On the hardness of approximating Sparsest-Cut

and Multicut

Shuchi Chawla, Robert Krauthgamer, Ravi Kumar, Yuval

Rabani, D. Sivakumar

Shuchi Chawla2 On the hardness of approximating Multicut & Sparsest Cut

Multicut

s1

t1

Goal: separate each si from ti

removing the fewest edges

s2

s4

s3

t3

t2

t4

Cost = 7

Shuchi Chawla3 On the hardness of approximating Multicut & Sparsest Cut

Sparsest Cut

Goal: find a cut that minimizes sparsity

For a set S, “demand” D(S)

= no. of pairs separated “capacity” C(S)

= no. of edges separated

Sparsity = C(S)/D(S)

s1

t1

s2

s4

s3

t3

t2

t4

Sparsity = 1/1 = 1

Shuchi Chawla4 On the hardness of approximating Multicut & Sparsest Cut

Approximating Multicut & Sparsest Cut

O(log n) for “uniform” demands [LR’88]

O(log n) via LPs [LLR’95, AR’98]

O(log n) for uniform demands via SDP [ARV’04]

O(log3/4n) [CGR’05], O(log n log log n) [ALN’05]

Nothing known!

Sparsest Cut

O(log n) approx via LPs [GVY’96]

APX-hard [DJPSY’94]

Integrality gap of O(log n) for LP & SDP [ACMM’05]

Multicut

Shuchi Chawla5 On the hardness of approximating Multicut & Sparsest Cut

Our results

• Use Khot’s 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 > 0• Stronger UGC (log log n)-hardness

Shuchi Chawla6 On the hardness of approximating Multicut & Sparsest Cut

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 < ?” is NP-hard

Shuchi Chawla7 On the hardness of approximating Multicut & Sparsest Cut

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 > or value < ?” is NP-hard[Khot’02]

Shuchi Chawla8 On the hardness of approximating Multicut & Sparsest Cut

The power of UGC

• Implies the following hardness results– Vertex-Cover 2 [KR’03]

– Max-cut GW = 0.878 [KKMO’04]

– Min 2-CNF Deletion

– Max-k-cut

– 2-Lin-mod-2

UGC: “Is value > or value < ?” is NP-hard[Khot’02]

. . .

Shuchi Chawla9 On the hardness of approximating Multicut & Sparsest Cut

1- 1/3 1- (/log n) solvable [Trevisan 05]

L() known NP-hard [FR 04]

1/k 1-k-0.1 solvable [Khot 02]

The plausibility of UGC

0 1

Conjecture is trueConjecture is plausible

(1) 1- (1) conjectured NP-hard [Khot 02]

k : # labels

n : # nodes

Strongest plausible version: 1/, 1/ < min ( k , log n )

Shuchi Chawla10 On the hardness of approximating Multicut & Sparsest Cut

Our results

• Use Khot’s 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 ( log 1/() )-hardness L-hardness for any constant L

> 0

• Stronger UGC ( log log n )-hardness ( k log n, , 1/log n )

Shuchi Chawla11 On the hardness of approximating Multicut & Sparsest Cut

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

Suppose: size of cut < x 2d-1

Then, a dimension h such that:fraction of edges cut along h >

2-(x)

[KKL88]:

Shuchi Chawla12 On the hardness of approximating Multicut & Sparsest Cut

Relating cuts to labels

( , , )

Shuchi Chawla13 On the hardness of approximating Multicut & Sparsest Cut

Picking labels for a vertex: # edges cut in dimension h

total # edges cut in cube

Prob[ label1 = h1 & label2 = h2 ]

>

Good Multicut good labeling

Suppose that “cross-edges” cannot be cut

Each cube must have exactly the same cut!

Prob[ label = h ] =

[ If cut < x 2d-1 ]2-x

x>

2-2x

x2

> for x = O(log 1/)

** * *

cut < log (1/) 2d-1 per cube

-fraction of edges can be satisfied

Conversely, a “NO”-instance of UG

cut > log (1/) 2d-1 per cube

Shuchi Chawla14 On the hardness of approximating Multicut & Sparsest Cut

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

Cost of cross-edges = n/m

Total cost 2d-1 n + m2d-1 n/m O(2d n) = O(2d) per cube

no. of nodes

no. of edges in UG

a “YES”-instance of UG

cut < 2d per cube

Shuchi Chawla15 On the hardness of approximating Multicut & Sparsest Cut

Revisiting the “NO” instance

• Cheapest multicut may cut cross-edges

• Cannot cut too many cross-edges on averageFor most cube-pairs, few edges cut Cuts on either side are similar, if not the

same

• Same analysis as before follows

Shuchi Chawla16 On the hardness of approximating Multicut & Sparsest Cut

A recap…

“NO”-instance of UG cut > log 1/(+) 2d-1 per cube

“YES”-instance of UG cut < 2d per cube

UGC: NP-hard to distinguish between “YES” and “NO” instances of UG

NP-hard to distinguish between whether cut < 2dn or cut > log 1/(+) 2d-1 n

( log 1/(+) )-hardness for Multicut

Shuchi Chawla18 On the hardness of approximating Multicut & Sparsest Cut

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!

Shuchi Chawla19 On the hardness of approximating Multicut & Sparsest Cut

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 … ?