46 research outputs found
Lowest Degree k-Spanner: Approximation and Hardness
A k-spanner is a subgraph in which distances are approximately preserved, up to some given stretch factor k. We focus on the following problem: Given a graph and a value k, can we find a k-spanner that minimizes the maximum degree? While reasonably strong bounds are known for some spanner problems, they almost all involve minimizing the total number of edges. Switching the objective to the degree introduces significant new challenges, and currently the only known approximation bound is an O~(Delta^(3-2*sqrt(2)))-approximation for the special case when k = 2 [Chlamtac, Dinitz, Krauthgamer FOCS 2012] (where Delta is the maximum degree in the input graph). In this paper we give the first non-trivial algorithm and polynomial-factor hardness of approximation for the case of general k. Specifically, we give an LP-based O~(Delta^((1-1/k)^2) )-approximation and prove that it is hard to approximate the optimum to within Delta^Omega(1/k) when the graph is undirected, and to within Delta^Omega(1) when it is directed
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)
Algorithms and Hardness Results for Compressing Graphs with Distance Constraints
Graphs have been widely utilized in network design and other applications. A natural question is, can we keep as few edges of the original graph as possible, but still make sure that the vertices are connected within certain distance constraints.
In this thesis, we will consider different versions of graph compression problems, including graph spanners, approximate distance oracles, and Steiner networks. Since these problems are all NP-hard problems, we will mostly focus on designing approximation algorithms and proving inapproximability results
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