72 research outputs found
Fault Tolerant Max-Cut
In this work, we initiate the study of fault tolerant Max-Cut, where given an edge-weighted undirected graph G = (V,E), the goal is to find a cut S ? V that maximizes the total weight of edges that cross S even after an adversary removes k vertices from G. We consider two types of adversaries: an adaptive adversary that sees the outcome of the random coin tosses used by the algorithm, and an oblivious adversary that does not. For any constant number of failures k we present an approximation of (0.878-?) against an adaptive adversary and of ?_{GW}? 0.8786 against an oblivious adversary (here ?_{GW} is the approximation achieved by the random hyperplane algorithm of [Goemans-Williamson J. ACM `95]). Additionally, we present a hardness of approximation of ?_{GW} against both types of adversaries, rendering our results (virtually) tight.
The non-linear nature of the fault tolerant objective makes the design and analysis of algorithms harder when compared to the classic Max-Cut. Hence, we employ approaches ranging from multi-objective optimization to LP duality and the ellipsoid algorithm to obtain our results
Optimal Vertex Fault Tolerant Spanners (for fixed stretch)
A -spanner of a graph is a sparse subgraph whose shortest path
distances match those of up to a multiplicative error . In this paper we
study spanners that are resistant to faults. A subgraph is an
vertex fault tolerant (VFT) -spanner if is a -spanner
of for any small set of vertices that might "fail." One
of the main questions in the area is: what is the minimum size of an fault
tolerant -spanner that holds for all node graphs (as a function of ,
and )? This question was first studied in the context of geometric
graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more
recently been considered in general undirected graphs [Chechik et al. STOC '09,
Dinitz and Krauthgamer PODC '11].
In this paper, we settle the question of the optimal size of a VFT spanner,
in the setting where the stretch factor is fixed. Specifically, we prove
that every (undirected, possibly weighted) -node graph has a
-spanner resilient to vertex faults with edges, and this is fully optimal (unless the famous Erdos Girth
Conjecture is false). Our lower bound even generalizes to imply that no data
structure capable of approximating similarly can
beat the space usage of our spanner in the worst case. We also consider the
edge fault tolerant (EFT) model, defined analogously with edge failures rather
than vertex failures. We show that the same spanner upper bound applies in this
setting. Our data structure lower bound extends to the case (and hence we
close the EFT problem for -approximations), but it falls to for . We leave it as an open problem to
close this gap.Comment: To appear in SODA 201
\~{O}ptimal Vertex Fault-Tolerant Spanners in \~{O}ptimal Time: Sequential, Distributed and Parallel
We (nearly) settle the time complexity for computing vertex fault-tolerant
(VFT) spanners with optimal sparsity (up to polylogarithmic factors). VFT
spanners are sparse subgraphs that preserve distance information, up to a small
multiplicative stretch, in the presence of vertex failures. These structures
were introduced by [Chechik et al., STOC 2009] and have received a lot of
attention since then. We provide algorithms for computing nearly optimal
-VFT spanners for any -vertex -edge graph, with near optimal running
time in several computational models:
- A randomized sequential algorithm with a runtime of
(i.e., independent in the number of faults ). The state-of-the-art time
bound is by [Bodwin, Dinitz and
Robelle, SODA 2021].
- A distributed congest algorithm of rounds. Improving
upon [Dinitz and Robelle, PODC 2020] that obtained FT spanners with
near-optimal sparsity in rounds.
- A PRAM (CRCW) algorithm with work and
depth. Prior bounds implied by [Dinitz and Krauthgamer, PODC 2011] obtained
sub-optimal FT spanners using work and
depth.
An immediate corollary provides the first nearly-optimal PRAM algorithm for
computing nearly optimal -\emph{vertex} connectivity certificates
using polylogarithmic depth and near-linear work. This improves the
state-of-the-art parallel bounds of depth and
work, by [Karger and Motwani, STOC'93].Comment: STOC 202
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