Document

**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

We study the approximability of computing the partition functions of two-state spin systems. The problem is parameterized by a 2×2 symmetric matrix. Previous results on this problem were restricted either to the case where the matrix has non-negative entries, or to the case where the diagonal entries are equal, i.e. Ising models. In this paper, we study the generalization to arbitrary 2×2 interaction matrices with real entries. We show that in some regions of the parameter space, it’s #P-hard to even determine the sign of the partition function, while in other regions there are fully polynomial approximation schemes for the partition function. Our results reveal several new computational phase transitions.

Yumou Fei, Leslie Ann Goldberg, and Pinyan Lu. Two-State Spin Systems with Negative Interactions. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 45:1-45:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fei_et_al:LIPIcs.ITCS.2024.45, author = {Fei, Yumou and Goldberg, Leslie Ann and Lu, Pinyan}, title = {{Two-State Spin Systems with Negative Interactions}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {45:1--45:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.45}, URN = {urn:nbn:de:0030-drops-195739}, doi = {10.4230/LIPIcs.ITCS.2024.45}, annote = {Keywords: Approximate Counting, Spin Systems, #P-Hardness, Randomized Algorithms} }

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RANDOM

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

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.

Andreas Galanis, Leslie Ann Goldberg, and Paulina Smolarova. Sampling from the Random Cluster Model on Random Regular Graphs at All Temperatures via Glauber Dynamics. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 64:1-64:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{galanis_et_al:LIPIcs.APPROX/RANDOM.2023.64, author = {Galanis, Andreas and Goldberg, Leslie Ann and Smolarova, Paulina}, title = {{Sampling from the Random Cluster Model on Random Regular Graphs at All Temperatures via Glauber Dynamics}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {64:1--64:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.64}, URN = {urn:nbn:de:0030-drops-188896}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.64}, annote = {Keywords: approximate counting, Glauber dynamics, random cluster model, approximate sampling, random regular graphs} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Given a class of graphs ℋ, the problem ⊕Sub(ℋ) is defined as follows. The input is a graph H ∈ ℋ together with an arbitrary graph G. The problem is to compute, modulo 2, the number of subgraphs of G that are isomorphic to H. The goal of this research is to determine for which classes ℋ the problem ⊕Sub(ℋ) is fixed-parameter tractable (FPT), i.e., solvable in time f(|H|)⋅|G|^O(1).
Curticapean, Dell, and Husfeldt (ESA 2021) conjectured that ⊕Sub(ℋ) is FPT if and only if the class of allowed patterns ℋ is matching splittable, which means that for some fixed B, every H ∈ ℋ can be turned into a matching (a graph in which every vertex has degree at most 1) by removing at most B vertices.
Assuming the randomised Exponential Time Hypothesis, we prove their conjecture for (I) all hereditary pattern classes ℋ, and (II) all tree pattern classes, i.e., all classes ℋ such that every H ∈ ℋ is a tree. We also establish almost tight fine-grained upper and lower bounds for the case of hereditary patterns (I).

Leslie Ann Goldberg and Marc Roth. Parameterised and Fine-Grained Subgraph Counting, Modulo 2. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 68:1-68:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{goldberg_et_al:LIPIcs.ICALP.2023.68, author = {Goldberg, Leslie Ann and Roth, Marc}, title = {{Parameterised and Fine-Grained Subgraph Counting, Modulo 2}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {68:1--68:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.68}, URN = {urn:nbn:de:0030-drops-181200}, doi = {10.4230/LIPIcs.ICALP.2023.68}, annote = {Keywords: modular counting, parameterised complexity, fine-grained complexity, subgraph counting} }

Document

**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

We study the problems of counting copies and induced copies of a small pattern graph H in a large host graph G. Recent work fully classified the complexity of those problems according to structural restrictions on the patterns H. In this work, we address the more challenging task of analysing the complexity for restricted patterns and restricted hosts. Specifically we ask which families of allowed patterns and hosts imply fixed-parameter tractability, i.e., the existence of an algorithm running in time f(H)⋅|G|^O(1) for some computable function f. Our main results present exhaustive and explicit complexity classifications for families that satisfy natural closure properties. Among others, we identify the problems of counting small matchings and independent sets in subgraph-closed graph classes 𝒢 as our central objects of study and establish the following crisp dichotomies as consequences of the Exponential Time Hypothesis:
- Counting k-matchings in a graph G ∈ 𝒢 is fixed-parameter tractable if and only if 𝒢 is nowhere dense.
- Counting k-independent sets in a graph G ∈ 𝒢 is fixed-parameter tractable if and only if 𝒢 is nowhere dense. Moreover, we obtain almost tight conditional lower bounds if 𝒢 is somewhere dense, i.e., not nowhere dense. These base cases of our classifications subsume a wide variety of previous results on the matching and independent set problem, such as counting k-matchings in bipartite graphs (Curticapean, Marx; FOCS 14), in F-colourable graphs (Roth, Wellnitz; SODA 20), and in degenerate graphs (Bressan, Roth; FOCS 21), as well as counting k-independent sets in bipartite graphs (Curticapean et al.; Algorithmica 19).
At the same time our proofs are much simpler: using structural characterisations of somewhere dense graphs, we show that a colourful version of a recent breakthrough technique for analysing pattern counting problems (Curticapean, Dell, Marx; STOC 17) applies to any subgraph-closed somewhere dense class of graphs, yielding a unified view of our current understanding of the complexity of subgraph counting.

Marco Bressan, Leslie Ann Goldberg, Kitty Meeks, and Marc Roth. Counting Subgraphs in Somewhere Dense Graphs. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bressan_et_al:LIPIcs.ITCS.2023.27, author = {Bressan, Marco and Goldberg, Leslie Ann and Meeks, Kitty and Roth, Marc}, title = {{Counting Subgraphs in Somewhere Dense Graphs}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.27}, URN = {urn:nbn:de:0030-drops-175304}, doi = {10.4230/LIPIcs.ITCS.2023.27}, annote = {Keywords: counting problems, somewhere dense graphs, parameterised complexity theory} }

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Invited Talk

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

This is an extended abstract of my talk at ICALP 2022, based on joint work with John Lapinskas.

Leslie Ann Goldberg. Some New (And Old) Results on Contention Resolution (Invited Talk). In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 3:1-3:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{goldberg:LIPIcs.ICALP.2022.3, author = {Goldberg, Leslie Ann}, title = {{Some New (And Old) Results on Contention Resolution}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {3:1--3:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.3}, URN = {urn:nbn:de:0030-drops-163444}, doi = {10.4230/LIPIcs.ICALP.2022.3}, annote = {Keywords: contention resolution, multiple access channel, randomised algorithms} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

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, 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.

Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, and Daniel Štefankovič. Fast Sampling via Spectral Independence Beyond Bounded-Degree Graphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 21:1-21:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bezakova_et_al:LIPIcs.ICALP.2022.21, author = {Bez\'{a}kov\'{a}, Ivona and Galanis, Andreas and Goldberg, Leslie Ann and \v{S}tefankovi\v{c}, Daniel}, title = {{Fast Sampling via Spectral Independence Beyond Bounded-Degree Graphs}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {21:1--21:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.21}, URN = {urn:nbn:de:0030-drops-163622}, doi = {10.4230/LIPIcs.ICALP.2022.21}, annote = {Keywords: Hard-core model, Random graphs, Markov chains} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We study the performance of Markov chains for the q-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 metastable phases for the q-state Potts model on the d-regular random graph for all integers q,d ≥ 3, and establish that for an interval of temperatures, 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. 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 large q and d ≥ 5.
Based on our 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 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.

Amin Coja-Oghlan, Andreas Galanis, Leslie Ann Goldberg, Jean Bernoulli Ravelomanana, Daniel Štefankovič, and Eric Vigoda. Metastability of the Potts Ferromagnet on Random Regular Graphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 45:1-45:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{cojaoghlan_et_al:LIPIcs.ICALP.2022.45, author = {Coja-Oghlan, Amin and Galanis, Andreas and Goldberg, Leslie Ann and Ravelomanana, Jean Bernoulli and \v{S}tefankovi\v{c}, Daniel and Vigoda, Eric}, title = {{Metastability of the Potts Ferromagnet on Random Regular Graphs}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {45:1--45:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.45}, URN = {urn:nbn:de:0030-drops-163865}, doi = {10.4230/LIPIcs.ICALP.2022.45}, annote = {Keywords: Markov chains, sampling, random regular graph, Potts model} }

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Invited Talk

**Published in:** LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

A homomorphism from a graph G to a graph H is a function from the vertices of G to the vertices of H that preserves the edges of G in the sense that every edge of G is mapped to an edge of H. By changing the target graph H, we can capture interesting structures in G. For example, homomorphisms from G to a k-clique H correspond to the proper k-colourings of G. There has been a lot of algorithmic work on the problem of (approximately) counting homomorphisms. The goal is to figure out for which graphs H the problem of approximately counting homomorphisms to H is algorithmically feasible. This talk will survey what is known. Despite much work, there are still plenty of open problems. We will discuss the problem of approximately counting list homomorphisms (where the input specifies, for each vertex of G, the list of vertices of H to which it can be mapped). Because the lists add extra expressibility, it is easier to prove that counting homomorphisms to a particular graph H is intractable. In fact, we have a full trichotomy (joint work with Galanis and Jerrum, 2017). Here, the complexity of homomorphism-counting is related to certain hereditary graph classes. The trichotomy will be explained in the talk - no prior knowledge of the area will be assumed. In more recent work, with Focke and Živn{ý}, we have investigated the complexity of counting retractions to H - this problem falls between homomorphism-counting and list-homomorphism counting. Here we have only a partial classification, which applies to all square-free graphs H. So again, there are plenty of open problems.

Leslie Ann Goldberg. Approximately Counting Graph Homomorphisms and Retractions (Invited Talk). In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goldberg:LIPIcs.FSTTCS.2021.3, author = {Goldberg, Leslie Ann}, title = {{Approximately Counting Graph Homomorphisms and Retractions}}, booktitle = {41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)}, pages = {3:1--3:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-215-0}, ISSN = {1868-8969}, year = {2021}, volume = {213}, editor = {Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.3}, URN = {urn:nbn:de:0030-drops-155146}, doi = {10.4230/LIPIcs.FSTTCS.2021.3}, annote = {Keywords: Graph homomorphisms, counting} }

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RANDOM

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

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.

Andreas Galanis, Leslie Ann Goldberg, and James Stewart. Fast Mixing via Polymers for Random Graphs with Unbounded Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 36:1-36:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{galanis_et_al:LIPIcs.APPROX/RANDOM.2021.36, author = {Galanis, Andreas and Goldberg, Leslie Ann and Stewart, James}, title = {{Fast Mixing via Polymers for Random Graphs with Unbounded Degree}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {36:1--36:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.36}, URN = {urn:nbn:de:0030-drops-147291}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.36}, annote = {Keywords: Markov chains, approximate counting, Potts model, expander graphs, random graphs} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

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 ≥ 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.

Andreas Galanis, Leslie Ann Goldberg, and Andrés Herrera-Poyatos. The Complexity of Approximating the Complex-Valued Potts Model. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 36:1-36:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{galanis_et_al:LIPIcs.MFCS.2020.36, author = {Galanis, Andreas and Goldberg, Leslie Ann and Herrera-Poyatos, Andr\'{e}s}, title = {{The Complexity of Approximating the Complex-Valued Potts Model}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {36:1--36:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.36}, URN = {urn:nbn:de:0030-drops-127038}, doi = {10.4230/LIPIcs.MFCS.2020.36}, annote = {Keywords: approximate counting, Potts model, Tutte polynomial, partition function, complex numbers} }

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**Published in:** LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

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 Jenssen 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 Õ(n²) time, where n is the size of the graph.

Andreas Galanis, Leslie Ann Goldberg, and James Stewart. Fast Algorithms for General Spin Systems on Bipartite Expanders. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 37:1-37:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{galanis_et_al:LIPIcs.MFCS.2020.37, author = {Galanis, Andreas and Goldberg, Leslie Ann and Stewart, James}, title = {{Fast Algorithms for General Spin Systems on Bipartite Expanders}}, booktitle = {45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)}, pages = {37:1--37:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-159-7}, ISSN = {1868-8969}, year = {2020}, volume = {170}, editor = {Esparza, Javier and Kr\'{a}l', Daniel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.37}, URN = {urn:nbn:de:0030-drops-127049}, doi = {10.4230/LIPIcs.MFCS.2020.37}, annote = {Keywords: bipartite expanders, approximate counting, spin systems} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

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 was due to Montanari and Shah and was based on the correlation decay method, which works up to densities (1+o_k(1))(2log k)/k, the Gibbs uniqueness threshold for the model. Instead, our algorithm harnesses a recent technique by Moitra to work for random formulas with much higher densities. 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.

Andreas Galanis, Leslie Ann Goldberg, Heng Guo, and Kuan Yang. Counting Solutions to Random CNF Formulas. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 53:1-53:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{galanis_et_al:LIPIcs.ICALP.2020.53, author = {Galanis, Andreas and Goldberg, Leslie Ann and Guo, Heng and Yang, Kuan}, title = {{Counting Solutions to Random CNF Formulas}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {53:1--53:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.53}, URN = {urn:nbn:de:0030-drops-124603}, doi = {10.4230/LIPIcs.ICALP.2020.53}, annote = {Keywords: random CNF formulas, approximate counting} }

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RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

For spin systems, such as the hard-core model on independent sets weighted by fugacity lambda>0, efficient algorithms for the associated approximate counting/sampling problems typically apply in the high-temperature region, corresponding to low fugacity. Recent work of Jenssen, Keevash and Perkins (2019) yields an FPTAS for approximating the partition function (and an efficient sampling algorithm) on bounded-degree (bipartite) expander graphs for the hard-core model at sufficiently high fugacity, and also the ferromagnetic Potts model at sufficiently low temperatures. 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. We present a simple discrete-time Markov chain for abstract polymer models, and present an elementary proof of rapid mixing of this new chain under sufficient decay of the polymer weights. Applying these general polymer results to the hard-core and ferromagnetic Potts models on bounded-degree (bipartite) expander graphs yields fast algorithms with running time 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 Delta)} for algorithms based on Barvinok’s polynomial interpolation method on graphs of maximum degree Delta. In addition, our approach via our polymer model Markov chain is conceptually simpler as it circumvents the zero-free analysis and the generalization to complex parameters. Finally, we 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 system Glauber dynamics restricted to even and odd or "red" dominant portions of the respective state spaces.

Zongchen Chen, Andreas Galanis, Leslie Ann Goldberg, Will Perkins, James Stewart, and Eric Vigoda. Fast Algorithms at Low Temperatures via Markov Chains. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 41:1-41:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chen_et_al:LIPIcs.APPROX-RANDOM.2019.41, author = {Chen, Zongchen and Galanis, Andreas and Goldberg, Leslie Ann and Perkins, Will and Stewart, James and Vigoda, Eric}, title = {{Fast Algorithms at Low Temperatures via Markov Chains}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {41:1--41:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.41}, URN = {urn:nbn:de:0030-drops-112560}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.41}, annote = {Keywords: Markov chains, approximate counting, Potts model, hard-core model, expander graphs} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

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

Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, and Daniel Štefankovič. The Complexity of Approximating the Matching Polynomial in the Complex Plane. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 22:1-22:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bezakova_et_al:LIPIcs.ICALP.2019.22, author = {Bez\'{a}kov\'{a}, Ivona and Galanis, Andreas and Goldberg, Leslie Ann and \v{S}tefankovi\v{c}, Daniel}, title = {{The Complexity of Approximating the Matching Polynomial in the Complex Plane}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {22:1--22:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.22}, URN = {urn:nbn:de:0030-drops-105983}, doi = {10.4230/LIPIcs.ICALP.2019.22}, annote = {Keywords: matchings, partition function, correlation decay, connective constant} }

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Invited Talk

**Published in:** LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

This paper is an extended abstract of my STACS 2019 talk "Computational Complexity and Partition Functions".

Leslie Ann Goldberg. Computational Complexity and Partition Functions (Invited Talk). In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 1:1-1:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{goldberg:LIPIcs.STACS.2019.1, author = {Goldberg, Leslie Ann}, title = {{Computational Complexity and Partition Functions}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {1:1--1:3}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.1}, URN = {urn:nbn:de:0030-drops-102405}, doi = {10.4230/LIPIcs.STACS.2019.1}, annote = {Keywords: partition functions, approximation} }

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**Published in:** LIPIcs, Volume 116, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

We consider the problem of sampling from the Potts model on random regular graphs. It is conjectured that sampling is possible when the temperature of the model is in the so-called uniqueness regime of the regular tree, but positive algorithmic results have been for the most part elusive. In this paper, for all integers q >= 3 and Delta >= 3, we develop algorithms that produce samples within error o(1) from the q-state Potts model on random Delta-regular graphs, whenever the temperature is in uniqueness, for both the ferromagnetic and antiferromagnetic cases.
The algorithm for the antiferromagnetic Potts model is based on iteratively adding the edges of the graph and resampling a bichromatic class that contains the endpoints of the newly added edge. Key to the algorithm is how to perform the resampling step efficiently since bichromatic classes can potentially induce linear-sized components. To this end, we exploit the tree uniqueness to show that the average growth of bichromatic components is typically small, which allows us to use correlation decay algorithms for the resampling step. While the precise uniqueness threshold on the tree is not known for general values of q and Delta in the antiferromagnetic case, our algorithm works throughout uniqueness regardless of its value.
In the case of the ferromagnetic Potts model, we are able to simplify the algorithm significantly by utilising the random-cluster representation of the model. In particular, we demonstrate that a percolation-type algorithm succeeds in sampling from the random-cluster model with parameters p,q on random Delta-regular graphs for all values of q >= 1 and p<p_c(q,Delta), where p_c(q,Delta) corresponds to a uniqueness threshold for the model on the Delta-regular tree. When restricted to integer values of q, this yields a simplified algorithm for the ferromagnetic Potts model on random Delta-regular graphs.

Antonio Blanca, Andreas Galanis, Leslie Ann Goldberg, Daniel Stefankovic, Eric Vigoda, and Kuan Yang. Sampling in Uniqueness from the Potts and Random-Cluster Models on Random Regular Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{blanca_et_al:LIPIcs.APPROX-RANDOM.2018.33, author = {Blanca, Antonio and Galanis, Andreas and Goldberg, Leslie Ann and Stefankovic, Daniel and Vigoda, Eric and Yang, Kuan}, title = {{Sampling in Uniqueness from the Potts and Random-Cluster Models on Random Regular Graphs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)}, pages = {33:1--33:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-085-9}, ISSN = {1868-8969}, year = {2018}, volume = {116}, editor = {Blais, Eric and Jansen, Klaus and D. P. Rolim, Jos\'{e} and Steurer, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2018.33}, URN = {urn:nbn:de:0030-drops-94371}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2018.33}, annote = {Keywords: sampling, Potts model, random regular graphs, phase transitions} }

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**Published in:** LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

The problem of (approximately) counting the independent sets of a bipartite graph (#BIS) is the canonical approximate counting problem that is complete in the intermediate complexity class #RHPi_1. It is believed that #BIS does not have an efficient approximation algorithm but also that it is not NP-hard. We study the robustness of the intermediate complexity of #BIS by considering variants of the problem parameterised by the size of the independent set. We map the complexity landscape for three problems, with respect to exact computation and approximation and with respect to conventional and parameterised complexity. The three problems are counting independent sets of a given size, counting independent sets with a given number of vertices in one vertex class and counting maximum independent sets amongst those with a given number of vertices in one vertex class. Among other things, we show that all of these problems are NP-hard to approximate within any polynomial ratio. (This is surprising because the corresponding problems without the size parameter are complete in #RHPi_1, and hence are not believed to be NP-hard.) We also show that the first problem is #W[1]-hard to solve exactly but admits an FPTRAS, whereas the other two are W[1]-hard to approximate even within any polynomial ratio. Finally, we show that, when restricted to graphs of bounded degree, all three problems have efficient exact fixed-parameter algorithms.

Radu Curticapean, Holger Dell, Fedor V. Fomin, Leslie Ann Goldberg, and John Lapinskas. A Fixed-Parameter Perspective on #BIS. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 13:1-13:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{curticapean_et_al:LIPIcs.IPEC.2017.13, author = {Curticapean, Radu and Dell, Holger and Fomin, Fedor V. and Goldberg, Leslie Ann and Lapinskas, John}, title = {{A Fixed-Parameter Perspective on #BIS}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {13:1--13:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-051-4}, ISSN = {1868-8969}, year = {2018}, volume = {89}, editor = {Lokshtanov, Daniel and Nishimura, Naomi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.13}, URN = {urn:nbn:de:0030-drops-85613}, doi = {10.4230/LIPIcs.IPEC.2017.13}, annote = {Keywords: Approximate counting, parameterised complexity, independent sets} }

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**Published in:** Dagstuhl Reports, Volume 7, Issue 8 (2018)

This report documents the program and the outcomes of Dagstuhl Seminar 17341 "Computational Counting". The seminar was held from 20th to 25th August 2017, at Schloss Dagstuhl -- Leibnitz Center for Informatics.
A total of 43 researchers from all over the world, with interests and expertise in different aspects of computational counting, actively participated in the meeting.

Ivona Bezáková, Leslie Ann Goldberg, and Mark R. Jerrum. Computational Counting (Dagstuhl Seminar 17341). In Dagstuhl Reports, Volume 7, Issue 8, pp. 23-44, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@Article{bezakova_et_al:DagRep.7.8.23, author = {Bez\'{a}kov\'{a}, Ivona and Goldberg, Leslie Ann and Jerrum, Mark R.}, title = {{Computational Counting (Dagstuhl Seminar 17341)}}, pages = {23--44}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2018}, volume = {7}, number = {8}, editor = {Bez\'{a}kov\'{a}, Ivona and Goldberg, Leslie Ann and Jerrum, Mark R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.7.8.23}, URN = {urn:nbn:de:0030-drops-84283}, doi = {10.4230/DagRep.7.8.23}, annote = {Keywords: approximation algorithms, computational complexity, counting problems, partition functions, phase transitions} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We study the complexity of approximate counting Constraint Satisfaction Problems (#CSPs) in a bounded degree setting. Specifically, given a Boolean constraint language Gamma and a degree bound Delta, we study the complexity of #CSP_Delta(Gamma), which is the problem of counting satisfying assignments to CSP instances with constraints from Gamma and whose variables can appear at most Delta times. Our main result shows that: (i) if every function in Gamma is affine, then #CSP_Delta(Gamma) is in FP for all Delta, (ii) otherwise, if every function in Gamma is in a class called IM_2, then for all sufficiently large Delta, #CSP_Delta(Gamma) 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 Delta, it is NP-hard to approximate the number of satisfying assignments of an instance of #CSP_Delta(Gamma), even within an exponential factor. Our result extends previous results, which apply only in the so-called "conservative" case.

Andreas Galanis, Leslie Ann Goldberg, and Kuan Yang. Approximating Partition Functions of Bounded-Degree Boolean Counting Constraint Satisfaction Problems. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{galanis_et_al:LIPIcs.ICALP.2017.27, author = {Galanis, Andreas and Goldberg, Leslie Ann and Yang, Kuan}, title = {{Approximating Partition Functions of Bounded-Degree Boolean Counting Constraint Satisfaction Problems}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.27}, URN = {urn:nbn:de:0030-drops-74099}, doi = {10.4230/LIPIcs.ICALP.2017.27}, annote = {Keywords: Constraint Satisfaction, Approximate Counting} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We study the problem of approximately evaluating the independent set polynomial of bounded-degree graphs at a point lambda or, equivalently, the problem of approximating the partition function of the hard-core model with activity lambda on graphs G of max degree D. For lambda>0, breakthrough results of Weitz and Sly established a computational transition from easy to hard at lambda_c(D)=(D-1)^(D-1)/(D-2)^D, which coincides with the tree uniqueness phase transition from statistical physics.
For lambda<0, the evaluation of the independent set polynomial is connected to the conditions of the Lovasz Local Lemma. Shearer identified the threshold lambda*(D)=(D-1)^(D-1)/D^D as the maximum value p such that every family of events with failure probability at most p and whose dependency graph has max degree D has nonempty intersection. Very recently, Patel and Regts, and Harvey et al. have independently designed FPTASes for approximating the partition function whenever |lambda|<lambda*(D).
Our main result establishes for the first time a computational transition at the Shearer threshold. We show that for all D>=3, for all lambda<-lambda*(D), it is NP-hard to approximate the partition function on graphs of maximum degree D, even within an exponential factor. Thus, our result, combined with the FPTASes for lambda>-lambda*(D), establishes a phase transition for negative activities. In fact, we now have the following picture for the problem of approximating the partition function with activity lambda on graphs G of max degree D.
1. For -lambda*(D)<lambda<lambda_c(D), the problem admits an FPTAS.
2. For lambda<-lambda*(D) or lambda>lambda_c(D), the problem is NP-hard.
Rather than the tree uniqueness threshold of the positive case, the phase transition for negative activities corresponds to the existence of zeros for the partition function of the tree below -lambda*(D).

Andreas Galanis, Leslie Ann Goldberg, and Daniel Stefankovic. Inapproximability of the Independent Set Polynomial Below the Shearer Threshold. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{galanis_et_al:LIPIcs.ICALP.2017.28, author = {Galanis, Andreas and Goldberg, Leslie Ann and Stefankovic, Daniel}, title = {{Inapproximability of the Independent Set Polynomial Below the Shearer Threshold}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {28:1--28:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.28}, URN = {urn:nbn:de:0030-drops-73962}, doi = {10.4230/LIPIcs.ICALP.2017.28}, annote = {Keywords: approximate counting, independent set polynomial, Shearer threshold} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

Approximate counting via correlation decay is the core algorithmic technique used in the sharp delineation of the computational phase transition that arises in the approximation of the partition function of anti-ferromagnetic two-spin models.
Previous analyses of correlation-decay algorithms implicitly depended on the occurrence of strong spatial mixing. This, roughly, means that one uses worst-case analysis of the recursive procedure that creates the sub-instances. In this paper, we develop a new analysis method that is more refined than the worst-case analysis. We take the shape of instances in the computation tree into consideration and we amortise against certain "bad" instances that are created as the recursion proceeds. This enables us to show correlation decay and to obtain an FPTAS even when strong spatial mixing fails.
We apply our technique to the problem of approximately counting independent sets in hypergraphs with degree upper-bound Delta and with a lower bound k on the arity of hyperedges. Liu and Lin gave an FPTAS for k >= 2 and Delta <= 5 (lack of strong spatial mixing was the obstacle preventing this algorithm from being generalised to Delta = 6). Our technique gives a tight result for Delta = 6, showing that there is an FPTAS for k >= 3 and Delta <= 6. The best previously-known approximation scheme for Delta = 6 is the Markov-chain simulation based FPRAS of Bordewich, Dyer and Karpinski, which only works for k >= 8.
Our technique also applies for larger values of k, giving an FPTAS for k >= 1.66 Delta. This bound is not as strong as existing randomised results, for technical reasons that are discussed in the paper. Nevertheless, it gives the first deterministic approximation schemes in this regime. We further demonstrate that in the hypergraph independent set model, approximating the partition function is NP-hard even within the uniqueness regime.

Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, Heng Guo, and Daniel Stefankovic. Approximation via Correlation Decay When Strong Spatial Mixing Fails. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 45:1-45:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bezakova_et_al:LIPIcs.ICALP.2016.45, author = {Bez\'{a}kov\'{a}, Ivona and Galanis, Andreas and Goldberg, Leslie Ann and Guo, Heng and Stefankovic, Daniel}, title = {{Approximation via Correlation Decay When Strong Spatial Mixing Fails}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {45:1--45:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.45}, URN = {urn:nbn:de:0030-drops-63257}, doi = {10.4230/LIPIcs.ICALP.2016.45}, annote = {Keywords: approximate counting, independent sets in hypergraphs, correlation decay} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We examine the computational complexity of approximately counting the list H-colourings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H. If H is an irreflexive bipartite graph or a reflexive complete graph then counting list H-colourings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexive proper interval graph then approximately counting list H-colourings is equivalent to #BIS, the problem of approximately counting independent sets in a bipartite graph. This is a well-studied problem which is believed to be of intermediate complexity - it is believed that it does not have an FPRAS, but that it is not as difficult as approximating the most difficult counting problems in #P. For every other graph H, approximately counting list H-colourings is complete for #P with respect to approximation-preserving reductions (so there is no FPRAS unless NP = RP). Two pleasing features of the trichotomy are (i) it has a natural formulation in terms of hereditary graph classes, and (ii) the proof is largely self-contained and does not require any universal algebra (unlike similar dichotomies in the weighted case). We are able to extend the hardness results to the bounded-degree setting, showing that all hardness results apply to input graphs with maximum degree at most 6.

Andreas Galanis, Leslie Ann Goldberg, and Mark Jerrum. A Complexity Trichotomy for Approximately Counting List H-Colourings. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{galanis_et_al:LIPIcs.ICALP.2016.46, author = {Galanis, Andreas and Goldberg, Leslie Ann and Jerrum, Mark}, title = {{A Complexity Trichotomy for Approximately Counting List H-Colourings}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {46:1--46:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.46}, URN = {urn:nbn:de:0030-drops-63262}, doi = {10.4230/LIPIcs.ICALP.2016.46}, annote = {Keywords: approximate counting, graph homomorphisms, list colourings} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

The Moran process, as studied by Lieberman, Hauert and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen uniformly at random) possesses a mutation, with fitness r > 1. All other individuals have fitness 1. During each step of the algorithm, an individual is chosen with probability proportional to its fitness, and its state (mutant or non-mutant) is passed on to an out-neighbour which is chosen uniformly at random. If the underlying graph is strongly connected then the algorithm will eventually reach fixation, in which all individuals are mutants, or extinction, in which no individuals are mutants. An infinite family of directed graphs is said to be strongly amplifying if, for every r > 1, the extinction probability tends to 0 as the number of vertices increases. Strong amplification is a rather surprising property - it means that in such graphs, the fixation probability of a uniformly-placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of r > 1 (independently of n). The name "strongly amplifying" comes from the fact that this selective advantage is "amplified". Strong amplifiers have received quite a bit of attention, and Lieberman et al. proposed two potentially strongly-amplifying families - superstars and metafunnels. Heuristic arguments have been published, arguing that there are infinite families of superstars that are strongly amplifying. The same has been claimed for metafunnels. We give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying. We call the graphs in the family "megastars". When the algorithm is run on an n-vertex graph in this family, starting with a uniformly-chosen mutant, the extinction probability is roughly n^{-1/2} (up to logarithmic factors). We prove that all infinite families of superstars and metafunnels have larger extinction probabilities (as a function of n). Finally, we prove that our analysis of megastars is fairly tight - there is no infinite family of megastars such that the Moran algorithm gives a smaller extinction probability (up to logarithmic factors). Also, we provide a counterexample which clarifies the literature concerning the isothermal theorem of Lieberman et al. A full version [Galanis/Göbel/Goldberg/Lapinskas/Richerby, Preprint] containing detailed proofs is available at http://arxiv.org/abs/1512.05632. Theorem-numbering here matches the full version.

Andreas Galanis, Andreas Göbel, Leslie Ann Goldberg, John Lapinskas, and David Richerby. Amplifiers for the Moran Process. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 62:1-62:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{galanis_et_al:LIPIcs.ICALP.2016.62, author = {Galanis, Andreas and G\"{o}bel, Andreas and Goldberg, Leslie Ann and Lapinskas, John and Richerby, David}, title = {{Amplifiers for the Moran Process}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {62:1--62:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.62}, URN = {urn:nbn:de:0030-drops-62227}, doi = {10.4230/LIPIcs.ICALP.2016.62}, annote = {Keywords: Moran process, randomised algorithm on graphs, evolutionary dynamics} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Counting independent sets on bipartite graphs (#BIS) is considered a canonical counting problem of intermediate approximation complexity. It is conjectured that #BIS neither has an FPRAS nor is as hard as #SAT to approximate. We study #BIS in the general framework of two-state spin systems in bipartite graphs. Such a system is parameterized by three numbers (beta,gamma,lambda), where beta (respectively gamma) represents the weight of an edge (or "interaction strength") whose endpoints are of the same 0 (respectively 1) spin, and lambda is the weight of a 1 vertex, also known as an "external field". By convention, the edge weight with unequal 0/1 end points and the vertex weight with spin 0 are both normalized to 1. The partition function of the special case beta=1, gamma=0, and lambda=1 counts the number of independent sets. We define two notions, nearly-independent phase-correlated spins and symmetry breaking. We prove that it is #BIS-hard to approximate the partition function of any two-spin system on bipartite graphs supporting these two notions.
As a consequence, we show that #BIS on graphs of degree at most 6 is as hard to approximate as #BIS~without degree bound. The degree bound 6 is the best possible as Weitz presented an FPTAS to count independent sets on graphs of maximum degree 5. This result extends to the hard-core model and to other anti-ferromagnetic two-spin models. In particular, for all antiferromagnetic two-spin systems, namely those satisfying beta*gamma<1, we prove that when the infinite (Delta-1)-ary tree lies in the non-uniqueness region then it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree Delta, except for the case beta=gamma and lambda=1. The exceptional case is precisely the antiferromagnetic Ising model without an external field, and we show that it has an FPRAS on bipartite graphs. Our inapproximability results match the approximability results of Li et al., who presented an FPTAS for general graphs of maximum degree Delta when the parameters lie in the uniqueness region.

Jin-Yi Cai, Andreas Galanis, Leslie Ann Goldberg, Heng Guo, Mark Jerrum, Daniel Stefankovic, and Eric Vigoda. #BIS-Hardness for 2-Spin Systems on Bipartite Bounded Degree Graphs in the Tree Non-uniqueness Region. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 582-595, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{cai_et_al:LIPIcs.APPROX-RANDOM.2014.582, author = {Cai, Jin-Yi and Galanis, Andreas and Goldberg, Leslie Ann and Guo, Heng and Jerrum, Mark and Stefankovic, Daniel and Vigoda, Eric}, title = {{#BIS-Hardness for 2-Spin Systems on Bipartite Bounded Degree Graphs in the Tree Non-uniqueness Region}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {582--595}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.582}, URN = {urn:nbn:de:0030-drops-47235}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.582}, annote = {Keywords: Spin systems, approximate counting, complexity, #BIS-hardness, phase transition} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is polynomial on an n-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we give the expected absorption time is blog n lower bound and an explicit quadratic upper bound. We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson.

Josep Díaz, Leslie Ann Goldberg, David Richerby, and Maria Serna. Absorption Time of the Moran Process. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 630-642, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{diaz_et_al:LIPIcs.APPROX-RANDOM.2014.630, author = {D{\'\i}az, Josep and Goldberg, Leslie Ann and Richerby, David and Serna, Maria}, title = {{Absorption Time of the Moran Process}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {630--642}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.630}, URN = {urn:nbn:de:0030-drops-47279}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.630}, annote = {Keywords: Moran Process} }

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**Published in:** LIPIcs, Volume 25, 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)

A homomorphism from a graph G to a graph H is a function from V(G) to V(H) that preserves edges. Many combinatorial structures that arise in mathematics and computer science can be represented naturally as graph homomorphisms and as weighted sums of graph homomorphisms. In this paper, we study the complexity of counting homomorphisms modulo 2. The complexity of modular counting was introduced by Papadimitriou and Zachos and it has been pioneered by Valiant who famously introduced a problem for which counting modulo 7 is easy but counting modulo 2 is intractable. Modular counting provides a rich setting in which to study the structure of homomorphism problems. In this case, the structure of the graph H has a big influence on the complexity of the problem. Thus, our approach is graph-theoretic. We give a complete solution for the class of cactus graphs, which are connected graphs in which every edge belongs to at most one cycle. Cactus graphs arise in many applications such as the modelling of wireless sensor networks and the comparison of genomes. We show that, for some cactus graphs H, counting homomorphisms to H modulo 2 can be done in polynomial time. For every other fixed cactus graph H, the problem is complete for the complexity class +P which is a wide complexity class to which every problem in the polynomial hierarchy can be reduced (using randomised reductions). Determining which H lead to tractable problems can be done in polynomial time. Our result builds upon the work of Faben and Jerrum, who gave a dichotomy for the case in which H is a tree.

Andreas Göbel, Leslie Ann Goldberg, and David Richerby. Counting Homomorphisms to Cactus Graphs Modulo 2. In 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 25, pp. 350-361, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{gobel_et_al:LIPIcs.STACS.2014.350, author = {G\"{o}bel, Andreas and Goldberg, Leslie Ann and Richerby, David}, title = {{Counting Homomorphisms to Cactus Graphs Modulo 2}}, booktitle = {31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014)}, pages = {350--361}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-65-1}, ISSN = {1868-8969}, year = {2014}, volume = {25}, editor = {Mayr, Ernst W. and Portier, Natacha}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2014.350}, URN = {urn:nbn:de:0030-drops-44700}, doi = {10.4230/LIPIcs.STACS.2014.350}, annote = {Keywords: modular counting, homomorphisms, cactus graph, graph algorithms} }

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**Published in:** Dagstuhl Reports, Volume 3, Issue 1 (2013)

Dagstuhl Seminar 13031 "Computational Counting" was held from 13th to 18th January 2013, at Schloss Dagstuhl -- Leibnitz Center for Informatics. A total of 43 researchers from all over the world, with interests and expertise in different aspects of computational counting, actively participated in the meeting.

Peter Bürgisser, Leslie Ann Goldberg, Mark Jerrum, and Pascal Koiran. Computational Counting (Dagstuhl Seminar 13031). In Dagstuhl Reports, Volume 3, Issue 1, pp. 47-66, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@Article{burgisser_et_al:DagRep.3.1.47, author = {B\"{u}rgisser, Peter and Goldberg, Leslie Ann and Jerrum, Mark and Koiran, Pascal}, title = {{Computational Counting (Dagstuhl Seminar 13031)}}, pages = {47--66}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2013}, volume = {3}, number = {1}, editor = {B\"{u}rgisser, Peter and Goldberg, Leslie Ann and Jerrum, Mark and Koiran, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.3.1.47}, URN = {urn:nbn:de:0030-drops-40087}, doi = {10.4230/DagRep.3.1.47}, annote = {Keywords: Computational complexity, counting problems, graph polynomials, holographic algorithms, statistical physics, constraint satisfaction} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

We study the complexity of approximation for a weighted counting constraint satisfaction problem #CSP(F). In the conservative case, where F contains all unary functions, a classification is known for the Boolean domain. We give a classification for problems with general finite domain. We define weak log-modularity and weak log-supermodularity, and show that #CSP(F) is in FP if F is weakly log-modular. Otherwise, it is at least as hard to approximate as #BIS, counting independent sets in bipartite graphs, which is believed to be intractable. We further sub-divide the #BIS-hard case. If F is weakly log-supermodular, we show that #CSP(F) is as easy as Boolean log-supermodular weighted #CSP. Otherwise, it is NP-hard to approximate. Finally, we give a trichotomy for the arity-2 case.
Then, #CSP(F) is in FP, is #BIS-equivalent, or is equivalent to #SAT, the problem of approximately counting satisfying assignments of a CNF Boolean formula.

Xi Chen, Martin Dyer, Leslie Ann Goldberg, Mark Jerrum, Pinyan Lu, Colin McQuillan, and David Richerby. The complexity of approximating conservative counting CSPs. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 148-159, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{chen_et_al:LIPIcs.STACS.2013.148, author = {Chen, Xi and Dyer, Martin and Goldberg, Leslie Ann and Jerrum, Mark and Lu, Pinyan and McQuillan, Colin and Richerby, David}, title = {{The complexity of approximating conservative counting CSPs}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {148--159}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.148}, URN = {urn:nbn:de:0030-drops-39303}, doi = {10.4230/LIPIcs.STACS.2013.148}, annote = {Keywords: counting constraint satisfaction problem, approximation, complexity} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

Motivated by a desire to understand the computational complexity of counting constraint satisfaction problems (counting CSPs), particularly the complexity of approximation, we study functional clones of functions on the Boolean domain, which are analogous to the familiar relational clones constituting Post's lattice. One of these clones is the collection of log-supermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In our study, we assume that non-negative unary functions (weights) are available. Given this, we prove that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial non-lsm function is computationally as hard as any problem in #P. Furthermore, any non-trivial functional clone (in a sense that will be made precise below) contains the binary function "implies". As a consequence, all non-trivial counting CSPs (with non-negative unary weights assumed to be available) are computationally at least as difficult as #BIS, the problem of counting independent sets in a bipartite graph. There is empirical evidence that #BIS is hard to solve, even approximately.

Andrei A. Bulatov, Martin Dyer, Leslie Ann Goldberg, and Mark Jerrum. Log-supermodular functions, functional clones and counting CSPs. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 302-313, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{bulatov_et_al:LIPIcs.STACS.2012.302, author = {Bulatov, Andrei A. and Dyer, Martin and Goldberg, Leslie Ann and Jerrum, Mark}, title = {{Log-supermodular functions, functional clones and counting CSPs}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {302--313}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.302}, URN = {urn:nbn:de:0030-drops-34078}, doi = {10.4230/LIPIcs.STACS.2012.302}, annote = {Keywords: counting constraint satisfaction problems, approximation, complexity} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10481, Computational Counting (2011)

From November 28 to December 3 2010, the Dagstuhl Seminar 10481 ``Computational Counting'' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available.

Peter Bürgisser, Leslie Ann Goldberg, and Mark Jerrum. 10481 Abstracts Collection – Computational Counting. In Computational Counting. Dagstuhl Seminar Proceedings, Volume 10481, pp. 1-15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{burgisser_et_al:DagSemProc.10481.1, author = {B\"{u}rgisser, Peter and Goldberg, Leslie Ann and Jerrum, Mark}, title = {{10481 Abstracts Collection – Computational Counting}}, booktitle = {Computational Counting}, pages = {1--15}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2011}, volume = {10481}, editor = {Peter B\"{u}rgisser and Leslie Ann Goldberg and Mark Jerrum}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10481.1}, URN = {urn:nbn:de:0030-drops-29453}, doi = {10.4230/DagSemProc.10481.1}, annote = {Keywords: Computational complexity, counting problems, holographic algorithms, statistical physics, constraint satisfaction} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 10481, Computational Counting (2011)

From November 28 to December 3 2010, the Dagstuhl seminar 10481
``Computational Counting'' was held in Schloss Dagstuhl – Leibnitz Center
for Informatics. 36 researchers from all over the world, with interests and
expertise in different aspects of computational counting, actively
participated in the meeting.

Peter Bürgisser, Leslie Ann Goldberg, and Mark Jerrum. 10481 Executive Summary – Computational Counting. In Computational Counting. Dagstuhl Seminar Proceedings, Volume 10481, pp. 1-3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{burgisser_et_al:DagSemProc.10481.2, author = {B\"{u}rgisser, Peter and Goldberg, Leslie Ann and Jerrum, Mark}, title = {{10481 Executive Summary – Computational Counting}}, booktitle = {Computational Counting}, pages = {1--3}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2011}, volume = {10481}, editor = {Peter B\"{u}rgisser and Leslie Ann Goldberg and Mark Jerrum}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.10481.2}, URN = {urn:nbn:de:0030-drops-29441}, doi = {10.4230/DagSemProc.10481.2}, annote = {Keywords: Computational complexity, counting problems, holographic algorithms, statistical physics, constraint satisfaction} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9371, Algorithmic Methods for Distributed Cooperative Systems (2010)

Consensus problems occur in many contexts and have therefore been intensively studied in the past. In the standard consensus problem there are n processes with possibly different input values and the goal is to eventually reach a point at which all processes commit to exactly one of these values. We are studying a slight variant of the consensus problem called the stabilizing consensus problem. In this problem, we do not require that each process commits to a final value at some point, but that eventually they arrive at a common value without necessarily being aware of that. This should work irrespective of the states in which the processes are starting. Coming up with a self-stabilizing rule is easy without adversarial involvement, but we allow some T-bounded adversary to manipulate any T processes at any time. In this situation, a perfect consensus is impossible to reach, so we only require that there is a time point t and value v so that at any point after t, all but up to O(T) processes agree on v, which we call an almost stable consensus. As we will demonstrate, there is a surprisingly simple rule for the standard message passing model that just needs O(log n loglog n) time for any sqrt{n}-bounded adversary and just O(log n) time without adversarial involvement, with high probability, to reach an (almost) stable consensus from any initial state. A stable consensus is reached, with high probability, in the absence of adversarial involvement.

Benjamin Doerr, Leslie Ann Goldberg, Lorenz Minder, Thomas Sauerwald, and Christian Scheideler. Stabilizing Consensus with the Power of Two Choices. In Algorithmic Methods for Distributed Cooperative Systems. Dagstuhl Seminar Proceedings, Volume 9371, pp. 1-21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{doerr_et_al:DagSemProc.09371.6, author = {Doerr, Benjamin and Goldberg, Leslie Ann and Minder, Lorenz and Sauerwald, Thomas and Scheideler, Christian}, title = {{Stabilizing Consensus with the Power of Two Choices}}, booktitle = {Algorithmic Methods for Distributed Cooperative Systems}, pages = {1--21}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {9371}, editor = {S\'{a}ndor Fekete and Stefan Fischer and Martin Riedmiller and Suri Subhash}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09371.6}, URN = {urn:nbn:de:0030-drops-24290}, doi = {10.4230/DagSemProc.09371.6}, annote = {Keywords: Distributed consensus} }

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**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

The degree of a CSP instance is the maximum number of times that a variable may appear in the scope of constraints. We consider the approximate counting problem for Boolean CSPs with bounded-degree instances, for constraint languages containing the two unary constant relations $\{0\}$ and $\{1\}$. When the maximum degree is at least $25$ we obtain a complete classification of the complexity of this problem. It is exactly solvable in polynomial-time if every relation in the constraint language is affine. It is equivalent to the problem of approximately counting independent sets in bipartite graphs if every relation can be expressed as conjunctions of $\{0\}$, $\{1\}$ and binary implication. Otherwise, there is no FPRAS unless $\NPtime = \RPtime$. For lower degree bounds, additional cases arise in which the complexity is related to the complexity of approximately counting independent sets in hypergraphs.

Martin Dyer, Leslie Ann Goldberg, Markus Jalsenius, and David Richerby. The Complexity of Approximating Bounded-Degree Boolean #CSP. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 323-334, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{dyer_et_al:LIPIcs.STACS.2010.2466, author = {Dyer, Martin and Goldberg, Leslie Ann and Jalsenius, Markus and Richerby, David}, title = {{The Complexity of Approximating Bounded-Degree Boolean #CSP}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {323--334}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2466}, URN = {urn:nbn:de:0030-drops-24669}, doi = {10.4230/LIPIcs.STACS.2010.2466}, annote = {Keywords: Boolean constraint satisfaction problem, generalized satisfiability, counting, approximation algorithms} }

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**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

\emph{Partition functions}, also known as \emph{homomorphism functions}, form a rich family of graph invariants that contain combinatorial invariants such as the number of $k$-colourings or the number of independent sets of a graph and also the partition functions of certain ``spin glass'' models of statistical physics such as the Ising model.
Building on earlier work by Dyer and Greenhill (2000) and Bulatov and Grohe (2005), we completely classify the computational complexity of partition functions. Our main result is a dichotomy theorem stating that every partition function is either computable in polynomial time or \#P-complete. Partition functions are described by symmetric matrices with real entries, and we prove that it is decidable in polynomial time in terms of the matrix whether a given partition function is in polynomial time or \#P-complete.
While in general it is very complicated to give an explicit algebraic or combinatorial description of the tractable cases, for partition functions described by a Hadamard matrices --- these turn out to be central in our proofs --- we obtain a simple algebraic tractability criterion, which says that the tractable cases are those ``representable'' by a quadratic polynomial over the field $\ensuremath{\mathbb{F}_2}$.

Leslie Ann Goldberg, Martin Grohe, Mark Jerrum, and Marc Thurley. A Complexity Dichotomy for Partition Functions with Mixed Signs. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 493-504, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{goldberg_et_al:LIPIcs.STACS.2009.1821, author = {Goldberg, Leslie Ann and Grohe, Martin and Jerrum, Mark and Thurley, Marc}, title = {{A Complexity Dichotomy for Partition Functions with Mixed Signs}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {493--504}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1821}, URN = {urn:nbn:de:0030-drops-18217}, doi = {10.4230/LIPIcs.STACS.2009.1821}, annote = {Keywords: Computational complexity} }

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