More Like this

1. Near-Linear Time Homomorphism Counting in Bounded Degeneracy Graphs: The Barrier of Long Induced Cycles

   https://doi.org/10.1137/1.9781611976465.138  

   Bera, Suman K. ; Pashanasangi, Noujan ; Seshadhri, C. ( January 2021 , Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms)

   null (Ed.)

   Counting homomorphisms of a constant sized pattern graph H in an input graph G is a fundamental computational problem. There is a rich history of studying the complexity of this problem, under various constraints on the input G and the pattern H. Given the significance of this problem and the large sizes of modern inputs, we investigate when near-linear time algorithms are possible. We focus on the case when the input graph has bounded degeneracy, a commonly studied and practically relevant class for homomorphism counting. It is known from previous work that for certain classes of H, H-homomorphisms can be counted exactly in near-linear time in bounded degeneracy graphs. Can we precisely characterize the patterns H for which near-linear time algorithms are possible? We completely resolve this problem, discovering a clean dichotomy using fine-grained complexity. Let m denote the number of edges in G. We prove the following: if the largest induced cycle in H has length at most 5, then there is an O(m log m) algorithm for counting H-homomorphisms in bounded degeneracy graphs. If the largest induced cycle in H has length at least 6, then (assuming standard fine-grained complexity conjectures) there is a constant γ > 0, such that there is no o(m1+γ) time algorithm for counting H-homomorphisms. 

   more » « less
2. Hardness of approximation in p via short cycle removal: cycle detection, distance oracles, and beyond

   https://doi.org/10.1145/3519935.3520066  

   Abboud, Amir ; Bringmann, Karl ; Khoury, Seri ; Zamir, Or ( June 2022 , STOC 2022: Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing)

   We present a new technique for efficiently removing almost all short cycles in a graph without unintentionally removing its triangles. Consequently, triangle finding problems do not become easy even in almost k-cycle free graphs, for any constant k≥ 4. Triangle finding is at the base of many conditional lower bounds in P, mainly for distance computation problems, and the existence of many 4- or 5-cycles in a worst-case instance had been the obstacle towards resolving major open questions. Hardness of approximation: Are there distance oracles with m1+o(1) preprocessing time and mo(1) query time that achieve a constant approximation? Existing algorithms with such desirable time bounds only achieve super-constant approximation factors, while only 3− factors were conditionally ruled out (Pătraşcu, Roditty, and Thorup; FOCS 2012). We prove that no O(1) approximations are possible, assuming the 3-SUM or APSP conjectures. In particular, we prove that k-approximations require Ω(m1+1/ck) time, which is tight up to the constant c. The lower bound holds even for the offline version where we are given the queries in advance, and extends to other problems such as dynamic shortest paths. The 4-Cycle problem: An infamous open question in fine-grained complexity is to establish any surprising consequences from a subquadratic or even linear-time algorithm for detecting a 4-cycle in a graph. This is arguably one of the simplest problems without a near-linear time algorithm nor a conditional lower bound. We prove that Ω(m1.1194) time is needed for k-cycle detection for all k≥ 4, unless we can detect a triangle in √n-degree graphs in O(n2−δ) time; a breakthrough that is not known to follow even from optimal matrix multiplication algorithms. 

   more » « less
3. Towards a Decomposition-Optimal Algorithm for Counting and Sampling Arbitrary Motifs in Sublinear Time

   Biswas, A. S. ; Eden, T. ; Rubinfeld, R. ( January 2021 , Random)

   Counting and uniformly sampling motifs in a graph are fundamental algorithmic tasks with numerous applications across multiple fields. Since these problems are computationally expensive, recent efforts have focused on devising sublinear-time algorithms for these problems. We consider the model where the algorithm gets a constant size motif H and query access to a graph G, where the allowed queries are degree, neighbor, and pair queries, as well as uniform edge sample queries. In the sampling task, the algorithm is required to output a uniformly distributed copy of H in G (if one exists), and in the counting task it is required to output a good estimate to the number of copies of H in G. Previous algorithms for the uniform sampling task were based on a decomposition of H into a collection of odd cycles and stars, denoted D∗(H) = {Ok1 , ...,Okq , Sp1 , ..., Spℓ19 }. 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 arbitrary motifs which, up to poly(log n) factors, for any motif H whose decomposition contains at least two components or at least one star, is always preferable. The main ingredient leading to this improvement is an improved uniform algorithm for sampling stars, which might be of independent interest, as it allows to sample vertices according to the p-th moment of the degree distribution. We further show how to use our sampling algorithm to get an approximate counting algorithm, with essentially the same complexity. Finally, we prove that this algorithm is decomposition-optimal for decompositions that contain at least one odd cycle. That is, we prove that for any decomposition D that contains at least one odd cycle, there exists a motif HD 30 with decomposition D, and a family of graphs G, so that in order to output a uniform copy of H in a uniformly chosen graph in G, the number of required queries matches our upper bound. These are the first lower bounds for motifs H with a nontrivial decomposition, i.e., motifs that have more than a single component in their decomposition. 

   more » « less
4. The Complexity of Average-Case Dynamic Subgraph Counting

   https://doi.org/10.1137/1.9781611977073.23  

   Henzinger, Monika ; Lincoln, Andrea ; and Saha, Barna ( January 2022 , Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA))

   Statistics of small subgraph counts such as triangles, four-cycles, and s-t paths of short lengths reveal important structural properties of the underlying graph. These problems have been widely studied in social network analysis. In most relevant applications, the graphs are not only massive but also change dynamically over time. Most of these problems become hard in the dynamic setting when considering the worst case. In this paper, we ask whether the question of small subgraph counting over dynamic graphs is hard also in the average case. We consider the simplest possible average case model where the updates follow an Erdős-Rényi graph: each update selects a pair of vertices (u, v) uniformly at random and flips the existence of the edge (u, v). We develop new lower bounds and matching algorithms in this model for counting four-cycles, counting triangles through a specified point s, or a random queried point, and st paths of length 3, 4 and 5. Our results indicate while computing st paths of length 3, and 4 are easy in the average case with O(1) update time (note that they are hard in the worst case), it becomes hard when considering st paths of length 5. We introduce new techniques which allow us to get average-case hardness for these graph problems from the worst-case hardness of the Online Matrix vector problem (OMv). Our techniques rely on recent advances in fine-grained average-case complexity. Our techniques advance this literature, giving the ability to prove new lower bounds on average-case dynamic algorithms. Read More: https://epubs.siam.org/doi/abs/10.1137/1.9781611977073.23 

   more » « less
5. Improved Merlin–Arthur Protocols for Central Problems in Fine-Grained Complexity

   https://doi.org/10.1007/s00453-023-01102-6  

   Akmal, Shyan ; Chen, Lijie ; Jin, Ce ; Raj, Malvika ; Williams, Ryan ( February 2023 , Algorithmica)

   In a Merlin–Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin–Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include:

   Certifying that a list ofnintegers has no 3-SUM solution can be done in Merlin–Arthur time$$\tilde{O}(n)$$$\tilde{O}\left(n\right)$. Previously, Carmosino et al. [ITCS 2016] showed that the problem has a nondeterministic algorithm running in$$\tilde{O}(n^{1.5})$$$\tilde{O}\left(n^{1.5}\right)$time (that is, there is a proof system with proofs of length$$\tilde{O}(n^{1.5})$$$\tilde{O}\left(n^{1.5}\right)$and a deterministic verifier running in$$\tilde{O}(n^{1.5})$$$\tilde{O}\left(n^{1.5}\right)$time).

   Counting the number ofk-cliques with total edge weight equal to zero in ann-node graph can be done in Merlin–Arthur time$${\tilde{O}}(n^{\lceil k/2\rceil })$$$\tilde{O}\left(n^{\lceil k/2\rceil }\right)$(where$$k\ge 3$$$k\ge 3$). For oddk, this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in anm-edge graph can be done in Merlin–Arthur time$${\tilde{O}}(m)$$$\tilde{O}\left(m\right)$. Previous Merlin–Arthur protocols by Williams [CCC’16] and Björklund and Kaski [PODC’16] could only countk-cliques in unweighted graphs, and had worse running times for smallk.

   Computing the All-Pairs Shortest Distances matrix for ann-node graph can be done in Merlin–Arthur time$$\tilde{O}(n^2)$$$\tilde{O}\left(n^2\right)$. Note this is optimal, as the matrix can have$$\Omega (n^2)$$$\Omega \left(n^2\right)$nonzero entries in general. Previously, Carmosino et al. [ITCS 2016] showed that this problem has an$$\tilde{O}(n^{2.94})$$$\tilde{O}\left(n^{2.94}\right)$nondeterministic time algorithm.

   Certifying that ann-variablek-CNF is unsatisfiable can be done in Merlin–Arthur time$$2^{n/2 - n/O(k)}$$$2^{n/2-n/O\left(k\right)}$. We also observe an algebrization barrier for the previous$$2^{n/2}\cdot \textrm{poly}(n)$$$2^{n/2}·\text{poly}\left(n\right)$-time Merlin–Arthur protocol of R. Williams [CCC’16] for$$\#$$$\#$SAT: in particular, his protocol algebrizes, and we observe there is no algebrizing protocol fork-UNSAT running in$$2^{n/2}/n^{\omega (1)}$$$2^{n/2}/n^{\omega \left(1\right)}$time. Therefore we have to exploit non-algebrizing properties to obtain our new protocol.

   Certifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time$$2^{4n/5}\cdot \textrm{poly}(n)$$$2^{4n/5}·\text{poly}\left(n\right)$. Previously, the only nontrivial result known along these lines was an Arthur–Merlin–Arthur protocol (where Merlin’s proof depends on some of Arthur’s coins) running in$$2^{2n/3}\cdot \textrm{poly}(n)$$$2^{2n/3}·\text{poly}\left(n\right)$time.

   Due to the centrality of these problems in fine-grained complexity, our results have consequences for many other problems of interest. For example, our work implies that certifying there is no Subset Sum solution tonintegers can be done in Merlin–Arthur time$$2^{n/3}\cdot \textrm{poly}(n)$$$2^{n/3}·\text{poly}\left(n\right)$, improving on the previous best protocol by Nederlof [IPL 2017] which took$$2^{0.49991n}\cdot \textrm{poly}(n)$$$2^{0.49991n}·\text{poly}\left(n\right)$time.

    

   more » « less