On the hardness of approximating SparsestCut and Multicut

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On the hardness of approximating Sparsest-Cut and
Multicut

• Shuchi Chawla, Robert Krauthgamer, Ravi Kumar,
  Yuval Rabani, D. Sivakumar

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Multicut
s1
s3
s2
t4
s4
Goal separate each si from ti removing the
fewest edges
t2
t3
t1
Cost 7
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Sparsest Cut
s1
s3
For a set S, demand D(S) no. of pairs
separated capacity C(S) no. of edges
separated
s2
t4
s4
Sparsity C(S)/D(S)
t2
Goal find a cut that minimizes sparsity
t3
t1
Sparsity 1/1 1
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Approximating Multicut Sparsest Cut
O(log n) approx via LPs GVY96 APX-hard
DJPSY94 Integrality gap of O(log n) for LP
SDP ACMM05
Multicut
O(log n) for uniform demands LR88 O(log n)
via LPs LLR95, AR98 O(?log n) for uniform
demands via SDP ARV04 O(log3/4n) CGR05,
O(?log n log log n) ALN05 Nothing known!
Sparsest Cut
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Our results

• Use Khots Unique Games Conjecture (UGC)
• A certain label cover problem is NP-hard to
  approximate
• The following holds for Multicut, Sparsest Cut
  and
• Min-2CNF? Deletion
• UGC ? L-hardness for any constant L gt 0
• Stronger UGC ? W(log log n)-hardness

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A label-cover game
Given A bipartite graph Set of labels for
each vertex Relation on labels for edges To
find A label for each vertex Maximize no.
of edges satisfied Value of game fraction of
edges satisfied by best solution
Is value ? or value lt ? ? is NP-hard
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Unique Games Conjecture
Given A bipartite graph Set of labels for
each vertex Bijection on labels for
edges To find A label for each vertex
Maximize no. of edges satisfied Value of game
fraction of edges satisfied by best solution
UGC Is value gt ??? or value lt ? ? is
NP-hard
Khot02
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The power of UGC

• Implies the following hardness results
• Vertex-Cover 2 ? ? KR03
• Max-cut ?GW 0.878 KKMO04
• Min 2-CNF Deletion
• Max-k-cut
• 2-Lin-mod-2

. . .
UGC Is value gt ??? or value lt ? ? is
NP-hard
Khot02
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The plausibility of UGC
k labels
?
?
n nodes
1
0
Strongest plausible version 1/?, 1/? lt min ( k
, log n )
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Our results

• Use Khots Unique Games Conjecture (UGC)
• A certain label cover problem is hard to
  approximate
• The following holds for Multicut, Sparsest Cut
  and
• Min-2CNF? Deletion
• UGC ? W( log 1/(d?) )-hardness
• ? L-hardness for any constant L gt 0
• Stronger UGC ? W( log log n )-hardness
• ( k ? log n, ?,? ? 1/log n )

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The key gadget

• Cheapest cut a dimension cut
• cost 2d-1
• Most expensive cut diagonal cut
• cost O(?d 2d)
• Cheap cuts lean heavily on few
  dimensions

KKL88
Suppose size of cut lt x 2d-1 Then, ? a
dimension h such that fraction of edges cut
along h gt 2-W(x)
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Relating cuts to labels
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Good Multicut ? good labeling
Suppose that cross-edges cannot be cut
Each cube must have exactly the same cut!
cut lt log (1/?) 2d-1 per cube ?
?-fraction of edges can be satisfied Conversely,
a NO-instance of UG ? cut gt log (1/?)
2d-1 per cube
Picking labels for a vertex
edges cut in dimension h
total edges cut in cube


Prob label1 h1 label2 h2 gt


Prob label h
2-2x x2
2-x x
gt
If cut lt x 2d-1
gt ? for x O(log 1/?)
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Good labeling ? good Multicut
Constructing a good cut given a label
assignment For every cube, pick the dimension
corresponding to the label of the vertex
What about unsatisfied edges? Remove the
corresponding cross-edges
a YES-instance of UG ? cut lt 2d per
cube
Cost of cross-edges n/?m
no. of nodes
no. of edges in UG
Total cost ? 2d-1 n ?m2d-1 n/?m
? O(2d n) O(2d) per cube
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Revisiting the NO instance

• Cheapest multicut may cut cross-edges
• Cannot cut too many cross-edges on average
• For most cube-pairs, few edges cut
• ? Cuts on either side are similar, if not the
  same
• Same analysis as before follows

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A recap

• NO-instance of UG ? cut gt log 1/(??) 2d-1 per
  cube
• YES-instance of UG ? cut lt
  2d per cube
• UGC NP-hard to distinguish between YES and
  NO instances of UG
• NP-hard to distinguish between whether
• cut lt 2dn or cut gt log 1/(??) 2d-1 n
• W( log 1/(??) )-hardness for Multicut

?
?
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Extensions to other problems

• Obvious extension to Min-CNF? Deletion
• Think of edges as 2-variable constraints
• Bi-criteria Multicut
• Allowed to separate only a ? ? ¼ frac of the
  demand-pairs
• Fourier analysis stays the same cheap cuts
  cutting ¼th of the pairs are close to dimension
  cuts
• Similar guarantee follows
• Sparsest Cut
• Simple extension of bi-criteria Multicut

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A related result

• Khot Vishnoi 05
• Independently obtain ?( min (1/?, log 1/?)1/6 )
  hardness based on the same assumption
• Use this to develop an integrality-gap instance
  for the Sparsest Cut SDP
• A graph with low SDP value and high actual value
• Implies that we cannot obtain a better than O(log
  log n)1/6 approximation using SDPs
• Independent of any assumptions!

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Open Problems

• Improving the hardness
• Fourier analysis is tight
• Prove/disprove UGC
• Reduction based on a general 2-prover system
• Improving the integrality gap for sparsest cut
• Hardness for uniform sparsest cut, min-bisection
  ?