34 research outputs found
Light Spanners
A -spanner of a weighted undirected graph , is a subgraph
such that for all . The sparseness of
the spanner can be measured by its size (the number of edges) and weight (the
sum of all edge weights), both being important measures of the spanner's
quality -- in this work we focus on the latter.
Specifically, it is shown that for any parameters and ,
any weighted graph on vertices admits a
-stretch spanner of weight at most , where is the weight of a minimum
spanning tree of . Our result is obtained via a novel analysis of the
classic greedy algorithm, and improves previous work by a factor of .Comment: 10 pages, 1 figure, to appear in ICALP 201
Exploration of Graphs with Excluded Minors
We study the online graph exploration problem proposed by Kalyanasundaram and Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This result encompasses and significantly extends the graph classes that were previously known to admit a constant competitive ratio. The main ingredient of our proof is that we find a connection between the performance of the particular exploration algorithm Blocking and the existence of light spanners. Conversely, we exploit this connection to construct light spanners of bounded genus graphs. In particular, we achieve a lightness that improves on the best known upper bound for genus g ? 1 and recovers the known tight bound for the planar case (g = 0)
Exploration of graphs with excluded minors
We study the online graph exploration problem proposed by Kalyanasundaram and
Pruhs (1994) and prove a constant competitive ratio on minor-free graphs. This
result encompasses and significantly extends the graph classes that were
previously known to admit a constant competitive ratio. The main ingredient of
our proof is that we find a connection between the performance of the
particular exploration algorithm Blocking and the existence of light spanners.
Conversely, we exploit this connection to construct light spanners of bounded
genus graphs. In particular, we achieve a lightness that improves on the best
known upper bound for genus g>0 and recovers the known tight bound for the
planar case (g=0).Comment: to appear at ESA 202
Light Spanners for High Dimensional Norms via Stochastic Decompositions
Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any n-point Euclidean metric has an O(t)-spanner with O~(n^{1+1/t^2}) edges, little is known.
In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of l_p with 1<p <=2. Second, our construction yields a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any n-point subset of l_p for 1<p <=2 has an O(t)-spanner with n^{1+O~(1/t^p)} edges and lightness n^{O~(1/t^p)}.
In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an O(t)-spanner with lightness O(n^{1/t}). We exhibit the following tradeoff: metrics with decomposability parameter nu=nu(t) admit an O(t)-spanner with lightness O~(nu^{1/t}). For example, n-point Euclidean metrics have nu <=n^{1/t}, metrics with doubling constant lambda have nu <=lambda, and graphs of genus g have nu <=g. While these families do admit a (1+epsilon)-spanner, its lightness depend exponentially on the dimension (resp. log g). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch
A Unified and Fine-Grained Approach for Light Spanners
Seminal works on light spanners from recent years provide near-optimal
tradeoffs between the stretch and lightness of spanners in general graphs,
minor-free graphs, and doubling metrics. In FOCS'19 the authors provided a
"truly optimal" tradeoff for Euclidean low-dimensional spaces. Some of these
papers employ inherently different techniques than others. Moreover, the
runtime of these constructions is rather high.
In this work, we present a unified and fine-grained approach for light
spanners. Besides the obvious theoretical importance of unification, we
demonstrate the power of our approach in obtaining (1) stronger lightness
bounds, and (2) faster construction times. Our results include:
_ -minor-free graphs: A truly optimal spanner construction and a fast
construction.
_ General graphs: A truly optimal spanner -- almost and a linear-time
construction with near-optimal lightness.
_ Low dimensional Euclidean spaces: We demonstrate that Steiner points help
in reducing the lightness of Euclidean -spanners almost
quadratically for .Comment: We split this paper into two papers: arXiv:2106.15596 and
arXiv:2111.1374
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Structural Results and Approximation Algorithms in Minor-free Graphs
Planarity has been successfully exploited to design faster and more accurate approximation algorithms for many graph optimization problems. The celebrated theorem of Kuratowski completely characterizes planar graphs as those excluding K_5 and K_{3,3} as minors. Kuratowski's theorem allows one to generalize planar graphs to H-minor-free graphs: those that exclude a fixed graph H as a minor. The deep results of Robertson and Seymour reveal many hidden structures in H-minor-free graphs, that have been used extensively in algorithmic designs. Relying on these structures, we design (i) an (efficient) polynomial time approximation scheme (PTAS) for two different variants of the traveling salesperson problem (TSP) and (ii) simple local search PTASes for r-dominating set and feedback vertex set problems. We then present several results concerning structures of planar graphs. Specifically, we make progresses on two conjectures on existence of large induced forests in planar graphs