Yet another algorithm for dense max cut - go greedy Claire Mathieu Warren Schudy (presenting) Brown University Computer Science SODA 2008

Max cut Splitting an area code in two… …to maximize long distance charges! 2-layer circuit board layout Research platform – e.g. first use of SDP in approximation algorithms

Standard greedy for Max-cut approx for general graphs (Animation done)

Dense graphs Definition: (n vertices) Poly-Time Approximation Schemes for dense graphs by: –Arora, Karger and Karpinski 95 –Fernandez de la Vega 96 –Goldreich, Goldwasser and Ron 98. –Frieze and Kannan 99 We prove the same theorem using a simpler algorithm (Animation done)

Take a random sample of vertices For all colorings of sampled vertices –Add remaining vertices greedily in random order Return best overall coloring found OPT Seeded greedy algorithm (Animation done) Constructed coloring Analyze when it guesses OPT

Our results Seeded greedy algorithm satisfies in time. The standard seedless greedy, when repeated times with random order, also works. Simpler proof than Alon, Fernandez de la Vega, Kannan, and Karpinski (2003) that the sample complexity of MaxCut is Results extend to weighted MAX-r-CSP (Animation done)

Talk outline Introduction (done) Analysis of seeded greedy: –Introduction of the smoothed coloring –Using the relation between the smoothed and constructed colorings to lower-bound the number of cut edges (profit) of the output Conclusions

Before choosing a random vertex, determine the greedy color for each Are we done updating S? No, because 1/3 of C was greedy, but only 1/7 of S was greedy! (Animation done) Constructed coloring C The Smoothed Coloring Smoothed coloring S (initialized to OPT) Time: 2 2 ½3 G G G G G

(Animation done) Next vertex… (Animation done) Update: Constructed coloring (C) Smoothed coloring (S) Time: 3 3 ½4 G G G G

4 (Animation done) Another vertex… (Animation done) Constructed coloring (C) Smoothed coloring (S) Time: 4 ½5 GG G Update:

(Animation done) Penultimate (Animation done) Constructed coloring (C) Smoothed coloring (S) Time: 6 5 GG Update:

(Animation done) Final vertex (Animation done) Smoothed coloring starts at OPT and ends at output Therefore it suffices to bound the change in profit of the smoothed coloring at each time step Constructed coloring (C) Smoothed coloring (S) Time: 76 G

(Animation done) S Changes Slowly (Animation done) Smoothed coloring (S)Time: 4 Time: 5 At most (fractional) vertices change color Consider each changing vertex separately (interactions negligible).

This vertex will gain a blue wedge and becomes. Net change: into Bounding the lost profit (Blue wins ties) (Animation done) Constructed coloring (C) Smoothed coloring (S) Time: 3

Finishing the proof (Animation done) By greedy

Conclusions Problem: dense weighted max cut and max-CSP Algorithm: seeded greedy Analysis: –Smoothed / extrapolated coloring –Martingale Bonus: simpler sample complexity proof

Questions? Acknowledgments: –Brown theory lunch and Claire Mathieu for comments on preliminary talks. (Animation done)