29 research outputs found
Approximating survivable networks with β-metric costs
AbstractThe Survivable Network Design (SND) problem seeks a minimum-cost subgraph that satisfies prescribed node-connectivity requirements. We consider SND on both directed and undirected complete graphs with β-metric costs when c(xz)⩽β[c(xy)+c(yz)] for all x,y,z∈V, which varies from uniform costs (β=1/2) to metric costs (β=1).For the k-Connected Subgraph (k-CS) problem our ratios are: 1+2βk(1−β)−12k−1 for undirected graphs, and 1+4β3k(1−3β2)−12k−1 for directed graphs and 12⩽β<13. For undirected graphs this improves the ratios β1−β of Böckenhauer et al. (2008) [3] and 2+βkn of Kortsarz and Nutov (2003) [11] for all k⩾4 and 12+3k−22(4k2−7k+2)⩽β⩽k2(k+1)2−2. We also show that SND admits the ratios 2β1−β for undirected graphs, and 4β31−3β2 for directed graphs with 1/2⩽β<1/3. For two important particular cases of SND, so-called Subset k-CS and Rooted SND, our ratios are 2β31−3β2 for directed graphs and β1−β for subset k-CS on undirected graphs
A Polynomial-time Algorithm for Outerplanar Diameter Improvement
The Outerplanar Diameter Improvement problem asks, given a graph and an
integer , whether it is possible to add edges to in a way that the
resulting graph is outerplanar and has diameter at most . We provide a
dynamic programming algorithm that solves this problem in polynomial time.
Outerplanar Diameter Improvement demonstrates several structural analogues to
the celebrated and challenging Planar Diameter Improvement problem, where the
resulting graph should, instead, be planar. The complexity status of this
latter problem is open.Comment: 24 page
Reachability Preservers: New Extremal Bounds and Approximation Algorithms
We abstract and study \emph{reachability preservers}, a graph-theoretic
primitive that has been implicit in prior work on network design. Given a
directed graph and a set of \emph{demand pairs} , a reachability preserver is a sparse subgraph that preserves
reachability between all demand pairs.
Our first contribution is a series of extremal bounds on the size of
reachability preservers. Our main result states that, for an -node graph and
demand pairs of the form for a small node subset ,
there is always a reachability preserver on edges. We
additionally give a lower bound construction demonstrating that this upper
bound characterizes the settings in which size reachability preservers
are generally possible, in a large range of parameters.
The second contribution of this paper is a new connection between extremal
graph sparsification results and classical Steiner Network Design problems.
Surprisingly, prior to this work, the osmosis of techniques between these two
fields had been superficial. This allows us to improve the state of the art
approximation algorithms for the most basic Steiner-type problem in directed
graphs from the of Chlamatac, Dinitz, Kortsarz, and
Laekhanukit (SODA'17) to .Comment: SODA '1
Complexity of the Steiner Network Problem with Respect to the Number of Terminals
In the Directed Steiner Network problem we are given an arc-weighted digraph
, a set of terminals , and an (unweighted) directed
request graph with . Our task is to output a subgraph of the minimum cost such that there is a directed path from to in
for all .
It is known that the problem can be solved in time
[Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time
even if is planar, unless Exponential-Time Hypothesis
(ETH) fails [Chitnis et al., SODA 2014]. However, as this reduction (and other
reductions showing hardness of the problem) only shows that the problem cannot
be solved in time unless ETH fails, there is a significant
gap in the complexity with respect to in the exponent.
We show that Directed Steiner Network is solvable in time , where is a constant depending solely on the
genus of and is a computable function. We complement this result by
showing that there is no algorithm for
any function for the problem on general graphs, unless ETH fails
On the Size and the Approximability of Minimum Temporally Connected Subgraphs
We consider temporal graphs with discrete time labels and investigate the
size and the approximability of minimum temporally connected spanning
subgraphs. We present a family of minimally connected temporal graphs with
vertices and edges, thus resolving an open question of (Kempe,
Kleinberg, Kumar, JCSS 64, 2002) about the existence of sparse temporal
connectivity certificates. Next, we consider the problem of computing a minimum
weight subset of temporal edges that preserve connectivity of a given temporal
graph either from a given vertex r (r-MTC problem) or among all vertex pairs
(MTC problem). We show that the approximability of r-MTC is closely related to
the approximability of Directed Steiner Tree and that r-MTC can be solved in
polynomial time if the underlying graph has bounded treewidth. We also show
that the best approximation ratio for MTC is at least and at most , for
any constant , where is the number of temporal edges and
is the maximum degree of the underlying graph. Furthermore, we prove
that the unweighted version of MTC is APX-hard and that MTC is efficiently
solvable in trees and -approximable in cycles
FPT Inapproximability of Directed Cut and Connectivity Problems
Cut problems and connectivity problems on digraphs are two well-studied classes of problems from the viewpoint of parameterized complexity. After a series of papers over the last decade, we now have (almost) tight bounds for the running time of several standard variants of these problems parameterized by two parameters: the number k of terminals and the size p of the solution. When there is evidence of FPT intractability, then the next natural alternative is to consider FPT approximations. In this paper, we show two types of results for directed cut and connectivity problems, building on existing results from the literature: first is to circumvent the hardness results for these problems by designing FPT approximation algorithms, or alternatively strengthen the existing hardness results by creating "gap-instances" under stronger hypotheses such as the (Gap-)Exponential Time Hypothesis (ETH). Formally, we show the following results:
Cutting paths between a set of terminal pairs, i.e., Directed Multicut: Pilipczuk and Wahlstrom [TOCT \u2718] showed that Directed Multicut is W[1]-hard when parameterized by p if k=4. We complement this by showing the following two results:
- Directed Multicut has a k/2-approximation in 2^{O(p^2)}* n^{O(1)} time (i.e., a 2-approximation if k=4),
- Under Gap-ETH, Directed Multicut does not admit an (59/58-epsilon)-approximation in f(p)* n^{O(1)} time, for any computable function f, even if k=4.
Connecting a set of terminal pairs, i.e., Directed Steiner Network (DSN): The DSN problem on general graphs is known to be W[1]-hard parameterized by p+k due to Guo et al. [SIDMA \u2711]. Dinur and Manurangsi [ITCS \u2718] further showed that there is no FPT k^{1/4-o(1)}-approximation algorithm parameterized by k, under Gap-ETH. Chitnis et al. [SODA \u2714] considered the restriction to special graph classes, but unfortunately this does not lead to FPT algorithms either: DSN on planar graphs is W[1]-hard parameterized by k. In this paper we consider the DSN_Planar problem which is an intermediate version: the graph is general, but we want to find a solution whose cost is at most that of an optimal planar solution (if one exists). We show the following lower bounds for DSN_Planar:
- DSN_Planar has no (2-epsilon)-approximation in FPT time parameterized by k, under Gap-ETH. This answers in the negative a question of Chitnis et al. [ESA \u2718].
- DSN_Planar is W[1]-hard parameterized by k+p. Moreover, under ETH, there is no (1+epsilon)-approximation for DSN_Planar in f(k,p,epsilon)* n^{o(k+sqrt{p+1/epsilon})} time for any computable function f.
Pairwise connecting a set of terminals, i.e., Strongly Connected Steiner Subgraph (SCSS): Guo et al. [SIDMA \u2711] showed that SCSS is W[1]-hard parameterized by p+k, while Chitnis et al. [SODA \u2714] showed that SCSS remains W[1]-hard parameterized by p, even if the input graph is planar. In this paper we consider the SCSS_Planar problem which is an intermediate version: the graph is general, but we want to find a solution whose cost is at most that of an optimal planar solution (if one exists). We show the following lower bounds for SCSS_Planar:
- SCSS_Planar is W[1]-hard parameterized by k+p. Moreover, under ETH, there is no (1+epsilon)-approximation for SCSS_Planar in f(k,p,epsilon)* n^{o(sqrt{k+p+1/epsilon})} time for any computable function f.
Previously, the only known FPT approximation results for SCSS applied to general graphs parameterized by k: a 2-approximation by Chitnis et al. [IPEC \u2713], and a matching (2-epsilon)-hardness under Gap-ETH by Chitnis et al. [ESA \u2718]
Parameterized approximation algorithms for bidirected steiner network problems
The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph G=(V,E) and a set {D}subseteq V x V of k demand pairs. The aim is to compute the cheapest network N subseteq G for which there is an s -> t path for each (s,t)in {D}. It is known that this problem is notoriously hard as there is no k^{1/4-o(1)}-approximation algorithm under Gap-ETH, even when parameterizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k.
For the bi-DSN_Planar problem, the aim is to compute a planar optimum solution N subseteq G in a bidirected graph G, i.e. for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of bi-DSN_Planar, unless FPT=W[1]. Additionally we study several generalizations of bi-DSN_Planar and obtain upper and lower bounds on obtainable runtimes parameterized by k.
One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network N subseteq G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists when parameterized by k [Chitnis et al., IPEC 2013]. We show a tight inapproximability result: under Gap-ETH there is no (2-{epsilon})-approximation algorithm parameterized by k (for any epsilon>0). To the best of our knowledge, this is the first example of a W[1]-hard problem admitting a non-trivial parameterized approximation factor which is also known to be tight! Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k