Lower Bounds for Subgraph Detection in the CONGEST Model

In the subgraph-freeness problem, we are given a constant-sized graph H, and wish to de- termine whether the network graph contains H as a subgraph or not. Until now, the only lower bounds on subgraph-freeness known for the CONGEST model were for cycles of length greater than 3; here we extend and generalize the cycle lower bound, and obtain polynomial lower bounds for subgraph-freeness in the CONGEST model for two classes of subgraphs. The first class contains any graph obtained by starting from a 2-connected graph H for which we already know a lower bound, and replacing the vertices of H by arbitrary connected graphs. We show that the lower bound on H carries over to the new graph. The second class is constructed by starting from a cycle Ck of length k ? 4, and constructing a graph H ? from Ck by replacing each edge {i, (i + 1) mod k} of the cycle with a connected graph Hi, subject to some constraints on the graphs H_{0}, . . .H_{k?1}. In this case we obtain a polynomial lower bound for the new graph H ?, depending on the size of the shortest cycle in H ? passing through the vertices of the original k-cycle

Approximating Minimum-Size 2-Edge-Connected and 2-Vertex-Connected Spanning Subgraphs

We study the unweighted 2-edge-connected and 2-vertex-connected spanning subgraph problems. A graph is 2-edge-connected if it is connected on removal of an edge, and it is 2-vertex-connected if it is connected on removal of a vertex. The problem of finding a minimum-size 2-edge-connected (or 2-vertex-connected) spanning subgraph of a given graph is NP-hard. We present a 4/3-approximation algorithm for unweighted 2ECSS on 3-vertex-connected input graphs, which matches the best known approximation ratio due to Sebő and Vygen for the general unweighted 2ECSS problem, but our analysis is with respect to the D2 lower bound. We also give a 17/12-approximation algorithm for unweighted 2VCSS on graphs of minimum degree at least 3, which is lower than the best known ratios of 3/2 by Garg, Santosh and Singla and 10/7 by Heeger and Vygen for the general unweighted 2VCSS problem. These algorithms are accompanied by new theorems about the known lower bounds

Approximating the Smallest Spanning Subgraph for 2-Edge-Connectivity in Directed Graphs

Let $G$ be a strongly connected directed graph. We consider the following three problems, where we wish to compute the smallest strongly connected spanning subgraph of $G$ that maintains respectively: the $2$-edge-connected blocks of $G$ (\textsf{2EC-B}); the $2$-edge-connected components of $G$ (\textsf{2EC-C}); both the $2$-edge-connected blocks and the $2$-edge-connected components of $G$ (\textsf{2EC-B-C}). All three problems are NP-hard, and thus we are interested in efficient approximation algorithms. For \textsf{2EC-C} we can obtain a $3/2$-approximation by combining previously known results. For \textsf{2EC-B} and \textsf{2EC-B-C}, we present new $4$-approximation algorithms that run in linear time. We also propose various heuristics to improve the size of the computed subgraphs in practice, and conduct a thorough experimental study to assess their merits in practical scenarios

Small Cuts and Connectivity Certificates: A Fault Tolerant Approach

We revisit classical connectivity problems in the {CONGEST} model of distributed computing. By using techniques from fault tolerant network design, we show improved constructions, some of which are even "local" (i.e., with O~(1) rounds) for problems that are closely related to hard global problems (i.e., with a lower bound of Omega(Diam+sqrt{n}) rounds). Distributed Minimum Cut: Nanongkai and Su presented a randomized algorithm for computing a (1+epsilon)-approximation of the minimum cut using O~(D +sqrt{n}) rounds where D is the diameter of the graph. For a sufficiently large minimum cut lambda=Omega(sqrt{n}), this is tight due to Das Sarma et al. [FOCS \u2711], Ghaffari and Kuhn [DISC \u2713]. - Small Cuts: A special setting that remains open is where the graph connectivity lambda is small (i.e., constant). The only lower bound for this case is Omega(D), with a matching bound known only for lambda <= 2 due to Pritchard and Thurimella [TALG \u2711]. Recently, Daga, Henzinger, Nanongkai and Saranurak [STOC \u2719] raised the open problem of computing the minimum cut in poly(D) rounds for any lambda=O(1). In this paper, we resolve this problem by presenting a surprisingly simple algorithm, that takes a completely different approach than the existing algorithms. Our algorithm has also the benefit that it computes all minimum cuts in the graph, and naturally extends to vertex cuts as well. At the heart of the algorithm is a graph sampling approach usually used in the context of fault tolerant (FT) design. - Deterministic Algorithms: While the existing distributed minimum cut algorithms are randomized, our algorithm can be made deterministic within the same round complexity. To obtain this, we introduce a novel definition of universal sets along with their efficient computation. This allows us to derandomize the FT graph sampling technique, which might be of independent interest. - Computation of all Edge Connectivities: We also consider the more general task of computing the edge connectivity of all the edges in the graph. In the output format, it is required that the endpoints u,v of every edge (u,v) learn the cardinality of the u-v cut in the graph. We provide the first sublinear algorithm for this problem for the case of constant connectivity values. Specifically, by using the recent notion of low-congestion cycle cover, combined with the sampling technique, we compute all edge connectivities in poly(D) * 2^{O(sqrt{log n log log n})} rounds. Sparse Certificates: For an n-vertex graph G and an integer lambda, a lambda-sparse certificate H is a subgraph H subseteq G with O(lambda n) edges which is lambda-connected iff G is lambda-connected. For D-diameter graphs, constructions of sparse certificates for lambda in {2,3} have been provided by Thurimella [J. Alg. \u2797] and Dori [PODC \u2718] respectively using O~(D) number of rounds. The problem of devising such certificates with o(D+sqrt{n}) rounds was left open by Dori [PODC \u2718] for any lambda >= 4. Using connections to fault tolerant spanners, we considerably improve the round complexity for any lambda in [1,n] and epsilon in (0,1), by showing a construction of (1-epsilon)lambda-sparse certificates with O(lambda n) edges using only O(1/epsilon^2 * log^{2+o(1)} n) rounds

A Partitioning Algorithm for Maximum Common Subgraph Problems

We introduce a new branch and bound algorithm for the maximum common subgraph and maximum common connected subgraph problems which is based around vertex labelling and partitioning. Our method in some ways resembles a traditional constraint programming approach, but uses a novel compact domain store and supporting inference algorithms which dramatically reduce the memory and computation requirements during search, and allow better dual viewpoint ordering heuristics to be calculated cheaply. Experiments show a speedup of more than an order of magnitude over the state of the art, and demonstrate that we can operate on much larger graphs without running out of memory