3,657 research outputs found
Minimal Disconnected Cuts in Planar Graphs
The problem of finding a disconnected cut in a graph is NP-hard in general but polynomial-time solvable on planar graphs. The problem of finding a minimal disconnected cut is also NP-hard but its computational complexity was not known for planar graphs. We show that it is polynomial-time solvable on 3-connected planar graphs but NP-hard for 2-connected planar graphs. Our technique for the first result is based on a structural characterization of minimal disconnected cuts in 3-connected inline image-free-minor graphs and on solving a topological minor problem in the dual. In addition we show that the problem of finding a minimal connected cut of size at least 3 is NP-hard for 2-connected apex graphs. Finally, we relax the notion of minimality and prove that the problem of finding a so-called semi-minimal disconnected cut is still polynomial-time solvable on planar graphs
Max -Flow Oracles and Negative Cycle Detection in Planar Digraphs
We study the maximum -flow oracle problem on planar directed graphs
where the goal is to design a data structure answering max -flow value (or
equivalently, min -cut value) queries for arbitrary source-target pairs
. For the case of polynomially bounded integer edge capacities, we
describe an exact max -flow oracle with truly subquadratic space and
preprocessing, and sublinear query time. Moreover, if
-approximate answers are acceptable, we obtain a static oracle
with near-linear preprocessing and query time and a
dynamic oracle supporting edge capacity updates and queries in
worst-case time.
To the best of our knowledge, for directed planar graphs, no (approximate)
max -flow oracles have been described even in the unweighted case, and
only trivial tradeoffs involving either no preprocessing or precomputing all
the possible answers have been known.
One key technical tool we develop on the way is a sublinear (in the number of
edges) algorithm for finding a negative cycle in so-called dense distance
graphs. By plugging it in earlier frameworks, we obtain improved bounds for
other fundamental problems on planar digraphs. In particular, we show: (1) a
deterministic time algorithm for negatively-weighted SSSP in
planar digraphs with integer edge weights at least . This improves upon the
previously known bounds in the important case of weights polynomial in , and
(2) an improved bound on finding a perfect matching in a
bipartite planar graph.Comment: Extended abstract to appear in SODA 202
Faster Shortest Paths in Dense Distance Graphs, with Applications
We show how to combine two techniques for efficiently computing shortest
paths in directed planar graphs. The first is the linear-time shortest-path
algorithm of Henzinger, Klein, Subramanian, and Rao [STOC'94]. The second is
Fakcharoenphol and Rao's algorithm [FOCS'01] for emulating Dijkstra's algorithm
on the dense distance graph (DDG). A DDG is defined for a decomposition of a
planar graph into regions of at most vertices each, for some parameter
. The vertex set of the DDG is the set of vertices
of that belong to more than one region (boundary vertices). The DDG has
arcs, such that distances in the DDG are equal to the distances in
. Fakcharoenphol and Rao's implementation of Dijkstra's algorithm on the DDG
(nicknamed FR-Dijkstra) runs in time, and is a
key component in many state-of-the-art planar graph algorithms for shortest
paths, minimum cuts, and maximum flows. By combining these two techniques we
remove the dependency in the running time of the shortest-path
algorithm, making it .
This work is part of a research agenda that aims to develop new techniques
that would lead to faster, possibly linear-time, algorithms for problems such
as minimum-cut, maximum-flow, and shortest paths with negative arc lengths. As
immediate applications, we show how to compute maximum flow in directed
weighted planar graphs in time, where is the minimum number
of edges on any path from the source to the sink. We also show how to compute
any part of the DDG that corresponds to a region with vertices and
boundary vertices in time, which is faster than has been
previously known for small values of
Decremental Single-Source Reachability in Planar Digraphs
In this paper we show a new algorithm for the decremental single-source
reachability problem in directed planar graphs. It processes any sequence of
edge deletions in total time and explicitly
maintains the set of vertices reachable from a fixed source vertex. Hence, if
all edges are eventually deleted, the amortized time of processing each edge
deletion is only , which improves upon a previously
known solution. We also show an algorithm for decremental
maintenance of strongly connected components in directed planar graphs with the
same total update time. These results constitute the first almost optimal (up
to polylogarithmic factors) algorithms for both problems.
To the best of our knowledge, these are the first dynamic algorithms with
polylogarithmic update times on general directed planar graphs for non-trivial
reachability-type problems, for which only polynomial bounds are known in
general graphs
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