Yet another algorithm for dense max cut - go greedy

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

2. 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 Customers Frequent calls Long distance charges: $2 $3

3. Standard greedy for Max-cut 10 21 01 • 0.5-approx for general graphs (Animation done)

4. 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)

5. Take a random sample of vertices For all colorings of sampled vertices Add remaining vertices greedily in random order Return best overall coloring found Seeded greedy algorithm Analyze when it guesses OPT OPT 01 10 Constructed coloring 22 12 01 (Animation done)

6. 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)

7. 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

8. The Smoothed Coloring Smoothed coloring S(initialized to OPT) 2 2 ½ 3 Constructed coloring C Time: 00 10 G G 10 G G 11 Before choosing a random vertex, determine the greedy color for each G Are we done updating S? No, because 1/3 of C was greedy, but only 1/7 of S was greedy! 01 (Animation done)

9. Next vertex… Smoothed coloring (S) 3 ½ 4 3 Time: 01 Constructed coloring (C) G 11 G G 12 01 G • Update: (Animation done) (Animation done)

10. Another vertex… Smoothed coloring (S) 4 4 ½ 5 Time: 01 Constructed coloring (C) G 11 G G 12 • Update: (Animation done) (Animation done)

11. Penultimate Smoothed coloring (S) 5 6 Time: Constructed coloring (C) 11 G G 22 • Update: (Animation done) (Animation done)

12. 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 Final vertex Smoothed coloring (S) 6 7 Time: Constructed coloring (C) 12 G (Animation done) (Animation done)

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

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

15. Finishing the proof By greedy (Animation done)

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

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