In this work, we consider the problem of sampling a $k$-clique in a graph from an almost uniform distribution in sublinear time in the general graph query model. Specifically the algorithm should output each $k$-clique with probability $(1\pm ε)/n_k$, where $n_k$ denotes the number of $k$-cliques in the graph and $ε$ is a given approximation parameter.
We prove that the query complexity of this problem is \[ Θ^*\left(\max\left\{ \left(\frac{(nα)^{k/2}}{ n_k}\right)^{\frac{1}{k-1}} ,\; \min\left\{nα,\frac{nα^{k-1}}{n_k} \right\}\right\}\right). \] where $n$ is the number of vertices in the graph, $α$ is its arboricity, and $Θ^*$ suppresses the dependence on $(\log n/ε)^{O(k)}$. Interestingly, this establishes a separation between approximate counting and approximate uniform sampling in the sublinear regime. For example, if $k=3$, $α= O(1)$, and $n_3$ (the number of triangles) is $Θ(n)$, then we get a lower bound of $Ω(n^{1/4})$ (for constant $ε$), while under these conditions, a $(1\pm ε)$-approximation of $n_3$ can be obtained by performing $\textrm{poly}(\log(n/ε))$ queries (Eden, Ron and Seshadhri, SODA20).
Our lower bound follows from a construction of a family of graphs with arboricity $α$ such that in each graph there are $n_k$ cliques (of size $k$), where one of these cliques is "hidden" and hence hard to sample. Our upper bound is based on defining a special auxiliary graph $H_k$, such that sampling edges almost uniformly in $H_k$ translates to sampling $k$-cliques almost uniformly in the original graph $G$. We then build on a known edge-sampling algorithm (Eden, Ron and Rosenbaum, ICALP19) to sample edges in $H_k$, where the challenge is simulate queries to $H_k$ while being given access only to $G$.
Given query access to an undirected graph $G$, we consider the problem of computing a $(1\pmε)$-approximation of the number of $k$-cliques in $G$. The standard query model for general graphs allows for degree queries, neighbor queries, and pair queries. Let $n$ be the number of vertices, $m$ be the number of edges, and $n_k$ be the number of $k$-cliques. Previous work by Eden, Ron and Seshadhri (STOC 2018) gives an $O^*(\frac{n}{n^{1/k}_k} + \frac{m^{k/2}}{n_k})$-time algorithm for this problem (we use $O^*(\cdot)$ to suppress $\poly(\log n, 1/ε, k^k)$ dependencies). Moreover, this bound is ne…
We study the problem of approximating the number of $k$-cliques in a graph when given query access to the graph. We consider the standard query model for general graphs via (1) degree queries, (2) neighbor queries and (3) pair queries. Let $n$ denote the number of vertices in the graph, $m$ the number of edges, and $C_k$ the number of $k$-cliques. We design an algorithm that outputs a $(1+\varepsilon)$-approximation (with high probability) for $C_k$, whose expected query complexity and running time are $O\left(\frac{n}{C_k^{1/k}}+\frac{m^{k/2}}{C_k}\right)\poly(\log n,1/\varepsilon,k)$. Hence,…
We study the planted clique problem in which a clique of size k is planted in an Erdos-Renyi graph G(n,1/2) and one is interested in recovering this planted clique. It is widely believed that it exhibits a statistical-computational gap when computational efficiency is equated with the existence of polynomial time algorithms. We study this problem under a more fine-grained computational lens and consider the following two questions. 1. Do there exist sublinear time algorithms for recovering the planted clique? 2. What is the smallest running time any algorithm can hope to have? We show that bec…
We present a simple sublinear-time algorithm for sampling an arbitrary subgraph $H$ \emph{exactly uniformly} from a graph $G$ with $m$ edges, to which the algorithm has access by performing the following types of queries: (1) degree queries, (2) neighbor queries, (3) pair queries and (4) edge sampling queries. The query complexity and running time of our algorithm are $\tilde{O}(\min\{m, \frac{m^{ρ(H)}}{\# H}\})$ and $\tilde{O}(\frac{m^{ρ(H)}}{\# H})$, respectively, where $ρ(H)$ is the fractional edge-cover of $H$ and $\# H$ is the number of copies of $H$ in $G$. For any clique on $r$ vertices…
The goal of the paper is to give fine-grained hardness results for the Subgraph Isomorphism (SI) problem for fixed size induced patterns $H$, based on the $k$-Clique hypothesis that the current best algorithms for Clique are optimal. Our first main result is that for any pattern graph $H$ that is a {\em core}, the SI problem for $H$ is at least as hard as $t$-Clique, where $t$ is the size of the largest clique minor of $H$. This improves (for cores) the previous known results [Dalirrooyfard-Vassilevska W. STOC'20] that the SI for $H$ is at least as hard as $k$-clique where $k$ is the size of t…
Quasi-cliques are dense incomplete subgraphs of a graph that generalize the notion of cliques. Enumerating quasi-cliques from a graph is a robust way to detect densely connected structures with applications to bio-informatics and social network analysis. However, enumerating quasi-cliques in a graph is a challenging problem, even harder than the problem of enumerating cliques. We consider the enumeration of top-k degree-based quasi-cliques, and make the following contributions: (1) We show that even the problem of detecting if a given quasi-clique is maximal (i.e. not contained within another …
In the subgraph counting problem, we are given a input graph $G(V, E)$ and a target graph $H$; the goal is to estimate the number of occurrences of $H$ in $G$. Our focus here is on designing sublinear-time algorithms for approximately counting occurrences of $H$ in $G$ in the setting where the algorithm is given query access to $G$. This problem has been studied in several recent papers which primarily focused on specific families of graphs $H$ such as triangles, cliques, and stars. However, not much is known about approximate counting of arbitrary graphs $H$. This is in sharp contrast to the …
We revisit the complexity of the classical Interval Scheduling in the dynamic setting. In this problem, the goal is to maintain a set of intervals under insertions and deletions and report the size of the maximum size subset of pairwise disjoint intervals after each update. Nontrivial approximation algorithms are known for this problem, for both the unweighted and weighted versions [Henzinger, Neumann, Wiese, SoCG 2020]. Surprisingly, it was not known if the general exact version admits an exact solution working in sublinear time, that is, without recomputing the answer after each update. Our …
In the Max $r$-SAT problem, the input is a CNF formula with $n$ variables where each clause is a disjunction of at most $r$ literals. The objective is to compute an assignment which satisfies as many of the clauses as possible. While there are a large number of polynomial-time approximation algorithms for this problem, we take the viewpoint of space complexity following [Biswas et al., Algorithmica 2021] and design sublinear-space approximation algorithms for the problem. We show that the classical algorithm of [Lieberherr and Specker, JACM 1981] can be implemented to run in $n^{O(1)}$ time wh…
Clique and near-clique counts are important graph properties with applications in graph generation, graph modeling, graph analytics, community detection among others. They are the archetypal examples of dense subgraphs. While there are several different definitions of near-cliques, most of them share the attribute that they are cliques that are missing a small number of edges. Clique counting is itself considered a challenging problem. Counting near-cliques is significantly harder more so since the search space for near-cliques is orders of magnitude larger than that of cliques. We give a form…
Maximal independent set (MIS), maximal matching (MM), and $(Δ+1)$-coloring in graphs of maximum degree $Δ$ are among the most prominent algorithmic graph theory problems. They are all solvable by a simple linear-time greedy algorithm and up until very recently this constituted the state-of-the-art. In SODA 2019, Assadi, Chen, and Khanna gave a randomized algorithm for $(Δ+1)$-coloring that runs in $\widetilde{O}(n\sqrt{n})$ time, which even for moderately dense graphs is sublinear in the input size. The work of Assadi et al. however contained a spoiler for MIS and MM: neither problems provably…
We consider the problem of sampling and approximately counting an arbitrary given motif $H$ in a graph $G$, where access to $G$ is given via queries: degree, neighbor, and pair, as well as uniform edge sample queries. Previous algorithms for these tasks were based on a decomposition of $H$ into a collection of odd cycles and stars, denoted $\mathcal{D}^*(H)=\{O_{k_1}, \ldots, O_{k_q}, S_{p_1}, \ldots, S_{p_\ell}\}$. These algorithms were shown to be optimal for the case where $H$ is a clique or an odd-length cycle, but no other lower bounds were known. We present a new algorithm for sampling a…
We present deterministic constant-round protocols for the graph connectivity problem in the model where each of the $n$ nodes of a graph receives a row of the adjacency matrix, and broadcasts a single sublinear size message to all other nodes. Communication rounds are synchronous. This model is sometimes called the broadcast congested clique. Specifically, we exhibit a deterministic protocol that computes the connected components of the input graph in $\lceil 1/ε\rceil$ rounds, each player communicating $\mathcal{O}(n^ε \cdot \log n)$ bits per round, with $0 < ε\leq 1$. We also provide a deter…
Estimating the length of the longest increasing subsequence (LIS) in an array is a problem of fundamental importance. Despite the significance of the LIS estimation problem and the amount of attention it has received, there are important aspects of the problem that are not yet fully understood. There are no better lower bounds for LIS estimation than the obvious bounds implied by testing monotonicity (for adaptive or nonadaptive algorithms). In this paper, we give the first nontrivial lower bound on the complexity of LIS estimation, and also provide novel algorithms that complement our lower b…
Estimating the number of subgraphs in data streams is a fundamental problem that has received great attention in the past decade. In this paper, we give improved streaming algorithms for approximately counting the number of occurrences of an arbitrary subgraph $H$, denoted $\# H$, when the input graph $G$ is represented as a stream of $m$ edges. To obtain our algorithms, we provide a generic transformation that converts constant-round sublinear-time graph algorithms in the query access model to constant-pass sublinear-space graph streaming algorithms. Using this transformation, we obtain the f…
We study the problem of coloring a given graph using a small number of colors in several well-established models of computation for big data. These include the data streaming model, the general graph query model, the massively parallel computation (MPC) model, and the CONGESTED-CLIQUE and the LOCAL models of distributed computation. On the one hand, we give algorithms with sublinear complexity, for the appropriate notion of complexity in each of these models. Our algorithms color a graph $G$ using about $κ(G)$ colors, where $κ(G)$ is the degeneracy of $G$: this parameter is closely related to …
Quality control in sublinear time: a case study via random graphs
School of Engineering and Applied Sciences, Harvard University（哈佛大学工程与应用科学学院）；Computer Science and Artificial Intelligence Laboratory, MIT（麻省理工学院计算机科学与人工智能实验室）
A polynomial Turing kernel for some parameterized problem $P$ is a polynomial-time algorithm that solves $P$ using queries to an oracle of $P$ whose sizes are upper-bounded by some polynomial in the parameter. Here the term "polynomial" refers to the bound on the query sizes, as the running time of any kernel is required to be polynomial. One of the most important open goals in parameterized complexity is to understand the applicability and limitations of polynomial Turing Kernels. As any fixed-parameter tractable problem admits a Turing kernel of some size, the focus has mostly being on deter…
Subgraph Isomorphism is a very basic graph problem, where given two graphs $G$ and $H$ one is to check whether $G$ is a subgraph of $H$. Despite its simple definition, the Subgraph Isomorphism problem turns out to be very broad, as it generalizes problems such as Clique, $r$-Coloring, Hamiltonicity, Set Packing and Bandwidth. However, for all of the mentioned problems $2^{\mathcal{O}(n)}$ time algorithms exist, so a natural and frequently asked question in the past was whether there exists a $2^{\mathcal{O}(n)}$ time algorithm for Subgraph Isomorphism. In the monograph of Fomin and Kratsch [Sp…
Over the past decade, there has been increasing interest in distributed/parallel algorithms for processing large-scale graphs. By now, we have quite fast algorithms -- usually sublogarithmic-time and often $poly(\log\log n)$-time, or even faster -- for a number of fundamental graph problems in the massively parallel computation (MPC) model. This model is a widely-adopted theoretical abstraction of MapReduce style settings, where a number of machines communicate in an all-to-all manner to process large-scale data. Contributing to this line of work on MPC graph algorithms, we present $poly(\log …
We investigate computational problems involving large weights through the lens of kernelization, which is a framework of polynomial-time preprocessing aimed at compressing the instance size. Our main focus is the weighted Clique problem, where we are given an edge-weighted graph and the goal is to detect a clique of total weight equal to a prescribed value. We show that the weighted variant, parameterized by the number of vertices $n$, is significantly harder than the unweighted problem by presenting an $O(n^{3 - \varepsilon})$ lower bound on the size of the kernel, under the assumption that N…
Bidimensionality is the most common technique to design subexponential-time parameterized algorithms on special classes of graphs, particularly planar graphs. The core engine behind it is a combinatorial lemma of Robertson, Seymour and Thomas that states that every planar graph either has a $\sqrt{k}\times \sqrt{k}$-grid as a minor, or its treewidth is $O(\sqrt{k})$. However, bidimensionality theory cannot be extended directly to several well-known classes of geometric graphs. Nevertheless, a relaxation of this lemma has been proven useful for unit disk graphs. Inspired by this, we prove a new…
In the distributed subgraph-freeness problem, we are given a graph $H$, and asked to determine whether the network graph contains $H$ as a subgraph or not. Subgraph-freeness is an extremely local problem: if the network had no bandwidth constraints, we could detect any subgraph $H$ in $|H|$ rounds, by having each node of the network learn its entire $|H|$-neighborhood. However, when bandwidth is limited, the problem becomes harder. Upper and lower bounds in the presence of congestion have been established for several classes of subgraphs, including cycles, trees, and more complicated subgraphs…
We study property testing in the context of distributed computing, under the classical CONGEST model. It is known that testing whether a graph is triangle-free can be done in a constant number of rounds, where the constant depends on how far the input graph is from being triangle-free. We show that, for every connected 4-node graph H, testing whether a graph is H-free can be done in a constant number of rounds too. The constant also depends on how far the input graph is from being H-free, and the dependence is identical to the one in the case of testing triangles. Hence, in particular, testing…
A bond of a graph $G$ is an inclusion-wise minimal disconnecting set of $G$, i.e., bonds are cut-sets that determine cuts $[S,V\setminus S]$ of $G$ such that $G[S]$ and $G[V\setminus S]$ are both connected. Given $s,t\in V(G)$, an $st$-bond of $G$ is a bond whose removal disconnects $s$ and $t$. Contrasting with the large number of studies related to maximum cuts, there are very few results regarding the largest bond of general graphs. In this paper, we aim to reduce this gap on the complexity of computing the largest bond and the largest $st$-bond of a graph. Although cuts and bonds are simil…
In recent years, crowdsourcing, aka human aided computation has emerged as an effective platform for solving problems that are considered complex for machines alone. Using human is time-consuming and costly due to monetary compensations. Therefore, a crowd based algorithm must judiciously use any information computed through an automated process, and ask minimum number of questions to the crowd adaptively. One such problem which has received significant attention is {\em entity resolution}. Formally, we are given a graph $G=(V,E)$ with unknown edge set $E$ where $G$ is a union of $k$ (again un…
The cut-set $\partial(S)$ of a graph $G=(V,E)$ is the set of edges that have one endpoint in $S\subset V$ and the other endpoint in $V\setminus S$, and whenever $G[S]$ is connected, the cut $[S,V\setminus S]$ of $G$ is called a connected cut. A bond of a graph $G$ is an inclusion-wise minimal disconnecting set of $G$, i.e., bonds are cut-sets that determine cuts $[S,V\setminus S]$ of $G$ such that $G[S]$ and $G[V\setminus S]$ are both connected. Contrasting with a large number of studies related to maximum cuts, there exist very few results regarding the largest bond of general graphs. In this…
