Clean Citation Style
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Göbel, Andreas; Lagodzinski, J. A. Gregor; Seidel, Karen Counting Homomorphisms to Trees Modulo a Prime. International Symposium on Mathematical Foundations of Computer Science (MFCS) 2018: 49:149:13
Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article we study the complexity of~\($\#_p\textsc{HomsTo}H$\), the problem of counting graph homomorphisms from an input graph to a graph \($H$\) modulo a prime number~\($p$\). Dyer and Greenhill proved a dichotomy stating that the tractability of nonmodular counting graph homomorphisms depends on the structure of the target graph. Many intractable cases in nonmodular counting become tractable in modular counting due to the common phenomenon of cancellation. In subsequent studies on counting modulo~\($2$\), however, the influence of the structure of~\($H$\) on the tractability was shown to persist, which yields similar dichotomies. Our main result states that for every tree~\($H$\) and every prime~\($p$\) the problem \($\#_p\textsc{HomsTo}H$\) is either polynomial time computable or \($\#_p\mathsf{P}$\)complete. This relates to the conjecture of Faben and Jerrum stating that this dichotomy holds for every graph \($H$\) when counting modulo~2. In contrast to previous results on modular counting, the tractable cases of \($\#_p\textsc{HomsTo}H$\) are essentially the same for all values of the modulo when \($H$\) is a tree. To prove this result, we study the structural properties of a homomorphism. As an important interim result, our study yields a dichotomy for the problem of counting weighted independent sets in a bipartite graph modulo some prime~\($p$\). These results are the first suggesting that such dichotomies hold not only for the onebit functions of the modulo~2 case but also for the modular counting functions of all primes~\($p$\).

Friedrich, Tobias; Göbel, Andreas; Quinzan, Francesco; Wagner, Markus Heavytailed Mutation Operators in SingleObjective Combinatorial Optimization. Parallel Problem Solving From Nature (PPSN) 2018

Göbel, Andreas; Lagodzinski, J. A. Gregor; Seidel, Karen Counting Homomorphisms to Trees Modulo a Prime. arXiv 2018

Bampas, Evangelos; Göbel, AndreasNikolas; Pagourtzis, Aris; Tentes, Aris On the connection between interval size functions and path counting. Computational Complexity 2017: 421467
We investigate the complexity of hard (\#Pcomplete) counting problems that have easy decision version. By ‘easy decision,’ we mean that deciding whether the result of counting is nonzero is in P. This property is shared by several wellknown problems, such as counting the number of perfect matchings in a given graph or counting the number of satisfying assignments of a given DNF formula. We focus on classes of such hardtocount easytodecide problems which emerged through two seemingly disparate approaches: one taken by Hemaspaandra et al. (SIAM J Comput 36(5):1264–1300, 2007), who defined classes of functions that count the size of intervals of ordered strings, and one followed by Kiayias et al. (Lect Notes Comput Sci 2563:453–463, 2001), who defined the classTotP, consisting of functions that count the total number of paths of NP computations. We provide inclusion and separation relations between TotP and interval size counting classes, by means of new classes that we define in this work. Our results imply that many known #Pcomplete problems with easy decision are contained in the classes defined by Hemaspaandra et al., but are unlikely to be complete for these classes under reductions under which these classes are downward closed, e.g., parsimonious reductions. This, applied to the #MONSAT problem, partially answers an open question of Hemaspaandra et al. We also define a new class of interval size functions which strictly contains FP and is strictly contained in TotP under reasonable complexitytheoretic assumptions. We show that this new class contains hard counting problems.

Galanis, Andreas; Göbel, Andreas; Goldberg, Leslie Ann; Lapinskas, John; Richerby, David Amplifiers for the Moran Process. Journal of the ACM 2017: 5:15:90
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 nonmutant) is passed on to an outneighbour 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. A formal definition is provided in the article. Strong amplification is a rather surprising propertyit 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 familiessuperstars 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. In this article, 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.

Göbel, Andreas Counting, Modular Counting and Graph Homomorphisms. 2017

Göbel, Andreas; Goldberg, Leslie Ann; Richerby, David Counting Homomorphisms to SquareFree Graphs, Modulo 2. Transactions on Computation Theory 2016: 12:112:29
We study the problem \( \oplus \)HomsToH of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph \(H\). A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (nonmodular) counting; thus, subtle dichotomy theorems can arise. We show the following dichotomy: for any \(H\) that contains no 4cycles, \(\oplus\)HomsToH is either in polynomial time or is \(\oplus\)Pcomplete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of treewidth2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs, including graphs of unbounded treewidth. In particular, we focus on squarefree graphs, which are graphs without 4cycles. These graphs arise frequently in combinatorics, for example, in connection with the strong perfect graph theorem and in certain graph algorithms. Previous dichotomy theorems required the graph to be treelike so that treelike decompositions could be exploited in the proof. We prove the conjecture for a much richer class of graphs by adopting a much more general approach.

Galanis, Andreas; Göbel, Andreas; Goldberg, LeslieAnn; Lapinskas, John; Richerby, David Amplifiers for the Moran Process. International Colloquium on Automata, Languages and Programming (ICALP) 2016: 62:162:13
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 nonmutant) is passed on to an outneighbour 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 uniformlyplaced 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 stronglyamplifying 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 nvertex graph in this family, starting with a uniformlychosen 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. Theoremnumbering here matches the full version.

Göbel, Andreas; Goldberg, Leslie Ann; McQuillan, Colin; Richerby, David; Yamakami, Tomoyuki Counting List Matrix Partitions of Graphs. SIAM Journal on Computing 2015: 10891118
Given a symmetric \(D\times D\) matrix \(M\) over \(0,1,*\), a list \(M\)partition of a graph \(G\) is a partition of the vertices of \(G\) into \(D\) parts which are associated with the rows of \(M\). The part of each vertex is chosen from a given list in such a way that no edge of \(G\) is mapped to a 0 in \(M\) and no nonedge of \(G\) is mapped to a 1 in \(M\). Many important graphtheoretic structures can be represented as list \(M\)partitions including graph colorings, split graphs, and homogeneous sets and pairs, which arise in the proofs of the weak and strong perfect graph conjectures. Thus, there has been quite a bit of work on determining for which matrices \(M\) computations involving list \(M\)partitions are tractable. This paper focuses on the problem of counting list \(M\)partitions, given a graph \(G\) and given a list for each vertex of \(G\). We identify a certain set of "tractable" matrices \(M\). We give an algorithm that counts list \(M\)partitions in polynomial time for every (fixed) matrix \(M\) in this set. The algorithm relies on data structures such as sparsedense partitions and subcube decompositions to reduce each problem instance to a sequence of problem instances in which the lists have a certain useful structure that restricts access to portions of \(M\) in which the interactions of 0s and 1s are controlled. We show how to solve the resulting restricted instances by converting them into particular counting constraint satisfaction problems (\({\#CSP}\)s), which we show how to solve using a constraint satisfaction technique known as arcconsistency. For every matrix \(M\) for which our algorithm fails, we show that the problem of counting list \(M\)partitions is \({\#P}\)complete. Furthermore, we give an explicit characterization of the dichotomy theorem: counting list \(M\)partitions is tractable (in \({FP}\)) if the matrix \(M\) has a structure called a derectangularizing sequence. If \(M\) has no derectangularizing sequence, we show that counting list \(M\)partitions is \({\#P}\)hard. We show that the metaproblem of determining whether a given matrix has a derectangularizing sequence is \({NP}\)complete. Finally, we show that list \(M\)partitions can be used to encode cardinality restrictions in \(M\)partitions problems, and we use this to give a polynomialtime algorithm for counting homogeneous pairs in graphs.

Göbel, Andreas; Goldberg, Leslie Ann; Richerby, David Counting Homomorphisms to SquareFree Graphs, Modulo 2. International Colloquium on Automata, Languages, and Programming (ICALP) 2015: 642653
We study the problem \( \oplus \)HomsToH of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph \(H\). A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (nonmodular) counting; thus, subtle dichotomy theorems can arise. We show the following dichotomy: for any \(H\) that contains no 4cycles, \(\oplus\)HomsToH is either in polynomial time or is \(\oplus\)Pcomplete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of treewidth2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs, including graphs of unbounded treewidth. In particular, we focus on squarefree graphs, which are graphs without 4cycles. These graphs arise frequently in combinatorics, for example, in connection with the strong perfect graph theorem and in certain graph algorithms. Previous dichotomy theorems required the graph to be treelike so that treelike decompositions could be exploited in the proof. We prove the conjecture for a much richer class of graphs by adopting a much more general approach.

Göbel, Andreas; Goldberg, Leslie Ann; Richerby, David The complexity of counting homomorphisms to cactus graphs modulo 2. Transactions on Computation Theory 2014: 17:117:29
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 in computer science can be represented naturally as graph homomorphisms and as weighted sums of graph homomorphisms. In this article, 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 graphtheoretic. 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 in the complexity class \(\oplus P\), 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.

Göbel, Andreas; Goldberg, Leslie Ann; McQuillan, Colin; Richerby, David; Yamakami, Tomoyuki Counting List Matrix Partitions of Graphs. Conference on Computational Complexity (CCC) 2014: 5665
Given a symmetric \(D \times D\) matrix \(M\) over \(0, 1, *}\), a list \(M\)partition of a graph \(G\) is a partition of the vertices of \(G\) into \(D\) parts which are associated with the rows of \(M\). The part of each vertex is chosen from a given list in such a way that no edge of \(G\) is mapped to a 0 in \(M\) and no nonedge of \(G\) is mapped to a 1 in \(M\). Many important graphtheoretic structures can be represented as list \(M\)partitions including graph colourings, split graphs and homogeneous sets, which arise in the proofs of the weak and strong perfect graph conjectures. Thus, there has been quite a bit of work on determining for which matrices \(M\) computations involving list \(M\)partitions are tractable. This paper focuses on the problem of counting list \(M\)partitions, given a graph \(G\) and given lists for each vertex of \(G\). We give an algorithm that solves this problem in polynomial time for every (fixed) matrix \(M\) for which the problem is tractable. The algorithm relies on data structures such as sparsedense partitions and sub cube decompositions to reduce each problem instance to a sequence of problem instances in which the lists have a certain useful structure that restricts access to portions of \(M\) in which the interactions of 0s and 1s is controlled. We show how to solve the resulting restricted instances by converting them into particular counting constraint satisfaction problems (\#CSPs) which we show how to solve using a constraint satisfaction technique known as "arcconsistency". For every matrix \(M\) for which our algorithm fails, we show that the problem of counting list \(M\)partitions is #Pcomplete. Furthermore, we give an explicit characterisation of the dichotomy theorem  counting list \(M\)partitions is tractable (in FP) if and only if the matrix \(M\) has a structure called a derectangularising sequence. Finally, we show that the metaproblem of determining whether a given matrix has a derectangularising sequence is NPcomplete.

Göbel, Andreas; Goldberg, Leslie Ann; Richerby, David Counting Homomorphisms to Cactus Graphs Modulo 2. Symposium on Theoretical Aspects of Computer Science (STACS) 2014: 350361
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 in computer science can be represented naturally as graph homomorphisms and as weighted sums of graph homomorphisms. In this article, 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 graphtheoretic. 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 in the complexity class \(\oplus P\), 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.