21 research outputs found
On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint
In the problem of minimum connected dominating set with routing cost
constraint, we are given a graph , and the goal is to find the
smallest connected dominating set of such that, for any two
non-adjacent vertices and in , the number of internal nodes on the
shortest path between and in the subgraph of induced by is at most times that in . For general graphs, the only
known previous approximability result is an -approximation algorithm
() for by Ding et al. For any constant , we
give an -approximation
algorithm. When , we give an -approximation
algorithm. Finally, we prove that, when , unless , for any constant , the problem admits no
polynomial-time -approximation algorithm, improving
upon the bound by Du et al. (albeit under a stronger hardness
assumption)
An FPT Algorithm for Minimum Additive Spanner Problem
For a positive integer t and a graph G, an additive t-spanner of G is a spanning subgraph in which the distance between every pair of vertices is at most the original distance plus t. The Minimum Additive t-Spanner Problem is to find an additive t-spanner with the minimum number of edges in a given graph, which is known to be NP-hard. Since we need to care about global properties of graphs when we deal with additive t-spanners, the Minimum Additive t-Spanner Problem is hard to handle and hence only few results are known for it. In this paper, we study the Minimum Additive t-Spanner Problem from the viewpoint of parameterized complexity. We formulate a parameterized version of the problem in which the number of removed edges is regarded as a parameter, and give a fixed-parameter algorithm for it. We also extend our result to the case with both a multiplicative approximation factor ? and an additive approximation parameter ?, which we call (?, ?)-spanners
Approximating Approximate Distance Oracles
Given a finite metric space (V,d), an approximate distance oracle is a data structure which, when queried on two points u,v in V, returns an approximation to the the actual distance between u and v which is within some bounded stretch factor of the true distance. There has been significant work on the tradeoff between the important parameters of approximate distance oracles (and in particular between the size, stretch, and query time), but in this paper we take a different point of view, that of per-instance optimization. If we are given an particular input metric space and stretch bound, can we find the smallest possible approximate distance oracle for that particular input? Since this question is not even well-defined, we restrict our attention to well-known classes of approximate distance oracles, and study whether we can optimize over those classes.
In particular, we give an O(log n)-approximation to the problem of finding the smallest stretch 3 Thorup-Zwick distance oracle, as well as the problem of finding the smallest Pv{a}trac{s}cu-Roditty distance oracle. We also prove a matching Omega(log n) lower bound for both problems, and an Omega(n^{frac{1}{k}-frac{1}{2^{k-1}}}) integrality gap for the more general stretch (2k-1) Thorup-Zwick distance oracle. We also consider the problem of approximating the best TZ or PR approximate distance oracle with outliers, and show that more advanced techniques (SDP relaxations in particular) allow us to optimize even in the presence of outliers
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
Spanner Approximations in Practice
A multiplicative ?-spanner H is a subgraph of G = (V,E) with the same vertices and fewer edges that preserves distances up to the factor ?, i.e., d_H(u,v) ? ?? d_G(u,v) for all vertices u, v. While many algorithms have been developed to find good spanners in terms of approximation guarantees, no experimental studies comparing different approaches exist. We implemented a rich selection of those algorithms and evaluate them on a variety of instances regarding, e.g., their running time, sparseness, lightness, and effective stretch
The Norms of Graph Spanners
A -spanner of a graph is a subgraph in which all distances are
preserved up to a multiplicative factor. A classical result of Alth\"ofer
et al. is that for every integer and every graph , there is a
-spanner of with at most edges. But for some
settings the more interesting notion is not the number of edges, but the
degrees of the nodes. This spurred interest in and study of spanners with small
maximum degree. However, this is not necessarily a robust enough objective: we
would like spanners that not only have small maximum degree, but also have
"few" nodes of "large" degree. To interpolate between these two extremes, in
this paper we initiate the study of graph spanners with respect to the
-norm of their degree vector, thus simultaneously modeling the number
of edges (the -norm) and the maximum degree (the -norm).
We give precise upper bounds for all ranges of and stretch : we prove
that the greedy -spanner has norm of at most , and that this bound is tight (assuming the Erd\H{o}s girth
conjecture). We also study universal lower bounds, allowing us to give
"generic" guarantees on the approximation ratio of the greedy algorithm which
generalize and interpolate between the known approximations for the
and norm. Finally, we show that at least in some situations,
the norm behaves fundamentally differently from or
: there are regimes ( and stretch in particular) where
the greedy spanner has a provably superior approximation to the generic
guarantee
Spanner Approximations in Practice
A multiplicative -spanner is a subgraph of with the
same vertices and fewer edges that preserves distances up to the factor
, i.e., for all vertices , .
While many algorithms have been developed to find good spanners in terms of
approximation guarantees, no experimental studies comparing different
approaches exist. We implemented a rich selection of those algorithms and
evaluate them on a variety of instances regarding, e.g., their running time,
sparseness, lightness, and effective stretch