourings. More precisely, we focus on the problem of approximately counting $q$-colourings on $K$-uniform hypergraphs with bounded degree $\Delta$. An efficient algorithm exists if $\Delta\lesssim \frac{q^{K/3-1}}{4^KK^2}$ (Jain, Pham, and Vuong, 2021; He, Sun, and Wu, 2021). Somewhat surprisingly however, a hardness bound is not known even for the easier problem of finding colourings. For the counting problem, the situation is even less clear and there is no evidence of the right constant controlling the growth of the exponent in terms of $K$. To this end, we first establish that for general $q$ computational hardness for finding a colouring on simple/linear hypergraphs occurs at $\Delta\gtrsim Kq^K$, almost matching the algorithm from the Lovasz Local Lemma. Our second and main contribution is to obtain a far more refined bound for the counting problem that goes well beyond the hardness of finding a colouring and which we conjecture that is asymptotically tight (up to constant factors). We show in particular that for all even $q\geq 4$ it is NP-hard to approximate the number of colourings when $\Delta\gtrsim q^{K/2}$. - Open Access
Approximating observables is as hard as counting
We study the computational complexity of estimating local observables for Gibbs distributions. A simple combinatorial example is the average size of an independent set in a graph. In a recent work, we established NP-hardness of approximating the average size of an independent set utilizing hardness of the corresponding optimization problem and the related phase transition behavior. Here, we instead consider settings where the underlying optimization problem is easily solvable. Our main contribution is to classify the complexity of approximating a wide class of observables via a generic reduction from approximate counting to the problem of estimating local observables. The key idea is to use the observables to interpolate the counting problem. Using this new approach, we are able to study observables on bipartite graphs where the underlying optimization problem is easy but the counting problem is believed to be hard. The most-well studied class of graphs that was excluded from previous hardness results were bipartite graphs. We establish hardness for estimating the average size of the independent set in bipartite graphs of maximum degree 6; more generally, we show tight hardness results for general vertex-edge observables for antiferromagnetic 2-spin systems on bipartite graphs. Our techniques go beyond 2-spin systems, and for the ferromagnetic Potts model we establish hardness of approximating the number of monochromatic edges in the same region as known hardness of approximate counting results.
Open AccessApproximating observables is as hard as counting
We study the computational complexity of estimating local observables for Gibbs distributions. A simple combinatorial example is the average size of an independent set in a graph. In a recent work, we established NP-hardness of approximating the average size of an independent set utilizing hardness of the corresponding optimization problem and the related phase transition behavior. Here, we instead consider settings where the underlying optimization problem is easily solvable. Our main contribution is to classify the complexity of approximating a wide class of observables via a generic reduction from approximate counting to the problem of estimating local observables. The key idea is to use the observables to interpolate the counting problem. Using this new approach, we are able to study observables on bipartite graphs where the underlying optimization problem is easy but the counting problem is believed to be hard. The most-well studied class of graphs that was excluded from previous hardness results were bipartite graphs. We establish hardness for estimating the average size of the independent set in bipartite graphs of maximum degree 6; more generally, we show tight hardness results for general vertex-edge observables for antiferromagnetic 2-spin systems on bipartite graphs. Our techniques go beyond 2-spin systems, and for the ferromagnetic Potts model we establish hardness of approximating the number of monochromatic edges in the same region as known hardness of approximate counting results.