19,490 research outputs found
Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm
Given an undirected graph and two disjoint vertex pairs and
, the Shortest two disjoint paths problem (S2DP) asks for the minimum
total length of two vertex disjoint paths connecting with , and
with , respectively.
We show that for cubic planar graphs there are NC algorithms, uniform
circuits of polynomial size and polylogarithmic depth, that compute the S2DP
and moreover also output the number of such minimum length path pairs.
Previously, to the best of our knowledge, no deterministic polynomial time
algorithm was known for S2DP in cubic planar graphs with arbitrary placement of
the terminals. In contrast, the randomized polynomial time algorithm by
Bj\"orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is
serial in nature, and cannot count the solutions.
Our results are built on an approach by Hirai and Namba, Algorithmica 2017,
for a generalisation of S2DP, and fast algorithms for counting perfect
matchings in planar graphs
A Local-To-Global Theorem for Congested Shortest Paths
P_k such that each P_i is a shortest path from s_i to t_i, and every node in the graph is on at most c paths P_i, or reporting that no such collection of paths exists. When c = k, there are no congestion constraints, and the problem can be solved easily by running a shortest path algorithm for each pair (s_i,t_i) independently. At the other extreme, when c = 1, the (k,c)-SPC problem is equivalent to the k-Disjoint Shortest Paths (k-DSP) problem, where the collection of shortest paths must be node-disjoint. For fixed k, k-DSP is polynomial-time solvable on DAGs and undirected graphs. Amiri and Wargalla interpolated between these two extreme values of c, to obtain an algorithm for (k,c)-SPC on DAGs that runs in polynomial time when k-c is constant.
In the same way, we prove that (k,c)-SPC can be solved in polynomial time on undirected graphs whenever k-c is constant. For directed graphs, because of our counterexample to the original theorem statement, our roundtrip local-to-global result does not imply such an algorithm (k,c)-SPC. Even without an algorithmic application, our proof for directed graphs may be of broader interest because it characterizes intriguing structural properties of shortest paths in directed graphs
A Local-to-Global Theorem for Congested Shortest Paths
Amiri and Wargalla (2020) proved the following local-to-global theorem in
directed acyclic graphs (DAGs): if is a weighted DAG such that for each
subset of 3 nodes there is a shortest path containing every node in ,
then there exists a pair of nodes such that there is a shortest
-path containing every node in .
We extend this theorem to general graphs. For undirected graphs, we prove
that the same theorem holds (up to a difference in the constant 3). For
directed graphs, we provide a counterexample to the theorem (for any constant),
and prove a roundtrip analogue of the theorem which shows there exists a pair
of nodes such that every node in is contained in the union of a
shortest -path and a shortest -path.
The original theorem for DAGs has an application to the -Shortest Paths
with Congestion (()-SPC) problem. In this problem, we are given a
weighted graph , together with node pairs ,
and a positive integer . We are tasked with finding paths such that each is a shortest path from to , and every
node in the graph is on at most paths , or reporting that no such
collection of paths exists.
When the problem is easily solved by finding shortest paths for each
pair independently. When , the -SPC problem recovers
the -Disjoint Shortest Paths (-DSP) problem, where the collection of
shortest paths must be node-disjoint. For fixed , -DSP can be solved in
polynomial time on DAGs and undirected graphs. Previous work shows that the
local-to-global theorem for DAGs implies that -SPC on DAGs whenever
is constant. In the same way, our work implies that -SPC can be
solved in polynomial time on undirected graphs whenever is constant.Comment: Updated to reflect reviewer comment
Shortest Vertex-Disjoint Two-Face Paths in Planar Graphs
Let be a directed planar graph of complexity~, each arc
having a nonnegative length. Let and~ be two distinct faces
of~; let be vertices incident with~; let
be vertices incident with~. We give an
algorithm to compute pairwise vertex-disjoint paths connecting
the pairs in~, with minimal total length, in
time
- …