The voter model is a classical stochastic process that models how opinions might spread through a network: at each step, every node lazily adopts the opinion of a random neighbour; eventually all nodes share the same opinion (consensus). Stronger connectivity should yield faster consensus. Berenbrink, Giakkoupis, Kermarrec, and Mallmann-Trenn (ICALP 2016) make this precise via the network's conductance: if the network has m edges, minimum degree d_min, and conductance at least ϕ, then the voter model reaches consensus in expected O(m/(d_minϕ)) steps. Their results extend to dynamic networks with fixed vertex degrees by considering the network's conductance at each time step. We introduce temporal conductance Φ, a more general connectivity measure for dynamic networks. Unlike static conductance, which collapses to 0 whenever some snapshot is disconnected, Φ captures connectivity through edges that appear at different times. We generalise the results of Berenbrink et al. from static conductance to temporal conductance, showing that the expected consensus time of the standard voter model is at most O(m/(d_minΦ)). Moreover, we prove that this bound is tight up to constant factors. We expect temporal conductance to be a useful primitive for analysing other dynamics on temporal networks, and potentially time-inhomogeneous Markov chains more generally.

For a graph G, a proper k-coloring of G is equitable if the sizes of any two color classes differ by at most one. The Equitable k-Coloring problem asks, for a given graph G and integer k, whether G admits an equitable k-coloring. Bodlaender and Fomin showed that it is polynomial-time solvable on graphs of bounded treewidth, while it remains -hard on cographs, and thus on graphs of constant clique-width. Fellows et al. showed that the problem becomes 𝖶[1]-hard when parameterized by tree-width (and hence clique-width) plus the number of colors k. We first show that, for every fixed k, counting equitable k-colorings is polynomial-time solvable on graph classes of bounded clique-width, given a clique-width expression. We then show that, under 𝖲𝖤𝖳𝖧, the dependence on clique-width in this algorithm is essentially optimal. As a consequence, our results provide a fairly tight picture of the complexity of Equitable k-Coloring with respect to the combined parameter k+clique-width. Second, we refine our clique-width algorithm for the linear setting. We show that there exists an algorithm, given an integer k≥ 1 and an n-vertex graph G together with a linear w-expression constructing G, computes the number of equitable k-colorings of G in time max{1,2^k-2}^w· n^k+O(1). Third, we consider a different structural restriction, namely the class of P_t-free graphs. A graph is called P_t-free if it does not contain the path on t vertices as an induced subgraph. This is a different setting from bounded clique-width; in particular, already P_5-free graphs have unbounded clique-width. Nevertheless, we show that for every P_t-free graph G, the number of equitable list 3-colorings of G can be computed in subexponential time.

Subgraph counting is a fundamental and well-studied problem whose computational complexity is well understood. Quite surprisingly, the hypergraph version of subgraph counting has been almost ignored. In this work, we address this gap by investigating the most basic sub-hypergraph counting problem: given a (small) hypergraph H and a (large) hypergraph G, compute the number of sub-hypergraphs of G isomorphic to H. Formally, for a family ℋ of hypergraphs, let #Sub(ℋ) be the restriction of the problem to H ∈ℋ; the induced variant #IndSub(ℋ) is defined analogously. Our main contribution is a complete classification of the complexity of these problems. Assuming the Exponential Time Hypothesis, we prove that #Sub(ℋ) is fixed-parameter tractable if and only if ℋ has bounded fractional co-independent edge-cover number, a novel graph parameter we introduce. Moreover, #IndSub(ℋ) is fixed-parameter tractable if and only if ℋ has bounded fractional edge-cover number. Both results subsume pre-existing results for graphs as special cases. We also show that the fixed-parameter tractable cases of #Sub(ℋ) and #IndSub(ℋ) are unlikely to be in polynomial time, unless respectively #P = P and Graph Isomorphism ∈ P. This shows a separation with the special case of graphs, where the fixed-parameter tractable cases are known to actually be in polynomial time.

We present a randomized algorithm for solving low-degree polynomial equation systems over finite fields faster than exhaustive search. In order to do so, we follow a line of work by Lokshtanov, Paturi, Tamaki, Williams, and Yu (SODA 2017), Bj\"orklund, Kaski, and Williams (ICALP 2019), and Dinur (SODA 2021). In particular, we generalize Dinur's algorithm for $\mathbb{F}_2$ to all finite fields, in particular the "symbolic interpolation" of Bj\"orklund, Kaski, and Williams, and we use an efficient trimmed multipoint evaluation and interpolation procedure for multivariate polynomials over finite fields by Van der Hoeven and Schost (AAECC 2013). The running time of our algorithm matches that of Dinur's algorithm for $\mathbb{F}_2$ and is significantly faster than the one of Lokshtanov et al. for $q>2$. We complement our results with tight conditional lower bounds that, surprisingly, we were not able to find in the literature. In particular, under the strong exponential time hypothesis, we prove that it is impossible to solve $n$-variate low-degree polynomial equation systems over $\mathbb{F}_q$ in time $O((q-\varepsilon)^{n})$. As a bonus, we show that under the counting version of the strong exponential time hypothesis, it is impossible to compute the number of roots of a single $n$-variate low-degree polynomial over $\mathbb{F}_q$ in time ${O((q-\varepsilon)^{n})}$; this generalizes a result of Williams (SOSA 2018) from $\mathbb{F}_2$ to all finite fields.

Graph representation learning is central for the application of machine learning (ML) models to complex graphs, such as social networks. Ensuring `fair' representations is essential, due to the societal implications and the use of sensitive personal data. In this paper, we demonstrate how the parametrization of the CrossWalk algorithm influences the ability to infer a sensitive attributes from node embeddings. By fine-tuning hyperparameters, we show that it is possible to either significantly enhance or obscure the detectability of these attributes. This functionality offers a valuable tool for improving the fairness of ML systems utilizing graph embeddings, making them adaptable to different fairness paradigms.

We study a query model of computation in which an n-vertex k-hypergraph can be accessed only via its independence oracle or via its colourful independence oracle, and each oracle query may incur a cost depending on the size of the query. In each of these models, we obtain oracle algorithms to approximately count the hypergraph's edges, and we unconditionally prove that no oracle algorithm for this problem can have significantly smaller worst-case oracle cost than our algorithms.

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).

In this article, we introduce a general framework for fine-grained reductions of approximate counting problems to their decision versions. (Thus, we use an oracle that decides whether any witness exists to multiplicatively approximate the number of witnesses with minimal overhead.) This mirrors a foundational result of Sipser (STOC 1983) and Stockmeyer (SICOMP 1985) in the polynomial-time setting, and a similar result of Müller (IWPEC 2006) in the FPT setting. Using our framework, we obtain such reductions for some of the most important problems in fine-grained complexity: the Orthogonal Vectors problem, 3SUM, and the Negative-Weight Triangle problem (which is closely related to All-Pairs Shortest Path). While all these problems have simple algorithms over which it is conjectured that no polynomial improvement is possible, our reductions would remain interesting even if these conjectures were proved; they have only polylogarithmic overhead and can therefore be applied to subpolynomial improvements such as the n3/ exp(Θ (√ log n))-time algorithm for the Negative-Weight Triangle problem due to Williams (STOC 2014). Our framework is also general enough to apply to versions of the problems for which more efficient algorithms are known. For example, the Orthogonal Vectors problem over GF(m)d for constant m can be solved in time n · poly (d) by a result of Williams and Yu (SODA 2014); our result implies that we can approximately count the number of orthogonal pairs with essentially the same running time. We also provide a fine-grained reduction from approximate #SAT to SAT. Suppose the Strong Exponential Time Hypothesis (SETH) is false, so that for some 1 < c < 2 and all k there is an O(cn)-time algorithm for k-SAT. Then we prove that for all k, there is an O(( c + o(1))n)-time algorithm for approximate #k-SAT. In particular, our result implies that the Exponential Time Hypothesis (ETH) is equivalent to the seemingly weaker statement that there is no algorithm to approximate #3-SAT to within a factor of 1+ɛ in time 2o(n)/ ɛ2 (taking ɛ > 0 as part of the input).

We systematically investigate the complexity of counting subgraph patterns modulo fixed integers. For example, it is known that the parity of the number of $k$-matchings can be determined in polynomial time by a simple reduction to the determinant. We generalize this to an $n^{f(t,s)}$-time algorithm to compute modulo $2^t$ the number of subgraph occurrences of patterns that are $s$ vertices away from being matchings. This shows that the known polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry over into the setting of counting modulo $2^t$. Complementing our algorithm, we also give a simple and self-contained proof that counting $k$-matchings modulo odd integers $q$ is Mod_q-W[1]-complete and prove that counting $k$-paths modulo $2$ is Parity-W[1]-complete, answering an open question by Björklund, Dell, and Husfeldt (ICALP 2015).

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 #RH\Pi_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 exhaustively 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 #RH\Pi_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.

Many graph parameters can be expressed as homomorphism counts to fixed target graphs; this includes the number of independent sets and the number of k-colorings for any fixed k. Dyer and Greenhill (RSA 2000) gave a sweeping complexity dichotomy for such problems, classifying which target graphs render the problem polynomial-time solvable or #P -hard. In this paper, we give a new and shorter proof of this theorem, with previously unknown tight lower bounds under the exponential-time hypothesis. We similarly strengthen complexity dichotomies by Focke, Goldberg, and Živný (SODA 2018) for counting surjective homomorphisms to fixed graphs. Both results crucially rely on our main contribution, a complexity dichotomy for evaluating linear combinations of homomorphism numbers to fixed graphs. In the terminology of Lovász (Colloquium Publications 2012), this amounts to counting homomorphisms to quantum graphs.

We consider the $\#\mathsf{W}[1]$-hard problem of counting all matchings with exactly $k$ edges in a given input graph $G$; we prove that it remains $\#\mathsf{W}[1]$-hard on graphs $G$ that are line graphs or bipartite graphs with degree $2$ on one side. In our proofs, we use that $k$-matchings in line graphs can be equivalently viewed as edge-injective homomorphisms from the disjoint union of $k$ length-$2$ paths into (arbitrary) host graphs. Here, a homomorphism from $H$ to $G$ is edge-injective if it maps any two distinct edges of $H$ to distinct edges in $G$. We show that edge-injective homomorphisms from a pattern graph $H$ can be counted in polynomial time if $H$ has bounded vertex-cover number after removing isolated edges. For hereditary classes $\mathcal{H}$ of pattern graphs, we complement this result: If the graphs in $\mathcal{H}$ have unbounded vertex-cover number even after deleting isolated edges, then counting edge-injective homomorphisms with patterns from $\mathcal{H}$ is $\#\mathsf{W}[1]$-hard. Our proofs rely on an edge-colored variant of Holant problems and a delicate interpolation argument; both may be of independent interest.

We consider the following natural "above guarantee" parameterization of the classical Longest Path problem: For given vertices s and t of a graph G, and an integer k, the problem Longest Detour asks for an (s,t)-path in G that is at least k longer than a shortest (s,t)-path. Using insights into structural graph theory, we prove that Longest Detour is fixed-parameter tractable (FPT) on undirected graphs and actually even admits a single-exponential algorithm, that is, one of running time exp(O(k)) poly(n). This matches (up to the base of the exponential) the best algorithms for finding a path of length at least k. Furthermore, we study the related problem Exact Detour that asks whether a graph G contains an (s,t)-path that is exactly k longer than a shortest (s,t)-path. For this problem, we obtain a randomized algorithm with running time about 2.746^k, and a deterministic algorithm with running time about 6.745^k, showing that this problem is FPT as well. Our algorithms for Exact Detour apply to both undirected and directed graphs.

Conjunctive queries select and are expected to return certain tuples from a relational database. We study the potentially easier problem of counting all selected tuples, rather than enumerating them. In particular, we are interested in the problem's parameterized and data complexity, where the query is considered to be small or fixed, and the database is considered to be large. We identify two structural parameters for conjunctive queries that capture their inherent complexity: The dominating star size and the linked matching number. If the dominating star size of a conjunctive query is large, then we show that counting solution tuples to the query is at least as hard as counting dominating sets, which yields a fine-grained complexity lower bound under the Strong Exponential Time Hypothesis as well as a #W[2]-hardness result. Moreover, if the linked matching number of a conjunctive query is large, then we show that the structure of the query is so rich that arbitrary queries up to a certain size can be encoded into it; this essentially establishes #A[2]-completeness. Using ideas stemming from Lovász, we lift complexity results from the class of conjunctive queries to arbitrary existential or universal formulas that might contain inequalities and negations on constraints over the free variables. As a consequence, we obtain a complexity classification that generalizes previous results of Chen, Durand, and Mengel (ToCS 2015; ICDT 2015; PODS 2016) for conjunctive queries and of Curticapean and Marx (FOCS 2014) for the subgraph counting problem. Our proof also relies on graph minors, and we show a strengthening of the Excluded-Grid-Theorem which might be of independent interest: If the linked matching number is large, then not only can we find a large grid somewhere in the graph, but we can find a large grid whose diagonal has disjoint paths leading into an assumed node-well-linked set.

Jaeger et al. (Math Proc Camb Philos Soc 108(1):35–53, 1990) proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: the evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahlén (in: ICALP 2010, vol. 6198, pp. 426–437, Springer, Berlin, Heidelberg, 2010) and Husfeldt and Taslaman (in: IPEC 2010, vol. 6478, pp. 192–203, Springer, Berlin, Heidelberg, 2010) in combination with the results of Curticapean (in: ICALP 2015, pp. 380–392, Springer, 2015), extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line $$y=1$$ , which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given n-vertex graph cannot be determined in time unless #ETH fails. Another dichotomy theorem we strengthen is the one of Creignou and Hermann (Inf Comput 125(1):1–12, 1996) for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that the #P-hard cases cannot be solved in time unless #ETH fails. The main ingredient is to prove that the number of independent sets in bipartite graphs with n vertices cannot be computed in time unless #ETH fails. In order to prove our results, we use the block interpolation idea by Curticapean and transfer it to systems of linear equations that might not directly correspond to interpolation.

In this paper, we relate a beautiful theory by Lov\'asz with a popular heuristic algorithm for the graph isomorphism problem, namely the color refinement algorithm and its $k$-dimensional generalization known as the Weisfeiler-Leman algorithm. We prove that two graphs $G$ and $H$ are indistinguishable by the color refinement algorithm if and only if, for all trees $T$, the number $\mathsf{hom}(T,G)$ of homomorphisms from $T$ to $G$ equals the corresponding number $\mathsf{hom}(T,H)$ for $H$. There is a natural system of linear equations whose nonnegative integer solutions correspond to the isomorphisms between two graphs. The non-negative real solutions to this system are called fractional isomorphisms, and two graphs are fractionally isomorphic if and only if the color refinement algorithm cannot distinguish them (Tinhofer 1986, 1991). We show that, if we drop the non-negativity constraints, that is, if we look for arbitrary real solutions, then a solution to the linear system exists if and only if, for all $t$, the two graphs have the same number of length-$t$ walks. We lift the results for trees to an equivalence between numbers of homomorphisms from graphs of tree width $k$, the $k$-dimensional Weisfeiler-Leman algorithm, and the level-$k$ Sherali-Adams relaxation of our linear program. We also obtain a partial result for graphs of path width $k$ and solutions to our system where we drop the non-negativity constraints. A consequence of our results is a quasi-linear time algorithm to decide whether, for two given graphs $G$ and $H$, there is a tree $T$ with $\mathsf{hom}(T,G)\neq\mathsf{hom}(T,H)$.

We devise an algorithm that approximately computes the number of paths of length k in a given directed graph with n vertices up to a multiplicative error of 1 ± ε. Our algorithm runs in time ε−2 4k(n+m) poly(k). The algorithm is based on associating with each vertex an element in the exterior (or, Grassmann) algebra, called an extensor, and then performing computations in this algebra. This connection to exterior algebra generalizes a number of previous approaches for the longest path problem and is of independent conceptual interest. Using this approach, we also obtain a deterministic 2k·poly(n) time algorithm to find a k-path in a given directed graph that is promised to have few of them. Our results and techniques generalize to the subgraph isomorphism problem when the subgraphs we are looking for have bounded pathwidth. Finally, we also obtain a randomized algorithm to detect k-multilinear terms in a multivariate polynomial given as a general algebraic circuit. To the best of our knowledge, this was previously only known for algebraic circuits not involving negative constants.

In this paper, we relate a beautiful theory by Lovász with a popular heuristic algorithm for the graph isomorphism problem, namely the color refinement algorithm and its k-dimensional generalization known as the Weisfeiler-Leman algorithm. We prove that two graphs G and H are indistinguishable by the color refinement algorithm if and only if, for all trees T, the number Hom(T,G) of homomorphisms from T to G equals the corresponding number Hom(T,H) for H. There is a natural system of linear equations whose nonnegative integer solutions correspond to the isomorphisms between two graphs. The nonnegative real solutions to this system are called fractional isomorphisms, and two graphs are fractionally isomorphic if and only if the color refinement algorithm cannot distinguish them (Tinhofer 1986, 1991). We show that, if we drop the nonnegativity constraints, that is, if we look for arbitrary real solutions, then a solution to the linear system exists if and only if, for all t, the two graphs have the same number of length-t walks. We lift the results for trees to an equivalence between numbers of homomorphisms from graphs of tree width k, the k-dimensional Weisfeiler-Leman algorithm, and the level-k Sherali-Adams relaxation of our linear program. We also obtain a partial result for graphs of bounded path width and solutions to our system where we drop the nonnegativity constraints. A consequence of our results is a quasi-linear time algorithm to decide whether, for two given graphs G and H, there is a tree T with Hom(T,G) = Hom(T,H).