Let @math be a directed graph. A in @math is a maximal vertex set @math with @math such that for each pair of distinct vertices @math , there exist two vertex-disjoint paths from @math to @math and two vertex-disjoint paths from @math to @math in @math . In contrast to the @math -vertex-connected components of @math , the subgraphs induced by the @math -directed blocks may consist of few or no edges. In this paper we present two algorithms for computing the @math -directed blocks of @math in @math time, where @math is the number of the strong articulation points of @math and @math is the number of the strong bridges of @math . Furthermore, we study two related concepts: the @math -strong blocks and the @math -edge blocks of @math . We give two algorithms for computing the @math -strong blocks of @math in @math time and we show that the @math -edge blocks of @math can be computed in @math time. In this paper we also study some optimization problems related to the strong articulation points and the @math -blocks of a directed graph. Given a strongly connected graph @math , find a minimum cardinality set @math such that @math is strongly connected and the strong articulation points of @math coincide with the strong articulation points of @math . This problem is called minimum strongly connected spanning subgraph with the same strong articulation points. We show that there is a linear time @math approximation algorithm for this NP-hard problem. We also consider the problem of finding a minimum strongly connected spanning subgraph with the same @math -blocks in a strongly connected graph @math . We present approximation algorithms for three versions of this problem, depending on the type of @math -blocks.
In independent work, Georgiadis, Italiano, Laura, and Parotsidis @cite_20 have studied @math -edge blocks and have given linear time algorithms for finding them. This is better than our results in Section .
@cite_20
2064493067 2272045681 2963972775 2962886649
We present a deterministic near-linear time algorithm that computes the edge-connectivity and finds a minimum cut for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. In near-linear time we can also construct the classic cactus representation of all minimum cuts. The previous fastest deterministic algorithm by Gabow from STOC'91 took O(m+λ2 n), where λ is the edge connectivity, but λ could be Ω(n). At STOC'96 Karger presented a randomized near linear time Monte Carlo algorithm for the minimum cut problem. As he points out, there is no better way of certifying the minimality of the returned cut than to use Gabow's slower deterministic algorithm and compare sizes. Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph G with minimum degree δ, producing a multigraph G with O(m δ) edges which preserves all minimum cuts of G with at least two vertices on each side. In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS'06. Normally such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed. We consider the maximum independent set problem on sparse graphs with maximum degree d. The best known result for the problem is an SDP based O(d log log d log d) approximation due to Halperin. It is also known that no o(d log2 d) approximation exists assuming the Unique Games Conjecture. We show the following two results: (i) The natural LP formulation for the problem strengthened by O(log4(d)) levels of the mixed-hierarchy has an integrality gap of O(d log2 d), where O(·) ignores some log log d factors. However, our proof is non-constructive, in particular it uses an entropy based approach due to Shearer, and does not give a O(d log2 d) approximation algorithm with sub-exponential running time. (ii) We give an O(d log d) approximation based on polylog(d)-levels of the mixed hierarchy that runs in nO(1) exp(logO(1) d) time, improving upon Halperin's bound by a modest log log d factor. Our algorithm is based on combining Halperin's approach together with an idea used by Ajtai, Erdos, Komlos and Szemeredi to show that Kr-free, degree-d graphs have independent sets of size Ωr(n log log d d). We present a deterministic algorithm that computes the edge-connectivity of a graph in near-linear time. This is for a simple undirected unweighted graph G with n vertices and m edges. This is the first o(mn) time deterministic algorithm for the problem. Our algorithm is easily extended to find a concrete minimum edge-cut. In fact, we can construct the classic cactus representation of all minimum cuts in near-linear time. The previous fastest deterministic algorithm by Gabow from STOC '91 took O(m+λ2 n), where λ is the edge connectivity, but λ can be as big as n−1. Karger presented a randomized near-linear time Monte Carlo algorithm for the minimum cut problem at STOC’96, but the returned cut is only minimum with high probability. Our main technical contribution is a near-linear time algorithm that contracts vertex sets of a simple input graph G with minimum degree Δ, producing a multigraph Ḡ with O(m Δ) edges, which preserves all minimum cuts of G with at least two vertices on each side. In our deterministic near-linear time algorithm, we will decompose the problem via low-conductance cuts found using PageRank a la Brin and Page (1998), as analyzed by Andersson, Chung, and Lang at FOCS’06. Normally, such algorithms for low-conductance cuts are randomized Monte Carlo algorithms, because they rely on guessing a good start vertex. However, in our case, we have so much structure that no guessing is needed. We present a polynomial-space algorithm that computes the number of independent sets of any input graph in time (O(1.1389^n) ) for graphs with maximum degree 3 and in time (O(1.2356^n) ) for general graphs, where n is the number of vertices. Together with the inclusion-exclusion approach of Bjorklund, Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster polynomial-space algorithm for the graph coloring problem with running time (O(2.2356^n) ). As a byproduct, we also obtain an exponential-space (O(1.2330^n) ) time algorithm for counting independent sets.