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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?

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Limits of Local Algorithms in Random Graphs

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  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

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