24 research outputs found
Deterministic Maximum Flows in Simple Graphs
In this paper we are interested in deterministically computing maximum flows in undirected simple graphs where edges have unit capacities. When the input graph has n vertices and m edges, and the maximum flow is known to be upper bounded by ? as prior knowledge, our algorithm has running time O?(m + n^{5/3}?^{1/2}); in the extreme case where ? = ?(n), our algorithm has running time O?(n^{2.17}). This always improves upon the previous best deterministic upper bound O?(n^{9/4}?^{1/8}) by [Duan, 2013]. Furthermore, when ? ? n^{0.67} our algorithm is faster than a classical upper bound of O(m + n?^{3/2}) by [Karger and Levin, 1998]
Faster Algorithms for Rooted Connectivity in Directed Graphs
We consider the fundamental problems of determining the rooted and global
edge and vertex connectivities (and computing the corresponding cuts) in
directed graphs. For rooted (and hence also global) edge connectivity with
small integer capacities we give a new randomized Monte Carlo algorithm that
runs in time . For rooted edge connectivity this is the first
algorithm to improve on the time bound in the dense-graph
high-connectivity regime. Our result relies on a simple combination of sampling
coupled with sparsification that appears new, and could lead to further
tradeoffs for directed graph connectivity problems.
We extend the edge connectivity ideas to rooted and global vertex
connectivity in directed graphs. We obtain a -approximation for
rooted vertex connectivity in time where is the
total vertex weight (assuming integral vertex weights); in particular this
yields an time randomized algorithm for unweighted
graphs. This translates to a time exact algorithm where
is the rooted connectivity. We build on this to obtain similar bounds
for global vertex connectivity.
Our results complement the known results for these problems in the low
connectivity regime due to work of Gabow [9] for edge connectivity from 1991,
and the very recent work of Nanongkai et al. [24] and Forster et al. [7] for
vertex connectivity
Spectral Sparsification via Bounded-Independence Sampling
We give a deterministic, nearly logarithmic-space algorithm for mild spectral
sparsification of undirected graphs. Given a weighted, undirected graph on
vertices described by a binary string of length , an integer , and an error parameter , our algorithm runs in space
where
and are the maximum and minimum edge
weights in , and produces a weighted graph with
edges that spectrally approximates , in
the sense of Spielmen and Teng [ST04], up to an error of .
Our algorithm is based on a new bounded-independence analysis of Spielman and
Srivastava's effective resistance based edge sampling algorithm [SS08] and uses
results from recent work on space-bounded Laplacian solvers [MRSV17]. In
particular, we demonstrate an inherent tradeoff (via upper and lower bounds)
between the amount of (bounded) independence used in the edge sampling
algorithm, denoted by above, and the resulting sparsity that can be
achieved.Comment: 37 page
Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Connectivity
Li and Panigrahi [Jason Li and Debmalya Panigrahi, 2020], in recent work, obtained the first deterministic algorithm for the global minimum cut of a weighted undirected graph that runs in time o(mn). They introduced an elegant and powerful technique to find isolating cuts for a terminal set in a graph via a small number of s-t minimum cut computations.
In this paper we generalize their isolating cut approach to the abstract setting of symmetric bisubmodular functions (which also capture symmetric submodular functions). Our generalization to bisubmodularity is motivated by applications to element connectivity and vertex connectivity. Utilizing the general framework and other ideas we obtain significantly faster randomized algorithms for computing global (and subset) connectivity in a number of settings including hypergraphs, element connectivity and vertex connectivity in graphs, and for symmetric submodular functions