Everywhere-sparse spanners via dense subgraphs

Eden Chlamtáč, Michael Dinitz, Robert Krauthgamer

Research output: Contribution to journalConference articlepeer-review

31 Scopus citations

Abstract

The significant progressg in constructing graph spanners that are sparse (small number of edges) or light (low total weight) has skipped spanners that are everywhere-sparse (small maximum degree). This disparity is in line with other network design problems, where the maximum-degree objective has been a notorious technical challenge. Our main result is for the Lowest Degree 2-Spanner (LD2S) problem, where the goal is to compute a 2-spanner of an input graph so as to minimize the maximum degree. We design a polynomial-time algorithm achieving approximation factor Õ(Δ3-2√2) ≈ Õ(Δ0.172), where Δ is the maximum degree of the input graph. The previous O(Δ1/4) - approximation was proved nearly two decades ago by Kortsarz and Peleg [SODA 1994, SICOMP 1998]. Our main conceptual contribution is to establish a formal connection between LD2S and a variant of the Densest k-Sub graph (DkS) problem. Specifically, we design for both problems strong relaxations based on the Sherali-Adams linear programming (LP) hierarchy, and show that "faithful" randomized rounding of the DkS-variant can be used to round LD2S solutions. Our notion of faithfulness intuitively means that all vertices and edges are chosen with probability proportional to their LP value, but the precise formulation is more subtle. Unfortunately, the best algorithms known for DkS use the Lovász-Schrijver LP hierarchy in a non-faithful way [Bhaskara, Charikar, Chlamtac, Feige, and Vijayaraghavan, STOC 2010]. Our main technical contribution is to overcome this shortcoming, while still matching the gap that arises in random graphs by planting a sub graph with same log-density.

Original languageEnglish
Article number6375355
Pages (from-to)758-767
Number of pages10
JournalProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
DOIs
StatePublished - 1 Dec 2012
Event53rd Annual IEEE Symposium on Foundations of Computer Science, FOCS 2012 - New Brunswick, NJ, United States
Duration: 20 Oct 201223 Oct 2012

Keywords

  • DkS
  • approximation
  • spanners

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