130 research outputs found
Computing the Girth of a Planar Graph in Linear Time
The girth of a graph is the minimum weight of all simple cycles of the graph.
We study the problem of determining the girth of an n-node unweighted
undirected planar graph. The first non-trivial algorithm for the problem, given
by Djidjev, runs in O(n^{5/4} log n) time. Chalermsook, Fakcharoenphol, and
Nanongkai reduced the running time to O(n log^2 n). Weimann and Yuster further
reduced the running time to O(n log n). In this paper, we solve the problem in
O(n) time.Comment: 20 pages, 7 figures, accepted to SIAM Journal on Computin
Stronger ILPs for the Graph Genus Problem
The minimum genus of a graph is an important question in graph theory and a key ingredient in several graph algorithms. However, its computation is NP-hard and turns out to be hard even in practice. Only recently, the first non-trivial approach - based on SAT and ILP (integer linear programming) models - has been presented, but it is unable to successfully tackle graphs of genus larger than 1 in practice.
Herein, we show how to improve the ILP formulation. The crucial ingredients are two-fold. First, we show that instead of modeling rotation schemes explicitly, it suffices to optimize over partitions of the (bidirected) arc set A of the graph. Second, we exploit the cycle structure of the graph, explicitly mapping short closed walks on A to faces in the embedding.
Besides the theoretical advantages of our models, we show their practical strength by a thorough experimental evaluation. Contrary to the previous approach, we are able to quickly solve many instances of genus > 1
Super-Fast 3-Ruling Sets
A -ruling set of a graph is a vertex-subset
that is independent and satisfies the property that every vertex is
at a distance of at most from some vertex in . A \textit{maximal
independent set (MIS)} is a 1-ruling set. The problem of computing an MIS on a
network is a fundamental problem in distributed algorithms and the fastest
algorithm for this problem is the -round algorithm due to Luby
(SICOMP 1986) and Alon et al. (J. Algorithms 1986) from more than 25 years ago.
Since then the problem has resisted all efforts to yield to a sub-logarithmic
algorithm. There has been recent progress on this problem, most importantly an
-round algorithm on graphs with
vertices and maximum degree , due to Barenboim et al. (Barenboim,
Elkin, Pettie, and Schneider, April 2012, arxiv 1202.1983; to appear FOCS
2012).
We approach the MIS problem from a different angle and ask if O(1)-ruling
sets can be computed much more efficiently than an MIS? As an answer to this
question, we show how to compute a 2-ruling set of an -vertex graph in
rounds. We also show that the above result can be improved
for special classes of graphs such as graphs with high girth, trees, and graphs
of bounded arboricity.
Our main technique involves randomized sparsification that rapidly reduces
the graph degree while ensuring that every deleted vertex is close to some
vertex that remains. This technique may have further applications in other
contexts, e.g., in designing sub-logarithmic distributed approximation
algorithms. Our results raise intriguing questions about how quickly an MIS (or
1-ruling sets) can be computed, given that 2-ruling sets can be computed in
sub-logarithmic rounds
Distributed distance-r covering problems on sparse high-girth graphs
We prove that the distance-r dominating set, distance-r connected dominating set,
distance-r vertex cover, and distance-r connected vertex cover problems admit constant
factor approximations in the CONGEST model of distributed computing in a constant
number of rounds on classes of sparse high-girth graphs. In this paper, sparse means
bounded expansion, and high-girth means girth at least 4r + 2. Our algorithm is quite
simple; however, the proof of its approximation guarantee is non-trivial. To complement
the algorithmic results, we show tightness of our approximation by providing a loosely
matching lower bound on rings.
Our result is the first to show the existence of constant-factor approximations in a constant
number of rounds in non-trivial classes of graphs for distance-r covering problems
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