1. Finding Almost-Perfect
Graph Bisections
Venkatesan Guruswami (CMU)
Yury Makarychev (TTI-C)
Prasad Raghavendra (Georgia Tech)
David Steurer (MSR)
Yuan Zhou (CMU)

2. Bipartite graph recognition
Depth-first search/breadth-first search
With some noise?
Given a bipartite graph with 1% noisy edges,
can we remove a small fraction of edges (10% say) to get a bipartite graph,
i.e. can we divide the vertices into two parts, so that 90% of the edges go accross the two parts?

3. MaxCut G=(V,E) cut(A, B) = edges(A, B) / |E| B = V - A where B = V - A
exact one of i, j in A :
edge (i, j) "on the cut"
MaxCut: find A, B such
that cut(A, B) is maximized
Bipartite graph recognition: MaxCut = 1 ?
Robust bipartite graph recognition: given MaxCut ≥ 0.99, to find cut(A, B) ≥ 0.9
B = V - A A
cut(A, B) = 4/5
subject to

4. c vs. s approximation for MaxCut
Given a graph with MaxCut value at least c, can we find a cut of value at least s ?
Robust bipartite graph recognition: given MaxCut ≥ 0.99, to find cut(A, B) ≥ 0.9
0.99 vs 0.9 approximation
"approximating almost perfect MaxCut"

5. Robust bipartite graph recognition
Task: given MaxCut ≥ 0.99, find cut(A, B) ≥ 0.9
We can always find cut(A, B) ≥ 1/2.
Assign each vertex -1, 1 randomly
For any edge (i, j), E[(1 - xixj)/2] = 1/2 vi vj

6. Robust bipartite graph recognition (cont'd)
Task: given MaxCut ≥ 0.99, find cut(A, B) ≥ 0.9
We can always find cut(A, B) ≥ 1/2.
Better than 1/2?
DFS/BFS/greedy?
Linear Programming?
No combinatorial algorithm known until very recent [KS11]
Natural LPs have big Integrality Gaps [VK07, STT07, CMM09]

7. Robust bipartite graph recognition (cont'd)
Task: given MaxCut ≥ 0.99, find cut(A, B) ≥ 0.9
We can always find cut(A, B) ≥ 1/2.
Better than 1/2?
The GW Semidefinite Programming relaxation [GW95]
0.878-approximation
Given MaxCut , can find a cut
vs approximation, tight under Unique Games Conjecture [Kho02, KKMO07, MOO10]
subject to

8. Robust satisfiability algorithms
Given an instance which can be satisfied by removing ε fraction of constraints, to make the instance satisfiable by removing g(ε) fraction of constraints
g(ε) -> 0 as ε -> 0
Examples
vs algorithm for MaxCut [GW95]
vs algorithm for Max2SAT [Zwick98]
vs algorithm for MaxHorn3SAT [Zwick98]

9. MaxBisection
G = (V, E)
Objective: A B
subject to

10. MaxBisection (cont'd) Approximating MaxBisection?
No easier than MaxCut
Reduction: take two copies of the MaxCut instance
G = (V, E)
Objective: A B

11. MaxBisection (cont'd) Approximating MaxBisection?
No easier than MaxCut
Strictly harder than MaxCut?
Approximation ratio: [FJ97], [Ye01], [HZ02], [FL06]
Approximating almost perfect solutions? Not known
G = (V, E)
Objective: A B

12. Finding almost-perfect MaxBisection
Question
Is there a vs approximation algorithm for MaxBisection, where ?
Answer. Yes.
Our result.
Theorem. There is a vs approximation algorithm for MaxBisection.
Theorem. Given a satisfiable MaxBisection instance, it is easy to find a (.49, .51)-balanced cut of value

13. Extension to MinBisection
minimize edges(A, B)/|V|, s.t. B = V - A, |B| = |A|
Our result
Theorem. There is a vs approximation algorithm for MaxBisection.
Theorem. Given a MinBisection instance of value , it is easy to find a (.49, .51)-balanced cut of value

14. The rest of this talk... Previous algorithms for MaxBisection.
Theorem. There is a vs approximation algorithm for MaxBisection.

15. Previous algorithms for MaxBisection

16. The GW algorithm for (almost perfect) MaxCut [GW95]
MaxCut objective
SDP relaxation
subject to
MaxCut = 2/3
subject to
SDP ≥ MaxCut
In this example:
SDP = 3/4 > MaxCut

17. The "rounding" algorithm
Lemma. We can (in poly time) get a cut of value when
Algorithm. Choose a random hyperplane, the hyperplane divides the vertices into two parts.
Analysis.
subject to

18. The "rounding" algorithm (cont'd)
Lemma. We can (in poly time) get a cut of value when
Algorithm. Choose a random hyperplane, the hyperplane divides the vertices into two parts.
Analysis.
implies for most edges (i, j), their SDP contribution is large
Claim. If , then
Therefore, the random hyperplane cuts many edges (in expectation)
subject to

19. The "rounding" algorithm (cont'd)
Claim. If , then
Proof.
vi, vj seperated by the hyperplane vi vj
vi, vj not seperated by the hyperplane

20. Known algorithms for MaxBisection
The standard SDP (used by all the previous algorithms)
Gives non-trivial approximation gaurantee
But does not help find almost perfect MaxBisection
, subject to
Bisection condition

21. Known algorithms for MaxBisection (cont'd)
The standard SDP (used by all the previous algorithms)
The "integrality gap"
, subject to
OPT < 0.9
SDP = 1

22. Known algorithms for MaxBisection (cont'd)
The standard SDP (used by all the previous algorithms)
The "integrality gap" : instances that OPT < 0.9, SDP = 1
Why is this a bad news for SDP?
Instances that OPT > 1 - ε, SDP > 1 - ε
Instances that OPT < 0.9, SDP > 1 - ε
SDP cannot tell whether an instance is almost satisfiable (OPT > 1 - ε) or not.
, subject to

23. Our approach

24. Theorem. There is a vs approximation algorithm for MaxBisection.

25. A simple fact Fact. -balanced cut of value bisection of value .
Proof. Get the bisection by moving fraction of random vertices from the large side to the small side.
fraction of cut edges affected : at most in expectation
Only need to find almost bisections.

26. Almost perfect MaxCuts on expanders
λ-expander: for each , such that ,
we have , where
G=(V,E) S

27. Almost perfect MaxCuts on expanders (cont'd)
λ-expander: for each , such that ,
we have , where
Key Observation. The (volume of) difference between two cuts on a λ-expander is at most
Proof. C X A B Y D

28. Almost perfect MaxCuts on expanders (cont'd)
λ-expander: for each , such that ,
we have , where
Key Observation. The (volume of) difference between two cuts on a λ-expander is at most
Approximating almost perfect MaxBisection on expanders is easy.
Just run the GW alg. to find the MaxCut.

29. The algorithm (sketch)
Decompose the graph into expanders
Discard all the inter-expander edges
Approximate OPT's behavior on each expander by finding MaxCut (GW)
Discard all the uncut edges
Combine the cuts on the expanders
Take one side from each cut to get an almost bisection. (subset sum)
Step 2:
find MaxCut
Step 3:
combine pieces
Step 1: decompose into expanders
G=(V,E)

30. Expander decomposition
Cheeger's inequality. Can (efficiently) find a cut of sparsity if the graph is not a -expander.
Corollary. A graph can be (efficiently) decomposed into expanders by removing edges (in fraction).
Proof.
If the graph is not an expander, divide it into small parts by sparsest cut (cheeger's inequality).
Process the small parts recursively.
G=(V,E)
λ-expander

31. The algorithm Decompose the graph into -expanders. Lose edges.
Apply GW algorithm on each expander to approximate OPT.
OPT(MaxBisection) =
GW finds cuts on these expanders
different from behavior of OPT
Lose edges.
Combine the cuts on the expanders (subset sum).
-balanced cut of value
a bisection of value

32. Theorem. There is a vs approximation algorithm for MaxBisection.
Proved:
Theorem. There is a vs approximation algorithm for MaxBisection.
Will prove:
short story

33. Eliminating the factor
Recall. Only need to find almost bisections ( close to a bisection)
Observation. Subset sum is "flexible with small items"
Making small items
more biased does not
change the solution
too much.
(101, 304)
(397, 201)
(8, 0)
(3, 5)
(8, 0)
(6, 2)
(6, 0)
(5, 1)
(5, 0)
(3, 2)
sum
(515, 515)

34. Eliminating the factor
Recall. Only need to find almost bisections ( close to a bisection)
Observation. Subset sum is "flexible with small items"
Making small items
more biased does not
change the solution
too much.
(101, 304)
(397, 201)
(8, 0)
(8, 0)
(6, 0)
(5, 0)
sum
(498, 505)

35. Eliminating the factor
Recall. Only need to find almost bisections ( close to a bisection)
Observation. Subset sum is "flexible with small items"
Making small items
more biased does not
change the solution
too much.
(101, 304)
(397, 201)
(8, 0)
(8, 0)
(6, 0)
(5, 0)
sum
(506, 505)

36. Eliminating the factor
Recall. Only need to find almost bisections ( close to a bisection)
Observation. Subset sum is "flexible with small items"
Making small items
more biased does not
change the solution
too much.
(101, 304)
(397, 201)
(8, 0)
(0, 8)
(6, 0)
(5, 0)
sum
(506, 513)

37. Eliminating the factor
Recall. Only need to find almost bisections ( close to a bisection)
Observation. Subset sum is "flexible with small items"
Making small items
more biased does not
change the solution
too much.
(101, 304)
(397, 201)
(8, 0)
(0, 8)
(6, 0)
(5, 0)
sum
(512, 513)

38. Eliminating the factor
Recall. Only need to find almost bisections ( close to a bisection)
Observation. Subset sum is "flexible with small items"
Making small items
more biased does not
change the solution
too much.
(101, 304)
(397, 201)
(8, 0)
(0, 8)
(6, 0)
(5, 0)
sum
(517, 513)

39. Eliminating the factor
Recall. Only need to find almost bisections ( close to a bisection)
Observation. Subset sum is "flexible with small items"
Making small items
more biased does not
change the solution
too much.
However, making small items
more balanced might be
a bad idea.
(200, 0)
(0, 2)
(0, 2)
100
copies
(0, 2)
sum
(200, 200)

40. Eliminating the factor
Recall. Only need to find almost bisections ( close to a bisection)
Observation. Subset sum is "flexible with small items"
Making small items
more biased does not
change the solution
too much.
However, making small items
more balanced might be
a bad idea.
(200, 0)
(1, 1)
(1, 1)
100
copies
(1, 1)
sum
(300, 100)

41. Eliminating the factor (cont'd)
Idea. Terminate early in the decomposition process. Decompose the graph into
-expanders (large items), or
subgraphs of vertices (small items).
Corollary. Only need to discard edges.
Lemma. We can find an almost bisection if the MaxCuts we get for small sets are more biased than those in OPT.

42. Finding a biased MaxCut
To find a cut that is as biased as OPT and as good as OPT (in terms of cut value).
Lemma. Given G=(V,E), if there exists a cut (X, Y) of value , then one can find a cut (A, B) of value , such that
MaxBisection
Biased MaxCut

43. The algorithm Decompose the graph into -expanders or small parts.
Lose edges.
Apply GW algorithm on each expander to approximate OPT.
Lose edges, different from OPT
Find biased MaxCuts in small parts.
Lose edges, at most less biased than OPT
Combine the cuts on the expanders and small parts (subset sum).
-balanced cut of value
a bisection of value

44. Finding a biased MaxCut -- A simpler task
Lemma. Given G=(V,E), if there exists a cut (X, Y) of value , then one can find a cut (A, B) of value , such that
SDP.
Claim. SDP ≥ |X|/|V|
--- Bias
maximize
subject to
--- Cut value

45. Rounding algorithm (sketch)
Goal: given SDP solution, to find a cut (A, B) such that
For most ( fraction) edges (i, j), we have
vi, vj are almost opposite to each other: vi ≈ - vj,
Indeed,

46. Rounding algorithm (sketch) (cont'd)
for most edges (i, j):
Project all vectors to v0
Divide v0 axis into intervals
length =
Most ( fraction )
edges' incident
vertices fall into
opposite intervals (good edges)
Discard all bad edges v0
I(-4) I(-3) I(-2) I(-1) I(1) I(2) I(3) I(4)

47. Rounding algorithm (sketch) (cont'd)
Let the cut (A, B) be
for each pair of intervals I(k) and I(-k),
let A include the one with more vertices,
B include the other
(A, B) cuts all
good edges v0

48. Rounding algorithm (sketch) (cont'd)
Let the cut (A, B) be
for each pair of intervals I(k) and I(-k),
let A include the one with more vertices,
B include the other
For each i in I(k) For each i in I(-k)

49. Finding a biased MaxCut
Lemma. Given G=(V,E), if there exists a cut (X, Y) of value , then one can find a cut (A, B) of value , such that
SDP.
--- Bias
maximize
subject to
--- Cut value
-triangle inequality

50. Future directions vs approximation?
"Global conditions" for other CSPs.
Balanced Unique Games?

51. The End. Any questions?

52. Eliminating the factor
Another key step.
Idea. Terminate early in the decomposition process. Decompose the graph into expanders or subgraphs of vertices.
Corollary. Only need to discard edges.
Lemma. We can find an almost bisection if the MaxCuts for small sets are more biased than those in OPT.
MaxBisection
Biased MaxCut

53. Finding a biased MaxCut
Lemma. Given G=(V,E), if there exists a cut (X, Y) of value , then one can find a cut (A, B) of value , such that
SDP.
Rounding. A hybrid of hyperplane and threshold rounding.
maximize
subject to
-triangle inequality

54. Future directions vs approximation?
"Global conditions" for other CSPs.
Balanced Unique Games?

55. The End. Any questions?