Consider 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. Several recent results (Dell and Lapinskas, STOC 2018; Dell, Lapinskas, and Meeks, SODA 2020) give efficient algorithms to approximately count the hypergraph’s edges in the colourful setting. These algorithms immediately imply fine-grained reductions from approximate counting to decision, with overhead only log^Θ(k) n over the running time n^α of the original decision algorithm, for many well-studied problems including k-Orthogonal Vectors, k-SUM, subgraph isomorphism problems including k-Clique and colourful-H, graph motifs, and k-variable first-order model checking. We explore the limits of what is achievable in this setting, obtaining unconditional lower bounds on the oracle cost of algorithms to approximately count the hypergraph’s edges in both the colourful and uncoloured settings. In both settings, we also obtain algorithms which essentially match these lower bounds; in the colourful setting, this requires significant changes to the algorithm of Dell, Lapinskas, and Meeks (SODA 2020) and reduces the total overhead to log^{Θ(k-α)}n. Our lower bound for the uncoloured setting shows that there is no fine-grained reduction from approximate counting to the corresponding uncoloured decision problem (except in the case α ≥ k-1): without an algorithm for the colourful decision problem, we cannot hope to avoid the much larger overhead of roughly n^{(k-α)²/4}. The uncoloured setting has previously been studied for the special case k = 2 (Peled, Ramamoorthy, Rashtchian, Sinha, ITCS 2018; Chen, Levi, and Waingarten, SODA 2020), and our work generalises the existing algorithms and lower bounds for this special case to k > 2 and to oracles with cost.

Ranking functions such as PageRank assign numeric values (ranks) to nodes of graphs, most notably the web graph. Node rankings are an integral part of Internet search algorithms, since they can be used to order the results of queries. However, these ranking functions are famously subject to attacks by spammers, who modify the web graph in order to give their own pages more rank.

In contention resolution, multiple processors are trying to coordinate to send discrete messages through a shared channel with sharply limited communication. If two processors inadvertently send at the same time, the messages collide and are not transmitted successfully. An important case is acknowledgement-based contention resolution, in which processors cannot listen to the channel at all; all they know is whether or not their own messages have got through. This situation arises frequently in both networking and cloud computing. The most common acknowledgement-based protocols in practice are backoff protocols — variants of binary exponential backoff are used in both Ethernet and TCP/IP, and both Google Drive and AWS instruct their users to implement it to handle busy periods. In queueing models, where each processor has a queue of messages, stable backoff protocols are already known (Håstad et al., SICOMP 1996). In queue-free models, where each processor has a single message but processors arrive randomly, it is a long-standing conjecture of Aldous (IEEE Trans. Inf. Theory 1987) that no stable backoff protocols exist for any positive arrival rate of processors. Despite exciting recent results for full-sensing protocols which assume far greater listening capabilities of the processors (see e.g. Bender et al. STOC 2020 or Chen et al. PODC 2021), this foundational question remains open; here instability is only known in general when the arrival rate of processors is at least 0.42 (Goldberg et al. SICOMP 2004). We prove Aldous's conjecture for all backoff protocols outside of a tightly-constrained special case using a new domination technique to get around the main difficulty, which is the strong dependencies between messages.

In this paper, we design efficient algorithms to approximately count the number of edges of a given $k$-hypergraph, and to sample an approximately uniform random edge. The hypergraph is not given explicitly and can be accessed only through its colorful independence oracle: The colorful independence oracle returns yes or no depending on whether a given subset of the vertices contains an edge that is colorful with respect to a given vertex-coloring. Our results extend and/or strengthen recent results in the graph oracle literature due to Beame et al. [ ACM Trans. Algorithms, 16 (2020), 52], Dell and Lapinskas [ Proceedings of STOC, ACM, 2018, pp. 281--288], and Bhattacharya et al. [ Proceedings of ISAAC, 2019]. Our results have consequences for approximate counting/sampling: We can turn certain kinds of decision algorithms into approximate counting/sampling algorithms without causing much overhead in the running time. We apply this approximate counting/sampling-to-decision reduction to key problems in fine-grained complexity (such as $k$-SUM, $k$-OV, and weighted $k$-Clique) and parameterized complexity (such as induced subgraphs of size $k$ or weight-$k$ solutions to constraint satisfaction problems).

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).

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 $$\mathsf {\#RH}\Pi _1$$ . 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 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 $$\mathsf {\#RH}\Pi _1$$ , 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.

In this paper, 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(√logn))-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 0 as part of the input). A full version of this paper containing detailed proofs is available at https://arxiv.org/abs/1707.04609.

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 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. A formal definition is provided in the article. 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. 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.