123 research outputs found
Exact Distance Oracles for Planar Graphs with Failing Vertices
We consider exact distance oracles for directed weighted planar graphs in the
presence of failing vertices. Given a source vertex , a target vertex
and a set of failed vertices, such an oracle returns the length of a
shortest -to- path that avoids all vertices in . We propose oracles
that can handle any number of failures. More specifically, for a directed
weighted planar graph with vertices, any constant , and for any , we propose an oracle of size
that answers queries in
time. In particular, we show an
-size, -query-time
oracle for any constant . This matches, up to polylogarithmic factors, the
fastest failure-free distance oracles with nearly linear space. For single
vertex failures (), our -size,
-query-time oracle improves over the previously best
known tradeoff of Baswana et al. [SODA 2012] by polynomial factors for , . For multiple failures, no planarity exploiting
results were previously known
On Computing Homological Hitting Sets
Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. In this paper, we initiate the algorithmic study of a high-dimensional cut problem. The problem we study, namely, Homological Hitting Set (HHS), is defined as follows: Given a nontrivial r-cycle z in a simplicial complex, find a set ? of r-dimensional simplices of minimum cardinality so that ? meets every cycle homologous to z. Our first result is that HHS admits a polynomial-time solution on triangulations of closed surfaces. Interestingly, the minimal solution is given in terms of the cocycles of the surface. Next, we provide an example of a 2-complex for which the (unique) minimal hitting set is not a cocycle. Furthermore, for general complexes, we show that HHS is W[1]-hard with respect to the solution size p. In contrast, on the positive side, we show that HHS admits an FPT algorithm with respect to p+?, where ? is the maximum degree of the Hasse graph of the complex ?
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
Topologically Trivial Closed Walks in Directed Surface Graphs
Let be a directed graph with vertices and edges, embedded on a
surface , possibly with boundary, with first Betti number . We
consider the complexity of finding closed directed walks in that are either
contractible (trivial in homotopy) or bounding (trivial in integer homology) in
. Specifically, we describe algorithms to determine whether contains a
simple contractible cycle in time, or a contractible closed walk in
time, or a bounding closed walk in time. Our
algorithms rely on subtle relationships between strong connectivity in and
in the dual graph ; our contractible-closed-walk algorithm also relies on
a seminal topological result of Hass and Scott. We also prove that detecting
simple bounding cycles is NP-hard.
We also describe three polynomial-time algorithms to compute shortest
contractible closed walks, depending on whether the fundamental group of the
surface is free, abelian, or hyperbolic. A key step in our algorithm for
hyperbolic surfaces is the construction of a context-free grammar with
non-terminals that generates all contractible closed walks of
length at most L, and only contractible closed walks, in a system of quads of
genus . Finally, we show that computing shortest simple contractible
cycles, shortest simple bounding cycles, and shortest bounding closed walks are
all NP-hard.Comment: 30 pages, 18 figures; fixed several minor bugs and added one figure.
An extended abstraction of this paper will appear at SOCG 201
- …