6 research outputs found
A Tight Algorithm for Strongly Connected Steiner Subgraph On Two Terminals With Demands
Given an edge-weighted directed graph on vertices and a set
of terminals, the objective of the \scss
(-SCSS) problem is to find an edge set of minimum weight such
that contains an path for each . In this paper, we investigate the computational complexity of a variant of
-SCSS where we have demands for the number of paths between each terminal
pair. Formally, the \sharinggeneral problem is defined as follows: given an
edge-weighted directed graph with weight function , two terminal vertices , and integers
; the objective is to find a set of paths from and paths from
such that is minimized,
where . For each , we show the following: The \sharing problem
can be solved in time. A matching lower bound for our algorithm: the
\sharing problem does not have an algorithm for any
computable function , unless the Exponential Time Hypothesis (ETH) fails.
Our algorithm for \sharing relies on a structural result regarding an optimal
solution followed by using the idea of a "token game" similar to that of
Feldman and Ruhl. We show with an example that the structural result does not
hold for the \sharinggeneral problem if . Therefore
\sharing is the most general problem one can attempt to solve with our
techniques.Comment: To appear in Algorithmica. An extended abstract appeared in IPEC '1
A tight lower bound for steiner orientation
In the STEINER ORIENTATION problem, the input is a mixed graph G (it has both directed and undirected edges) and a set of k terminal pairs T. The question is whether we can orient the undirected edges in a way such that there is a directed s⇝t path for each terminal pair (s,t)∈T. Arkin and Hassin [DAM’02] showed that the STEINER ORIENTATION problem is NP-complete. They also gave a polynomial time algorithm for the special case when k=2
.
From the viewpoint of exact algorithms, Cygan, Kortsarz and Nutov [ESA’12, SIDMA’13] designed an XP algorithm running in nO(k) time for all k≥1. Pilipczuk and Wahlström [SODA ’16] showed that the STEINER ORIENTATION problem is W[1]-hard parameterized by k. As a byproduct of their reduction, they were able to show that under the Exponential Time Hypothesis (ETH) of Impagliazzo, Paturi and Zane [JCSS’01] the STEINER ORIENTATION problem does not admit an f(k)⋅no(k/logk) algorithm for any computable function f. That is, the nO(k) algorithm of Cygan et al. is almost optimal.
In this paper, we give a short and easy proof that the nO(k) algorithm of Cygan et al. is asymptotically optimal, even if the input graph has genus 1. Formally, we show that the STEINER ORIENTATION problem is W[1]-hard parameterized by the number k of terminal pairs, and, under ETH, cannot be solved in f(k)⋅no(k) time for any function f even if the underlying undirected graph has genus 1. We give a reduction from the GRID TILING problem which has turned out to be very useful in proving W[1]-hardness of several problems on planar graphs. As a result of our work, the main remaining open question is whether STEINER ORIENTATION admits the “square-root phenomenon” on planar graphs (graphs with genus 0): can one obtain an algorithm running in time f(k)⋅nO(k√) for PLANAR STEINER ORIENTATION, or does the lower bound of f(k)⋅no(k) also translate to planar graphs
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