More Like this

1. Time-Message Trade-Offs in Distributed Algorithms

   https://doi.org/10.4230/LIPIcs.DISC.2018.32  

   Gmyr, Robert ; Pandurangan, Gopal ( January 2018 , 32nd International Symposium on Distributed Computing (DISC 2018))

   This paper focuses on showing time-message trade-offs in distributed algorithms for fundamental problems such as leader election, broadcast, spanning tree (ST), minimum spanning tree (MST), minimum cut, and many graph verification problems. We consider the synchronous CONGEST distributed computing model and assume that each node has initial knowledge of itself and the identifiers of its neighbors - the so-called KT_1 model - a well-studied model that also naturally arises in many applications. Recently, it has been established that one can obtain (almost) singularly optimal algorithms, i.e., algorithms that have simultaneously optimal time and message complexity (up to polylogarithmic factors), for many fundamental problems in the standard KT_0 model (where nodes have only local knowledge of themselves and not their neighbors). The situation is less clear in the KT_1 model. In this paper, we present several new distributed algorithms in the KT_1 model that trade off between time and message complexity. Our distributed algorithms are based on a uniform and general approach which involves constructing a sparsified spanning subgraph of the original graph - called a danner - that trades off the number of edges with the diameter of the sparsifier. In particular, a key ingredient of our approach is amore »distributed randomized algorithm that, given a graph G and any delta in [0,1], with high probability constructs a danner that has diameter O~(D + n^{1-delta}) and O~(min{m,n^{1+delta}}) edges in O~(n^{1-delta}) rounds while using O~(min{m,n^{1+delta}}) messages, where n, m, and D are the number of nodes, edges, and the diameter of G, respectively. Using our danner construction, we present a family of distributed randomized algorithms for various fundamental problems that exhibit a trade-off between message and time complexity and that improve over previous results. Specifically, we show the following results (all hold with high probability) in the KT_1 model, which subsume and improve over prior bounds in the KT_1 model (King et al., PODC 2014 and Awerbuch et al., JACM 1990) and the KT_0 model (Kutten et al., JACM 2015, Pandurangan et al., STOC 2017 and Elkin, PODC 2017): 1) Leader Election, Broadcast, and ST. These problems can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,1]. 2) MST and Connectivity. These problems can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,0.5]. In particular, for delta = 0.5 we obtain a distributed MST algorithm that runs in optimal O~(D+sqrt{n}) rounds and uses O~(min{m,n^{3/2}}) messages. We note that this improves over the singularly optimal algorithm in the KT_0 model that uses O~(D+sqrt{n}) rounds and O~(m) messages. 3) Minimum Cut. O(log n)-approximate minimum cut can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,0.5]. 4) Graph Verification Problems such as Bipartiteness, Spanning Subgraph etc. These can be solved in O~(D+n^{1-delta}) rounds using O~(min{m,n^{1+delta}}) messages for any delta in [0,0.5].« less
2. Bipartite Matching in Nearly-linear Time on Moderately Dense Graphs

   https://doi.org/10.1109/FOCS46700.2020.00090  

   van den Brand, Jan ; Lee, Yin-Tat ; Nanongkai, Danupon ; Peng, Richard ; Saranurak, Thatchaphol ; Sidford, Aaron ; Song, Zhao ; Wang, Di ( November 2020 , 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020)

   We present an $\tilde O(m+n^{1.5})$-time randomized algorithm for maximum cardinality bipartite matching and related problems (e.g. transshipment, negative-weight shortest paths, and optimal transport) on $m$-edge, $n$-node graphs. For maximum cardinality bipartite matching on moderately dense graphs, i.e. $m = \Omega(n^{1.5})$, our algorithm runs in time nearly linear in the input size and constitutes the first improvement over the classic $O(m\sqrt{n})$-time [Dinic 1970; Hopcroft-Karp 1971; Karzanov 1973] and $\tilde O(n^\omega)$-time algorithms [Ibarra-Moran 1981] (where currently $\omega\approx 2.373$). On sparser graphs, i.e. when $m = n^{9/8 + \delta}$ for any constant $\delta>0$, our result improves upon the recent advances of [Madry 2013] and [Liu-Sidford 2020b, 2020a] which achieve an $\tilde O(m^{4/3+o(1)})$ runtime. We obtain these results by combining and advancing recent lines of research in interior point methods (IPMs) and dynamic graph algorithms. First, we simplify and improve the IPM of [v.d.Brand-Lee-Sidford-Song 2020], providing a general primal-dual IPM framework and new sampling-based techniques for handling infeasibility induced by approximate linear system solvers. Second, we provide a simple sublinear-time algorithm for detecting and sampling high-energy edges in electric flows on expanders and show that when combined with recent advances in dynamic expander decompositions, this yields efficient data structures for maintaining the iterates ofmore »both [v.d.Brand~et~al.] and our new IPMs. Combining this general machinery yields a simpler $\tilde O(n \sqrt{m})$ time algorithm for matching based on the logarithmic barrier function, and our state-of-the-art $\tilde O(m+n^{1.5})$ time algorithm for matching based on the [Lee-Sidford 2014] barrier (as regularized in [v.d.Brand~et~al.]).« less
3. Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers

   https://doi.org/10.1109/FOCS46700.2020.00109  

   Chen, Li ; Goranci, Gramoz ; Henzinger, Monika ; Peng, Richard ; Saranurak, Thatchaphol ( November 2020 , 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020)

   We present a general framework of designing efficient dynamic approximate algorithms for optimization on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers, gives data structures that maintain approximate solutions in sub-linear update and query time. We illustrate the applicability of our paradigm to the following problems. (1) A fully-dynamic algorithm that approximates all-pair maximum-flows/minimum-cuts up to a nearly logarithmic factor in $\tilde{O}(n^{2/3})$ amortized time against an oblivious adversary, and $\tilde{O}(m^{3/4})$ time against an adaptive adversary. (2) An incremental data structure that maintains $O(1)$-approximate shortest path in $n^{o(1)}$ time per operation, as well as fully dynamic approximate all-pair shortest path and transshipment in $\tilde{O}(n^{2/3+o(1)})$ amortized time per operation. (3) A fully-dynamic algorithm that approximates all-pair effective resistance up to an $(1+\eps)$ factor in $\tilde{O}(n^{2/3+o(1)} \epsilon^{-O(1)})$ amortized update time per operation. The key tool behind result (1) is the dynamic maintenance of an algorithmic construction due to Madry [FOCS' 10], which partitions a graph into a collection of simpler graph structures (known as j-trees) and approximately captures the cut-flow and metric structure of the graph. The $O(1)$-approximation guarantee of (2) is by adapting the distance oracles by [Thorup-Zwick JACM `05].more »Result (3) is obtained by invoking the random-walk based spectral vertex sparsifier by [Durfee et al. STOC `19] in a hierarchical manner, while carefully keeping track of the recourse among levels in the hierarchy.« less
4. 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-Pairsmore »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.

   « less
5. Towards Tighter Space Bounds for Counting Triangles and Other Substructures in Graph Streams

   Bera, Suman K ; Chakrabarti, Amit ( March 2017 , 34th Symposium on Theoretical Aspects of Computer Science)

   We revisit the much-studied problem of space-efficiently estimating the number of triangles in a graph stream, and extensions of this problem to counting fixed-sized cliques and cycles, obtaining a number of new upper and lower bounds. For the important special case of counting triangles, we give a $4$-pass, $(1\pm\varepsilon)$-approximate, randomized algorithm that needs at most $\widetilde{O}(\varepsilon^{-2}\cdot m^{3/2}/T)$ space, where $m$ is the number of edges and $T$ is a promised lower bound on the number of triangles. This matches the space bound of a very recent algorithm (McGregor et al., PODS 2016), with an arguably simpler and more general technique. We give an improved multi-pass lower bound of $\Omega(\min\{m^{3/2}/T, m/\sqrt{T}\})$, applicable at essentially all densities $\Omega(n) \le m \le O(n^2)$. We also prove other multi-pass lower bounds in terms of various structural parameters of the input graph. Together, our results resolve a couple of open questions raised in recent work (Braverman et al., ICALP 2013). Our presentation emphasizes more general frameworks, for both upper and lower bounds. We give a sampling algorithm for counting arbitrary subgraphs and then improve it via combinatorial means in the special cases of counting odd cliques and odd cycles. Our results show that these problemsmore »are considerably easier in the cash-register streaming model than in the turnstile model, where previous work had focused (Manjunath et al., ESA 2011; Kane et al., ICALP 2012). We use Tur{\'a}n graphs and related gadgets to derive lower bounds for counting cliques and cycles, with triangle-counting lower bounds following as a corollary.« less