On the hardness of approximating Sparsest-Cut and Multicut

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

2. Multicut s3 s1 s2 t4 s4 Goal: separate each si from ti removing the fewest edges t2 t3 t1 Cost = 7 Shuchi Chawla

3. Sparsest Cut s3 s1 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 Shuchi Chawla

4. Approximating Multicut & Sparsest Cut Multicut O(log n) approx via LPs [GVY’96] APX-hard [DJPSY’94] Integrality gap of O(log n) for LP & SDP [ACMM’05] 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! 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 W(log log n)-hardness 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 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] Shuchi Chawla

8. 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 Chawla

9. Conjecture is plausible Conjecture is true (1) 1- (1) conjectured NP-hard [Khot 02] 1/k 1-k-0.1 solvable [Khot 02] L()  known NP-hard [FR 04] 1- 1/3 1- (/log n) solvable [Trevisan 05] The plausibility of UGC k : # labels   n : # nodes 1 0 Strongest plausible version: 1/, 1/ < min ( k , log n ) 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  W( log 1/(d+) )-hardness  L-hardness for any constant L > 0 • Stronger UGC W( log log n )-hardness ( k  log n, ,  1/log n ) Shuchi Chawla

11. 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 < x 2d-1 Then,  a dimension h such that: fraction of edges cut along h > 2-W(x) Shuchi Chawla

12. ( , , ) Relating cuts to labels Shuchi Chawla

13. Good Multicut  good labeling Suppose that “cross-edges” cannot be cut Each cube must have exactly the same cut! 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 Picking labels for a vertex: # edges cut in dimension h total # edges cut in cube * * Prob[ label1 = h1 & label2 =h2 ] > * * Prob[ label = h ] = 2-2x x2 2-x x > [ If cut < x 2d-1 ] >  for x = O(log 1/) 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 a “YES”-instance of UG  cut < 2d per cube What about unsatisfied edges? Remove the corresponding cross-edges 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 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 Shuchi Chawla

16. 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 W( log 1/(+) )-hardness for Multicut   Shuchi Chawla

17. 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 Chawla

18. 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 … ? Shuchi Chawla