TY - GEN
T1 - On strong diameter padded decompositions
AU - Filtser, Arnold
N1 - Publisher Copyright:
© Arnold Filtser.
PY - 2019/9/1
Y1 - 2019/9/1
N2 - Given a weighted graph G = (V, E, w), a partition of V is ∆-bounded if the diameter of each cluster is bounded by ∆. A distribution over ∆-bounded partitions is a β-padded decomposition if every ball of radius γ∆ is contained in a single cluster with probability at least e−β·γ. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee. Formerly, it was proven that Kr free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric dG has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong O(r)-padded decompositions for Kr free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim), Õ(ddim))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.
AB - Given a weighted graph G = (V, E, w), a partition of V is ∆-bounded if the diameter of each cluster is bounded by ∆. A distribution over ∆-bounded partitions is a β-padded decomposition if every ball of radius γ∆ is contained in a single cluster with probability at least e−β·γ. The weak diameter of a cluster C is measured w.r.t. distances in G, while the strong diameter is measured w.r.t. distances in the induced graph G[C]. The decomposition is weak/strong according to the diameter guarantee. Formerly, it was proven that Kr free graphs admit weak decompositions with padding parameter O(r), while for strong decompositions only O(r2) padding parameter was known. Furthermore, for the case of a graph G, for which the induced shortest path metric dG has doubling dimension ddim, a weak O(ddim)-padded decomposition was constructed, which is also known to be tight. For the case of strong diameter, nothing was known. We construct strong O(r)-padded decompositions for Kr free graphs, matching the state of the art for weak decompositions. Similarly, for graphs with doubling dimension ddim we construct a strong O(ddim)-padded decomposition, which is also tight. We use this decomposition to construct (O(ddim), Õ(ddim))-sparse cover scheme for such graphs. Our new decompositions and cover have implications to approximating unique games, the construction of light and sparse spanners, and for path reporting distance oracles.
KW - Distance oracles
KW - Doubling dimension
KW - Minor free graphs
KW - Padded decomposition
KW - Spanners
KW - Sparse cover
KW - Strong diameter
KW - Unique games
UR - https://www.scopus.com/pages/publications/85072860508
U2 - 10.4230/LIPIcs.APPROX-RANDOM.2019.6
DO - 10.4230/LIPIcs.APPROX-RANDOM.2019.6
M3 - Conference contribution
AN - SCOPUS:85072860508
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2019
A2 - Achlioptas, Dimitris
A2 - Vegh, Laszlo A.
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 22nd International Conference on Approximation Algorithms for Combinatorial Optimization Problems and 23rd International Conference on Randomization and Computation, APPROX/RANDOM 2019
Y2 - 20 September 2019 through 22 September 2019
ER -