On the hardness of approximating Sparsest-Cut and Multicut Shuchi Chawla, Robert Krauthgamer, Ravi Kumar, Yuval Rabani, D. Sivakumar

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 2 Multicut s1s1 t1t1 Goal: separate each s i from t i removing the fewest edges s2s2 s4s4 s3s3 t3t3 t2t2 t4t4 Cost = 7

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 3 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) s1s1 t1t1 s2s2 s4s4 s3s3 t3t3 t2t2 t4t4 Sparsity = 1/1 = 1

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 4 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(log 3/4 n) [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

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 5 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

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 6 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

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 7 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]

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 8 The power of UGC Implies the following hardness results –Vertex-Cover2   [KR’03] –Max-cut  GW = [KKMO’04] –Min 2-CNF Deletion –Max-k-cut –2-Lin-mod-2 UGC: “Is value >  or value <  ?” is NP-hard [Khot’02]...

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla  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 true Conjecture is plausible  (1) (1) 1 -  ( 1 ) conjectured NP-hard [Khot 02] k : # labels n : # nodes Strongest plausible version: 1 / , 1 /  < min ( k, log n )  

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 10 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 )

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 11 The key gadget Cheapest cut – a “dimension cut” cost = 2 d-1 Most expensive cut – “diagonal cut” cost = O(  d 2 d ) Cheap cuts lean heavily on few dimensions Suppose:size of cut < x 2 d-1 Then,  a dimension h such that: fraction of edges cut along h > 2 -  (x) [KKL88]:

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 12 Relating cuts to labels (,, )

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 13 Picking labels for a vertex: # edges cut in dimension h total # edges cut in cube Prob[ label 1 = h 1 & label 2 = h 2 ] > 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 2 d-1 ] 2 -x x > 2 -2x x 2 >  for x = O(log 1 /  ) * ** * cut < log ( 1 /  ) 2 d-1 per cube   -fraction of edges can be satisfied Conversely, a “NO”-instance of UG  cut > log ( 1 /  ) 2 d-1 per cube

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 14 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  2 d-1 n +  m2 d-1 n/  m  O(2 d n) = O(2 d ) per cube no. of nodes no. of edges in UG a “YES”-instance of UG  cut < 2 d per cube

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 15 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

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 16 A recap… “NO”-instance of UG  cut > log 1/(  +  ) 2 d-1 per cube “YES”-instance of UG  cut < 2 d per cube UGC:NP-hard to distinguish between “YES” and “NO” instances of UG NP-hard to distinguish between whether cut log 1/(  +  ) 2 d-1 n  ( log 1/(  +  ) )-hardness for Multicut  

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 17 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 ¼ th of the pairs are close to dimension cuts –Similar guarantee follows Sparsest Cut –Simple extension of bi-criteria Multicut

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 18 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!

On the hardness of approximating Multicut & Sparsest Cut Shuchi Chawla 19 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 … ?