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Algorithms on large graphs

Happy Birthday Ravi!. Algorithms on large graphs. L á szl ó Lov á sz Eötvös Lor ánd University, Budapest . The Weak Regularity Lemma. Cut norm of matrix A   n x n :. Cut distance of two graphs with V ( G ) = V ( G ’):. (extends to edge-weighted). The Weak Regularity Lemma.

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Algorithms on large graphs

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  1. Happy Birthday Ravi! Algorithms on large graphs LászlóLovász EötvösLorándUniversity, Budapest

  2. The Weak Regularity Lemma Cut norm of matrix Anxn: Cut distance of two graphs with V(G) = V(G’): (extends to edge-weighted)

  3. The Weak Regularity Lemma Avereged graph GP(P partition of V(G)) 0 Template graph G/P 1/2 1 1 1/2 2/5 0 1/5 1/2 1 2/5

  4. The Weak Regularity Lemma For every graph G and every >0 there is a partition  with and Frieze – Kannan 1999

  5. Algorithms for large graphs How is the graph given? • - Graph is HUGE. • - Not known explicitly, not even thenumber of nodes. Idealize: define minimum amount of info.

  6. Algorithms for large graphs Dense case: cn2 edges. - We can sample a uniform random node a bounded number of times, and see edges between sampled nodes. „Property testing”, constant time algorithms: Arora-Karger-Karpinski,Goldreich-Goldwasser-Ron, Rubinfeld-Sudan,Alon-Fischer-Krivelevich-Szegedy, Fischer, Frieze-Kannan, Alon-Shapira

  7. Algorithms for large graphs Parameter estimation: edge density, triangle density, maximum cut Property testing: is the graph bipartite? triangle-free? perfect? Computing a constant size encoding Computing a structure: find a maximum cut, regularity partition,... Computing a structure: find a maximum cut, regularity partition,... The partition (cut,...) can be computed in polynomial time. For every node, we can determine in constant timewhich class it belongs to

  8. Representative set Representative set of nodes: bounded size, (almost) every node is “similar” to one of the nodes in the set When are two nodes similar? Neighbors? Same neighborhood?

  9. Similaritydistance of nodes w t u s v This is a metric, computable inthe samplingmodel

  10. Representative set Strong representative setU: foranytwonodesins,tU, dsim(s,t) >  forall nodess, dsim(U,s)   Average representative setU: foranytwonodess,tU, dsim(s,t) >  fora random node s, Edsim(U,s)  2

  11. Representative sets and regularity partitions If P = {S1, . . . , Sk} is a weak regularity partition with error, then we can select nodes viSi such thatS = {v1, . . . , vk} is an average representative set with error< 4. If SVis an average representative set with error, then the Voronoi cells of S form a weak regularity partition with error < 8. L-Szegedy

  12. Representative sets and regularity partitions Voronoi diagram = weakregularitypartition

  13. Representative sets Everygraph has an average representative set withat most nodes. Everygraph has a strongrepresentativeset withat most nodes. Alon If S  V(G) and dsim(u,v)> for all u,vS, then

  14. Representative sets Example: every average representative set has nodes. dimension1/ angle 

  15. Representative sets and regularity partitions For every graph G and >0 there are ui, vi {0,1}V(G) and ai such that Frieze-Kannan

  16. Howtocomputea (weak) regularity partition? Construct weak representative setU Eachnode is in sameclassasclosestrepresentative.

  17. Howtocompute a maximum cut? - Construct representative set - Compute weights in template graph(use sampling) - Compute max cut in template graph Eachnode is onsamesideasclosestrepresentative. (Different algorithm implicit by Frieze-Kannan.)

  18. Howtocompute a maximum matching? Given a bigraph with bipartition {U,W} (|U|=|W|=n) and c[0,1], find a maximum subgraph with all degrees at most c|U|.

  19. Nondeterministically estimable parameters Divine help: coloring the nodes, orienting and coloring the edges G: directed, (edge)-colored graph G’: forget orientation, delete some colors, forget coloring; shadow of G g:parameterdefined ondirected, colored graphs g’(H)=max{g(G): G’=H}; shadow of g fnondeterministically estimable: f=g’,wheregis an estimable parameterof colored directed graphs.

  20. Nondeterministically estimable parameters Examples: density of maximum cut Goldreich-Goldwasser-Ron edit distance from a testable property Fischer- Newman the graph contains a subgraph G’ with all degrees cn and |E(G’)| an2

  21. Nondeterministically estimable parameters Every nondeterministically estimable graph pproperty is testable. Every nondeterministically estimable graph paratemeter is estimable. L-Vesztergombi L-Vesztergombi N=NP fordense property testing Proofviagraph limit theory: pure existence proof of an algorithm...

  22. Howtocompute a maximum matching? More generally, how to compute a witness in non-deterministic property testing?

  23. Happy Birthday Ravi!

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