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1. How to get Hardness of approximation results from Integrality gaps Guy KindlerWeizmann Institute

2. About this talk We’ll try to understand some notions, and their relations: • Combinatorial optimization problems • Approximation: relaxation by semi-definite programs. • Integrality gaps • Hardness of approximation Main example: the Max-Cut problem

3. Combinatorial optimization problems Input, search space, objective function Example: MAX-CUT

4. Example: MAX-CUT • input: G = (V,E) • Search space: Partition V=(C, Cc) • Objective function: w(C) = fraction of cut edges • The MAX-CUT Problem:Find mc(G)=maxC{w(C)} • [Karp ’72]: MAX-CUT is NP-complete

5. Example: MAX-CUT • input: G = (V,E) • Search space: Partition V=(C, Cc) • Objective function: w(C) = fraction of cut edges • The MAX-CUT Problem:Find mc(G)=maxC{w(C)} • -approximation: Output S, s.t. mc(G) ¸ S ¸ ¢mc(G). • History: ½-approximation easy, was best record for long time.

6. Semi-definite Relaxation Introducing geometry into combinatorial optimization [GW ’95]

7. Arithmetization v G=(V,E): u xv xu Problem: We can’t maximize quadratic functions,even over convex domains.

8. Relaxation by geometric embedding v G=(V,E): u xv xu Problem: We can’t maximize quadratic functions,even over convex domains.

9. Relaxation by geometric embedding v G=(V,E): u xv xu Problem: We can’t maximize quadratic functions,even over convex domains.

10. Relaxation by geometric embedding v G=(V,E): u xv xu Problem: We can’t maximize quadratic functions,even over convex domains.

11. xv xu Relaxation by geometric embedding Semi-definiterelaxation v G=(V,E): u Now we’re maximizing a linear function over a convex domain! (unit sphere in Rn)

12. v G=(V,E): u Relaxation by geometric embedding Is this really a relaxation? (unit sphere in Rn)

13. v G=(V,E): u xv xu Relaxation by geometric embedding Is this really a relaxation? (unit sphere in Rn)

14. v G=(V,E): u Analysis by randomized rounding xv xu xu (unit sphere in Rn)

15. xv xv xu <xu,xv> xu donation to (*) Analysis by randomized rounding arccos(<xu,xv>) • So (unit sphere in Rn)

16. xv <xu,xv> arccos(<xu,xv>) xu donation to (*) Analysis by randomized rounding • So

17. Analysis by randomized rounding • So mc(G)¸ GW¢rmc(G) ¸ GW¢mc(G) L.h.s. is tight, iff all inner products are ρ*

18. An 0.879 approximation for Max-Cut Is GW the best constant here? • The [GW ’95] algorithm: • Given G, compute rmc(G) • Let S=GW¢rmc(G) • Output S. mc(G)¸ S¸ GW¢mc(G) mc(G)¸ GW¢rmc(G) ¸ GW¢mc(G) Is there a graph where this occurs? L.h.s. is tight, iff all inner products are ρ*

19. Integrality gap Measuring the quality of the geometric approximation [FS ’96]

20. The integrality gap of Max-Cut On instance G: w rmc(G) mc(G) r-S(G) S(G)

21. The integrality gap of Max-Cut ¸ GW =GW for some G* On instance G: w rmc(G) The [GW ’95] algorithm: Given G, output IG¢rmc(G) mc(G) r-S(G) S(G)

22. The integrality gap of Max-Cut =GW for some G* On instance G: • Using ¢rmc(G) to approximate mc(G), the factor =IG cannot be improved! • On G* the algorithm computes mc(G*) perfectly! w rmc(G) The [GW ’95] algorithm: Given G, output IG¢rmc(G) mc(G) r-S(G) S(G)

23. (unit sphere in Rq) The Feige-Schechtman graph, G* arccos(ρ*) Vertices: Sn Edges: u~v iff <u,v>=* • rmc(F)¸(1-*)/2 • [FS]: mc(G*)=arccos(ρ*)/ • so

24. From IG to hardness A geometric trick may actually be inherent [KKMO ’05]

25. Thoughts about integrality gaps *under some reasonable comlexity theoretic assumptions • The integrality gap is an artifact of the relaxation. • The relaxation does great on an instance for which the integrality gap is achieved. • And yet, sometimes the integrality gap is provably* the best approximation factor achievable: • [KKMO ’04]: under UGC, GW is optimal for max-cut. • [HK ’03], [HKKSW], [KO ’06], [ABHS ’05], [AN ’02]

26. Thoughts about integrality gaps And the IG instance G* is used in the hardness proof • The integrality gap is an artifact of the relaxation. • An algorithm does great on an instance for which the integrality gap is achieved. • And yet, sometimes the integrality gap is provably* the best approximation factor achievable: • [KKMO ’04]: Under UGC, GW is optimal for max-cut. • [HK ’03], [HKKSW], [KO ’06], [ABHS ’05], [AN ’02]

27. A recipe for proving hardness Take the instance G*: w rmc(G*) mc(G*) r-S(G*) S(G*)

28. A recipe for proving hardness Add ‘teeth’ to S(G*)which achieve rmc(G*). w rmc(G*) mc(G*) r-S(G*) S(G*)

29. A recipe for proving hardness Now combine several instances, such that finding a point which belongs to all teeth becomes a hard combinatorialoptimization problem. w rmc(G*) mc(G*) r-S(G*) S(G*)

30. A recipe for proving hardness Now combine several instances, such that finding a point which belongs to all teeth becomes a hard combinatorialoptimization problem. w

31. A recipe for proving hardness Determining whether mc(G`)=mc(G*) or whethermc(G`)=rmc(G*) is intractable. Factor of hardness: mc(G*)/rmc(G*)=IG !! w

32. Vertices: Sn Edges: u~v iff <u,v>=* Adding teeth to Feige-Schechtman (unit sphere in Rq)

33. xi w.p. ½ + ½ρ -xi w.p. ½ - ½ρ y: yi = (unit sphere in Rq) Adding teeth to Feige-Schechtman Vertices: {-1,1}n Edges: u~v iff <u,v>=*: Vertices: Sn Edges: u~v iff <u,v>=* a random edge (x,y): x~{-1,1}n, • E[<x,y>] = ρ

34. Adding teeth to Feige-Schechtman • mc = arccos(ρ)/ ? • For C(x) = x7, • w(C) = Pedge[x7 ≠ y7] = (1-ρ)/2 !!

35. Adding teeth to Feige-Schechtman • mc = arccos(ρ)/ ? • For C(x) = x7, • w(C) = Pedge[x7 ≠ y7] = (1-ρ)/2 !! • For C(x) = sign(xi) = Maj(x), w(C) = P[Maj(x)≠Maj(y)]≈ (arccos ρ)/π no influential coordinates

36. Adding teeth to Feige-Schechtman • mc = arccos(ρ)/ ? • [for axis parallel cut]:w(C)=(1-ρ)/2 • [MOO ‘05]:If i, Ii(C)<, w(C) (arccos ρ)/π +o(1) • Ratio between weight of ‘teeth’ cuts and regular cuts is ≈ GW(for ρ = ρ*)

37. Conclusion We tried to understand some notions, and their relations: • Combinatorial optimization problems • Approximation: relaxation by semi-definite programs. • Integrality gaps • Hardness of approximation