The Complexity of Approximately Counting Tree Homomorphisms

We study two computational problems, parameterised by a fixed tree H. #HomsTo(H) is the problem of counting homomorphisms from an input graph G to H. #WHomsTo(H) is the problem of counting weighted homomorphisms to H, given an input graph G and a weight function for each vertex v of G. Even though H is a tree, these problems turn out to be sufficiently rich to capture all of the known approximation behaviour in #P. We give a complete trichotomy for #WHomsTo(H). If H is a star then #WHomsTo(H) is in FP. If H is not a star but it does not contain a certain induced subgraph J_3 then #WHomsTo(H) is equivalent under approximation-preserving (AP) reductions to #BIS, the problem of counting independent sets in a bipartite graph. This problem is complete for the class #RHPi_1 under AP-reductions. Finally, if H contains an induced J_3 then #WHomsTo(H) is equivalent under AP-reductions to #SAT, the problem of counting satisfying assignments to a CNF Boolean formula. Thus, #WHomsTo(H) is complete for #P under AP-reductions. The results are similar for #HomsTo(H) except that a rich structure emerges if H contains an induced J_3. We show that there are trees H for which #HomsTo(H) is #SAT-equivalent (disproving a plausible conjecture of Kelk). There is an interesting connection between these homomorphism-counting problems and the problem of approximating the partition function of the ferromagnetic Potts model. In particular, we show that for a family of graphs J_q, parameterised by a positive integer q, the problem #HomsTo(H) is AP-interreducible with the problem of approximating the partition function of the q-state Potts model. It was not previously known that the Potts model had a homomorphism-counting interpretation. We use this connection to obtain some additional upper bounds for the approximation complexity of #HomsTo(J_q)

The Complexity of Change

Many combinatorial problems can be formulated as "Can I transform configuration 1 into configuration 2, if certain transformations only are allowed?". An example of such a question is: given two k-colourings of a graph, can I transform the first k-colouring into the second one, by recolouring one vertex at a time, and always maintaining a proper k-colouring? Another example is: given two solutions of a SAT-instance, can I transform the first solution into the second one, by changing the truth value one variable at a time, and always maintaining a solution of the SAT-instance? Other examples can be found in many classical puzzles, such as the 15-Puzzle and Rubik's Cube. In this survey we shall give an overview of some older and more recent work on this type of problem. The emphasis will be on the computational complexity of the problems: how hard is it to decide if a certain transformation is possible or not?Comment: 28 pages, 6 figure

Counting Constraint Satisfaction Problems

This chapter surveys counting Constraint Satisfaction Problems (counting CSPs, or #CSPs) and their computational complexity. It aims to provide an introduction to the main concepts and techniques, and present a representative selection of results and open problems. It does not cover holants, which are the subject of a separate chapter

The complexity of approximating bounded-degree Boolean #CSP

AbstractThe degree of a CSP instance is the maximum number of times that any variable appears in the scopes of constraints. We consider the approximate counting problem for Boolean CSP with bounded-degree instances, for constraint languages containing the two unary constant relations {0} and {1}. When the maximum allowed degree is large enough (at least 6) 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 NP=RP. For lower degree bounds, additional cases arise, where the complexity is related to the complexity of approximately counting independent sets in hypergraphs

The Complexity of Approximately Counting Retractions

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

Reconfigurations of Combinatorial Problems: Graph Colouring and Hamiltonian Cycle

We explore algorithmic aspects of two known combinatorial problems, Graph Colouring and Hamiltonian Cycle, by examining properties of their solution space. One can model the set of solutions of a combinatorial problem $P$ by the solution graph $R(P)$, where vertices are solutions of $P$ and there is an edge between two vertices, when the two corresponding solutions satisfy an adjacency reconfiguration rule. For example, we can define the reconfiguration rule for graph colouring to be that two solutions are adjacent when they differ in colour in exactly one vertex. The exploration of the properties of the solution graph $R(P)$ can give rise to interesting questions. The connectivity of $R(P)$ is the most prominent question in this research area. This is reasonable, since the main motivation for modelling combinatorial solutions as a graph is to be able to transform one into the other in a stepwise fashion, by following paths between solutions in the graph. Connectivity questions can be made binary, that is expressed as decision problems which accept a 'yes' or 'no' answer. For example, given two specific solutions, is there a path between them? Is the graph of solutions $R(P)$ connected? In this thesis, we first show that the diameter of the solution graph $R_{l}(G)$ of vertex $l$-colourings of k-colourable chordal and chordal bipartite graphs $G$ is $O(n^2)$, where $l > k$ and n is the number of vertices of $G$. Then, we formulate a decision problem on the connectivity of the graph colouring solution graph, where we allow extra colours to be used in order to enforce a path between two colourings with no path between them. We give some results for general instances and we also explore what kind of graphs pose a challenge to determine the complexity of the problem for general instances. Finally, we give a linear algorithm which decides whether there is a path between two solutions of the Hamiltonian Cycle Problem for graphs of maximum degree five, and thus providing insights towards the complexity classification of the decision problem

Counting Independent Sets and Colorings on Random Regular Bipartite Graphs

We give a fully polynomial-time approximation scheme (FPTAS) to count the number of independent sets on almost every Delta-regular bipartite graph if Delta >= 53. In the weighted case, for all sufficiently large integers Delta and weight parameters lambda = Omega~ (1/(Delta)), we also obtain an FPTAS on almost every Delta-regular bipartite graph. Our technique is based on the recent work of Jenssen, Keevash and Perkins (SODA, 2019) and we also apply it to confirm an open question raised there: For all q >= 3 and sufficiently large integers Delta=Delta(q), there is an FPTAS to count the number of q-colorings on almost every Delta-regular bipartite graph

Approximately counting and sampling small witnesses using a colourful decision oracle

In this paper, we prove "black box" results for turning algorithms which decide whether or not a witness exists into algorithms to approximately count the number of witnesses, or to sample from the set of witnesses approximately uniformly, with essentially the same running time. We do so by extending the framework of Dell and Lapinskas (STOC 2018), which covers decision problems that can be expressed as edge detection in bipartite graphs given limited oracle access; our framework covers problems which can be expressed as edge detection in arbitrary k-hypergraphs given limited oracle access. (Simulating this oracle generally corresponds to invoking a decision algorithm.) This includes many key problems in both the fine-grained setting (such as k-SUM, k-OV and weighted k-Clique) and the parameterised setting (such as induced subgraphs of size k or weight-k solutions to CSPs). From an algorithmic standpoint, our results will make the development of new approximate counting algorithms substantially easier; indeed, it already yields a new state-of-the-art algorithm for approximately counting graph motifs, improving on Jerrum and Meeks (JCSS 2015) unless the input graph is very dense and the desired motif very small. Our k-hypergraph reduction framework generalises and strengthens results in the graph oracle literature due to Beame et al. (ITCS 2018) and Bhattacharya et al. (CoRR abs/1808.00691)