Exact Distance Oracles for Planar Graphs with Failing Vertices

We consider exact distance oracles for directed weighted planar graphs in the presence of failing vertices. Given a source vertex $u$, a target vertex $v$ and a set $X$ of $k$ failed vertices, such an oracle returns the length of a shortest $u$-to-$v$ path that avoids all vertices in $X$. We propose oracles that can handle any number $k$ of failures. More specifically, for a directed weighted planar graph with $n$ vertices, any constant $k$, and for any $q \in [1,\sqrt n]$, we propose an oracle of size $\tilde{\mathcal{O}}(\frac{n^{k+3/2}}{q^{2k+1}})$ that answers queries in $\tilde{\mathcal{O}}(q)$ time. In particular, we show an $\tilde{\mathcal{O}}(n)$-size, $\tilde{\mathcal{O}}(\sqrt{n})$-query-time oracle for any constant $k$. This matches, up to polylogarithmic factors, the fastest failure-free distance oracles with nearly linear space. For single vertex failures ($k=1$), our $\tilde{\mathcal{O}}(\frac{n^{5/2}}{q^3})$-size, $\tilde{\mathcal{O}}(q)$-query-time oracle improves over the previously best known tradeoff of Baswana et al. [SODA 2012] by polynomial factors for $q = \Omega(n^t)$, $t \in (1/4,1/2]$. For multiple failures, no planarity exploiting results were previously known

Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs

As massive graphs become more prevalent, there is a rapidly growing need for scalable algorithms that solve classical graph problems, such as maximum matching and minimum vertex cover, on large datasets. For massive inputs, several different computational models have been introduced, including the streaming model, the distributed communication model, and the massively parallel computation (MPC) model that is a common abstraction of MapReduce-style computation. In each model, algorithms are analyzed in terms of resources such as space used or rounds of communication needed, in addition to the more traditional approximation ratio. In this paper, we give a single unified approach that yields better approximation algorithms for matching and vertex cover in all these models. The highlights include: * The first one pass, significantly-better-than-2-approximation for matching in random arrival streams that uses subquadratic space, namely a $(1.5+\epsilon)$-approximation streaming algorithm that uses $O(n^{1.5})$ space for constant $\epsilon > 0$. * The first 2-round, better-than-2-approximation for matching in the MPC model that uses subquadratic space per machine, namely a $(1.5+\epsilon)$-approximation algorithm with $O(\sqrt{mn} + n)$ memory per machine for constant $\epsilon > 0$. By building on our unified approach, we further develop parallel algorithms in the MPC model that give a $(1 + \epsilon)$-approximation to matching and an $O(1)$-approximation to vertex cover in only $O(\log\log{n})$ MPC rounds and $O(n/poly\log{(n)})$ memory per machine. These results settle multiple open questions posed in the recent paper of Czumaj~et.al. [STOC 2018]

When Algorithms for Maximal Independent Set and Maximal Matching Run in Sublinear Time

Maximal independent set (MIS), maximal matching (MM), and (Delta+1)-(vertex) coloring in graphs of maximum degree Delta 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 (Delta+1)-coloring that runs in 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 admits a sublinear-time algorithm in general graphs. In this work, we dig deeper into the possibility of achieving sublinear-time algorithms for MIS and MM. The neighborhood independence number of a graph G, denoted by beta(G), is the size of the largest independent set in the neighborhood of any vertex. We identify beta(G) as the "right" parameter to measure the runtime of MIS and MM algorithms: Although graphs of bounded neighborhood independence may be very dense (clique is one example), we prove that carefully chosen variants of greedy algorithms for MIS and MM run in O(n beta(G)) and O(n log{n} * beta(G)) time respectively on any n-vertex graph G. We complement this positive result by observing that a simple extension of the lower bound of Assadi et al. implies that Omega(n beta(G)) time is also necessary for any algorithm to either problem for all values of beta(G) from 1 to Theta(n). We note that our algorithm for MIS is deterministic while for MM we use randomization which we prove is unavoidable: any deterministic algorithm for MM requires Omega(n^2) time even for beta(G) = 2. Graphs with bounded neighborhood independence, already for constant beta = beta(G), constitute a rich family of possibly dense graphs, including line graphs, proper interval graphs, unit-disk graphs, claw-free graphs, and graphs of bounded growth. Our results suggest that even though MIS and MM do not admit sublinear-time algorithms in general graphs, one can still solve both problems in sublinear time for a wide range of beta(G) << n. Finally, by observing that the lower bound of Omega(n sqrt{n}) time for (Delta+1)-coloring due to Assadi et al. applies to graphs of (small) constant neighborhood independence, we unveil an intriguing separation between the time complexity of MIS and MM, and that of (Delta+1)-coloring: while the time complexity of MIS and MM is strictly higher than that of (Delta+1) coloring in general graphs, the exact opposite relation holds for graphs with small neighborhood independence

Brief announcement: (1+)-approximate shortest paths in dynamic streams

Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of 2k - 1 versus n1+1/k , for an integer parameter k. (In fact, existing solutions also incur an extra factor of 1+ in the stretch for weighted graphs, and an additional factor of logO(1) n in the space.) The only existing solution of the second type uses n1/2-O(1/k) passes over the stream (for space O(n1+1/k )), and applies only to unweighted graphs

Expander Decomposition in Dynamic Streams

In this paper we initiate the study of expander decompositions of a graph G = (V, E) in the streaming model of computation. The goal is to find a partitioning ? of vertices V such that the subgraphs of G induced by the clusters C ? ? are good expanders, while the number of intercluster edges is small. Expander decompositions are classically constructed by a recursively applying balanced sparse cuts to the input graph. In this paper we give the first implementation of such a recursive sparsest cut process using small space in the dynamic streaming model. Our main algorithmic tool is a new type of cut sparsifier that we refer to as a power cut sparsifier - it preserves cuts in any given vertex induced subgraph (or, any cluster in a fixed partition of V) to within a (?, ?)-multiplicative/additive error with high probability. The power cut sparsifier uses O?(n/??) space and edges, which we show is asymptotically tight up to polylogarithmic factors in n for constant ?