5 research outputs found
Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems
A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, input is a set T of n points in the Euclidean plane (R^2) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r in T is a rectilinear Steiner tree S for T such that the path in S from r to any point t in T is a shortest path. In the Rectilinear Steiner Arborescense problem the input is a set T of n points in R^2, and a root r in T, the task is to find an rectilinear Steiner arborescence for T, rooted at r of smallest possible total length. In this paper, we give the first subexponential time algorithms for both problems. Our algorithms are deterministic and run in 2^{O(sqrt{n}log n)} time
Subexponential Algorithms for Rectilinear Steiner Tree and Arborescence Problems
A rectilinear Steiner tree for a set K of points in the plane is a tree that connects k using horizontal and vertical lines. In the Rectilinear Steiner Tree problem, the input is a set K={z1,z2,…, zn} of n points in the Euclidean plane (R2), and the goal is to find a rectilinear Steiner tree for k of smallest possible total length. A rectilinear Steiner arborescence for a set k of points and a root r ∈ K is a rectilinear Steiner tree T for K such that the path in T from r to any point z ∈ K is a shortest path. In the Rectilinear Steiner Arborescence problem, the input is a set K of n points in R2, and a root r ∈ K, and the task is to find a rectilinear Steiner arborescence for K, rooted at r of smallest possible total length. In this article, we design deterministic algorithms for these problems that run in 2O(√ nlog n) time
Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane
Given a set of points with their pairwise distances, the traveling
salesman problem (TSP) asks for a shortest tour that visits each point exactly
once. A TSP instance is rectilinear when the points lie in the plane and the
distance considered between two points is the distance. In this paper, a
fixed-parameter algorithm for the Rectilinear TSP is presented and relies on
techniques for solving TSP on bounded-treewidth graphs. It proves that the
problem can be solved in where denotes the
number of horizontal lines containing the points of . The same technique can
be directly applied to the problem of finding a shortest rectilinear Steiner
tree that interconnects the points of providing a
time complexity. Both bounds improve over the best time bounds known for these
problems.Comment: 24 pages, 13 figures, 6 table
Tight bounds for planar strongly connected Steiner subgraph with fixed number of terminals (and extensions)
(see paper for full abstract)
Given a vertex-weighted directed graph and a set of terminals, the objective of the SCSS problem is to find a
vertex set of minimum weight such that contains a
path for each . The problem is NP-hard, but
Feldman and Ruhl [FOCS '99; SICOMP '06] gave a novel algorithm for
the SCSS problem, where is the number of vertices in the graph and is
the number of terminals. We explore how much easier the problem becomes on
planar directed graphs:
- Our main algorithmic result is a algorithm
for planar SCSS, which is an improvement of a factor of in the
exponent over the algorithm of Feldman and Ruhl.
- Our main hardness result is a matching lower bound for our algorithm: we
show that planar SCSS does not have an algorithm
for any computable function , unless the Exponential Time Hypothesis (ETH)
fails.
The following additional results put our upper and lower bounds in context:
- In general graphs, we cannot hope for such a dramatic improvement over the
algorithm of Feldman and Ruhl: assuming ETH, SCSS in general graphs
does not have an algorithm for any computable
function .
- Feldman and Ruhl generalized their algorithm to the more general
Directed Steiner Network (DSN) problem; here the task is to find a subgraph of
minimum weight such that for every source there is a path to the
corresponding terminal . We show that, assuming ETH, there is no
time algorithm for DSN on acyclic planar graphs.Comment: To appear in SICOMP. Extended abstract in SODA 2014. This version has
a new author (Andreas Emil Feldmann), and the algorithm is faster and
considerably simplified as compared to conference versio