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Spanning and Sparsifying Graphs: Efficient Sparse Approximations for Optimization Problems

This paper explores how undirected graphs can be approximated by sparse graphs, focusing on both distance-based and cut-based approximations. Key concepts include minimizing stretch and congestion while ensuring that sparse graphs are spanning trees. With applications across NP-hard optimization problems, such as flow and cut problems, the study outlines methods for achieving efficient approximations using spanners and probabilistic approaches. This research is significant for optimizing network designs and support for various algorithmic applications in graph theory.

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Spanning and Sparsifying Graphs: Efficient Sparse Approximations for Optimization Problems

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  1. Spanning and Sparsifying Rajmohan Rajaraman Northeastern University, Boston May 2012 Spanning and Sparsifying

  2. Sparse Approximations of Graphs • How well can an undirected graph G = (V,E) be approximated by a sparse graph H? • Distance-based approximation: • Distance in H should be at least distance in G • Stretch = distance in H/distance in G • Minimize stretch • Cut-based approximation: • Capacity of cut in H should be at most capacity in G • Congestion = capacity in G/capacity in H • Minimize congestion Spanning and Sparsifying

  3. What kind of sparse graphs? • Minimize number of edges while satisfying a required stretch or congestion constraint • Achieve very small stretch or congestion with O(n) or Õ(n) edges • Require that sparse graph H be a tree • Require that H be a spanning tree of G Spanning and Sparsifying

  4. Applications • Many problems can be solved much better and/or much faster on sparsifier H • Many NP-hard optimization problems can be solved optimally on trees • The running time of many flow/cut problems depend significantly on the number of edges • If H is a good approximation of G, then solution for H or some refinement is likely a good solution for G Spanning and Sparsifying

  5. Distance-based Approximations • Spanners [Peleg-Schaeffer 89]: • Sparse subgraphs that approximate all pairwise distances • (α,β)-spanner: • For each pair (u,v), distH(u,v) is at most α×distG(u,v) + β • Multiplicative spanner: Focus on α • (1+ε, (log(k)/ε)log(k))-spanners exist with O(kn1+1/k) edges [Elkin-Peleg 04,Thorup-Zwick 06] • Additive spanner: Focus on β • Conjecture: (1,2k-2)-spanners with O(n1+1/k) edges exist [Woodruff 06, Baswana-Kavitha-Mehlhorn-Pettie 05,…] Spanning and Sparsifying

  6. Approximations using Trees • What can be approximated using the sparsest graphs? • Maximum stretch of any deterministic selection is Ω(n) • Probabilistic approximation: • There exists a probability distribution of trees such that the expected stretch for any pair is O(log(n)) • [Bartal 96, 98], …, [Fakcharoenphol-Rao-Talwar 03] • Õ(log(n)) achievable even if tree required to be a spanning tree [Elkin-Emek-Spielman-Teng 05, Abraham-Bartal-Neiman 09,…] Spanning and Sparsifying

  7. Major Tool in Network Optimization • When objective function is a linear combination of distances: • Can reduce the problem on a general graph to that of solving on a tree • Cost of O(log(n)) in approximation • Metrical Task Systems [Bartal 96] • Group Steiner Tree [Garg-Konjevod-Ravi 98] Spanning and Sparsifying

  8. Cut-Based Approximations • Is there a tree whose cuts well-approximate the ones in the graph? • [Räcke 03, 08] showed for two notions of probabilistic approximation that this is true • There exists a probability distribution over capacitated trees such that the expected congestion of every cut is O(log(n)) • Applications: • Minimum bisection • Minimum k-multicut • Oblivious routing Spanning and Sparsifying

  9. Graph Sparsifiers • Is there a sparse subgraph H of G, with capacities on edges, such that: • For every cut, capacity in H is within (1 ± ε) factor of the capacity in G • Graph sparsifiers with O(n) edges exist! • [Benczur-Karger 02] • [Spielman-Srivastava 08] provide stronger spectral sparsifiers • [Fung-Hariharan-Harvey-Panigrahy11] provide a general framework Spanning and Sparsifying

  10. Applications of Sparsifiers • Faster (1 ± ε)-approximation algorithms for flow-cut problems • Maximum flow and minimum cut [Benczur-Karger 02, …, Madry 10] • Graph partitioning [Khandekar-Rao-Vazirani 09] • Improved algorithms for linear system solvers [Spielman-Teng 04, 08] Spanning and Sparsifying

  11. Graph Sparsification via Random Sampling • Sample each edge with a certain probability • Uniform probability will not always work • Assign weight of selected edge to be the inverse of its probability • Ensures that in expectation, we are all set! • Real challenge: • How do we guarantee approximation for every one of the exponential number of cuts? • Three key components: • Non-uniform probability chosen captures “importance” of the cut (several measures have been proposed) • Distribution of the number of cuts of a particular size [Karger 99] • Chernoff bounds Spanning and Sparsifying

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