1 / 17

Limits of Local Algorithms in Random Graphs

Limits of Local Algorithms in Random Graphs. Madhu Sudan MSR. Joint work with David Gamarnik (MIT). Main Result. Background: Almost surely, r andom -regular graph on vertices has independent set of size for . Can you find such a large independent set?

Download Presentation

Limits of Local Algorithms in Random Graphs

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Limits of Local Algorithms in Random Graphs Madhu Sudan MSR Joint work with David Gamarnik (MIT) Local Algorithms on Random Graphs

  2. Main Result Local Algorithms on Random Graphs • Background: Almost surely, random -regular graph on vertices has independent set of size for . • Can you find such a large independent set? • Greedy finds one of half this size. • Our Theorem: “Local algorithms” can not. In fact they fall short by a constant factor.

  3. Definition: Local Algorithms Local Algorithms on Random Graphs • Informally: Local algorithms • Input = Communication network. • Wish to use local communication to compute some property of input. • In our case – large independent set in graph. • Allowed to use randomness, generated locally.

  4. Formally Local Algorithms on Random Graphs • (Randomized) Decision Algorithm: • : Determines if • is a weighting, say in on vertices • Correctness: • s.t. or . • Locality: • is -local if whenever -local weighted neighborhood around in and in are identical.

  5. Locality Locality Local Algorithms on Random Graphs • Locality in distributed algorithms • Usually algorithms try to compute some function of input graph, on the graph itself. • Algorithm uses data available topologically locally. • Leads to our model • Locality a la Codes/Property Testing • Locality simply refers to number of queries to input. • More general model. • We can’t/don’t deal with it.

  6. Motivations for our work Local Algorithms on Random Graphs • Paucity of “complexity” results for random graphs. Major exceptions: • Rossman: /Monotone complexity of planted clique. • Feige-Krauthgamer/Meka-Wigderson: SDP relaxations. • Physicists explanation of complexity • Clustering/Shattering explain inability of algorithms. • Graph Limit theory • Local characteristics of (random) graphs predict global properties (nearly).

  7. Motivations (contd.) Local Algorithms on Random Graphs Specific conjecture [Hatami-Lovasz-Szegedy]: As -local algorithms should find independent sets of cardinality . Refuted by our theorem.

  8. Proof Local Algorithms on Random Graphs • Part I: • A clustering phenomenon for independent sets in random graphs [Inspired by Coja-Oglan]. • Part II: • Locality Continuity (Clustering). Both parts simple.

  9. Clustering Phenomena Local Algorithms on Random Graphs • Generally: • When you look at “near-optimal” solutions, then they are very structured. • topology of solutions highly disconnected (in Hamming space. • In our context • Consider graph on independent sets (of size with if • Highly disconnected?

  10. Clustering Theorem Local Algorithms on Random Graphs • Theorem: s.t.: • Almost surely over , of size , • Proof: • Compute expected number of independent sets with forbidden intersection and note it is • Second moment proves concentration. • Implies Clustering.

  11. Locality (Clustering) Local Algorithms on Random Graphs • Main Idea: • Fix -local function , that usually produces independent sets of size • Sample weights twice: , and then ; -correlatedly. • Let and . • Prove: • whp, • whp, • s.t.

  12. Size of Ind. Set Local Algorithms on Random Graphs • Claim: Size of independent set produced by local algorithms is concentrated. • Let (where = infinite tree of degree ) • W.p. 1-o(1), size of ind. set produced • Proof: • Most neighborhoods are trees Expectation. • Most neighborhoods are disjoint Chebychev.

  13. -correlated distributions Local Algorithms on Random Graphs • Pick , independently. • Let w.p. and otherwise, independently for each . • Let ] • As in previous argument: • concentrated around expectation.

  14. Continuity of Local Algorithms on Random Graphs • Fix , and consider • Above expression is some polynomial in , of degree at most • In particular, it is continuous as function of • =Expectation over is also continuous. • Suffices to show

  15. Continuity (contd.) Local Algorithms on Random Graphs • ] • Follows from calculations (also naturally) that • Conclude: • whp, • whp, • s.t.

  16. Conclusions Local Algorithms on Random Graphs • “Clustering” is an obstacle? • Answer: • At least to local algorithms. • Local algorithms behave continuously, forcing non-clustering of solutions. • Open questions: • Barrier to local algorithms in general sense? • To other complexity classes?

  17. Thank You Local Algorithms on Random Graphs

More Related