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Avner Magen Joint work with Costis Georgiou, Toni Pitassi and Iannis Tourlakis

Tight integrality gaps for vertex-cover semidefinite relaxations in the Lov á sz-Schrijver Hierarchy. Avner Magen Joint work with Costis Georgiou, Toni Pitassi and Iannis Tourlakis University of Toronto. Minimum Vertex Cover. Finding minimum size VC is NP -hard

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Avner Magen Joint work with Costis Georgiou, Toni Pitassi and Iannis Tourlakis

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  1. Tight integrality gaps forvertex-cover semidefinite relaxations in the Lovász-Schrijver Hierarchy Avner Magen Joint work with Costis Georgiou, Toni Pitassi and Iannis Tourlakis University of Toronto

  2. Minimum Vertex Cover Finding minimum size VC is NP-hard Exist simple 2-approximations All known algs are 2  o(1) approximations! Probabilistically checkable proofs (PCPs)  No poly-time 1.36 approximation [Dinur-Safra’02] Unique Games Conjecture [Khot’02]  No poly-time 2   approximation [Khot-Regev’03] Alternative (concrete) approach [ABL’02, ABLT’06]: Rule out approximations by large subfamilies of algorithms

  3. Linear Programming approach Easy to see IG ≤ 2 min SiV vi vi + vj ≥ 1, ij E vi  {0,1} for Kn :IG =2 1/n 0 ≤ vi ≤ 1 True Optimum Integrality Gap: max Optimal Fractional Solution

  4. Semidefinite Programming Relaxations SDP: the ultimate remedy? Vertex Cover on G = (V,E) Tighter relaxation?Smaller integrality gap? Kleinberg-Goemans’98: Integrality gap 2  o(1) min SiV (1 + v0 ·vi)/2 (v0  vi) · (v0  vj) = 0, ij  E || vi ||2 = 1, vi  Rn+1 min SiV (1 + x0xi)/2 (x0  xi)(x0  xj) = 0, ij  E |xi| = 1 Charikar’02: Gap still 2  o(1) Hatami-M-Markakis’06: Integrality gap still 2  o(1), even with “pentagonal” inequalities Clearly holds in integral case vi  {1,1} (v0  vi) · (v0  vj)  0, i,j (vi  vj) · (vi  vk)  0, i,j

  5. Systematic Approach: Lovász-Schrijver Liftings [LS’91] • Procedures LS0, LS, LS+ for tightening linear relaxations • Integral hull in≤ n rounds • Optimize over rth round relaxation in nO(r) time Very powerful algorithms obtained through smallnumber of rounds: • GW’94, KZ’97, ARV’04 algorithms “poly-time” in LS+ • All NP in “exponential time” May view super-constant rounds lower bounds in LS+ models as evidence about inapproximability • Has PSD constraint • Sequence of tighter and tighter SDPs “Lift” to obtain SDP Relaxation Initial Linear Relaxation “Project” back to obtain tighter LP Integral Hull n2 variables n variables

  6. Previous Lower Bounds for Vertex Cover – without SDP constraints (LS) • [ABLT’06]: Int. gap 2  o(1) after W(log n)LS rounds • [Tourlakis’06]: Int. gap 1.5  o(1) after W(log2 n)LS rounds • [STT’06b]: Int. gap 2  o(1) after W(n)LS rounds

  7. Status of SDP variant LS+ Stronger: one round already • Implies clique constraint • More generally, gives n-θ(G) lower bound on VC (so sparse graph are generally not good) • Gives rise to SDPs in the “lift” phases.

  8. Integrality gap of 7/6 for LS+ (STT06a) PCP world: Hastad 0.5-hardness for MAX3XOR and the FGLSS reduction imply 7/6-hardness for VC AAT05 proved matching LB (for int. gap) in LS+ world for MAX3XOR STT06b using further ideas from FO06, extend AAT MAX3XOR LB to prove 7/6 int. gap for linear rounds graph family: FGLSS reduction on random MAX3XOR instances Int. gap 7/6 already after one round

  9. Vertex Cover in LS: results so far ABLT ’02,STT ’07 NO YES YES STT ’06 YES NO YES Charikar ’02 YES YES NO New result YES YES YES

  10. Main Result Theorem: Int. gap 2  o(1) for SDPs resulting after (√log n/log log n)LS+ rounds One LS+ round tighter than [C’02] SDP SDPs ruled out incomparable to SDPs with (generalized) triangle and pentagonal inequalities (e.g., [HMM’06]) Theorem: Int. gap 2  O(1/√log n/log log n)after O(1)LS+ rounds Karakostas [K’05] SDP gives 2  (1/√log n) approximation Use same graph families as [KG’98], [C’02], [HMM’06] SDP solutions rely on sequence of polynomials applying tensor operations on vectors

  11. LS+ lift-and-project: the quick guide Homogenization: coneK min SiV xi Convert vertex cover LP into an SDP? Multiply linear inequalities to get valid quadratic constraints. Crucially, add integrality conditions: (x0  xi)xi = 0 E.g., Linearize: replace products xixj with linear variables Yij Lifted SDP in (n + 1)2 variables Project resultingconvex body back onto n + 1 variables Y0i xi + xj  1 (i,j)  E 0  xi  1 i  V xi + xj  x0  0 (i,j)  E xi  0 i  V x0  xi  0 i  V (x0 = 1) (x0 = 1) (x0  xi)xi = 0 Y0i = Yii = xi vk · (vi + vj  v0)  0 ij  E (v0  vi) · (v0  vj) = 0 ij  E (v0  vi) · (v0  vj)  0 Yik + Yjk Y0k  0 ij  E Y00  Y0i  Y0j + Yij = 0 ij  E Y00  Y0i  Y0j + Yij  0 xk(xi + xj  x0)  0 ij  E (x0  xi)(x0 – xj) = 0 ij  E (x0  xi)(x0  xj)  0 xk(xi + xj  x0)  0 ij  E (x0  xi)(xi + xj  x0)  0 ij  E (x0  xi)(x0  xj)  0 xk(xi + xj  x0)  0 ij  E (x0  xi)(xj – x0)  0 ij  E (x0  xi)(x0  xj)  0 Yei ,Y(e0ei)  K Y is PSD

  12. How LS and LS+ tighten VC Relaxation min SiV xi xi + xj ≥ 1, ij  E 0 ≤ xi ≤ 1 One round of LS precisely adds “odd-cycle constraints”: • For all cycles C in G of odd length, SiCxi ≥ (|C|+1)/2 x1 + x2 + x3 ≥ 2 One round of LS+ adds more: • Clique constraints: For all cliques K in G, SiKxi ≥ |K| – 1 vs. x1 + x2 + x3 ≥ 3/2

  13. Deriving the clique constraints in LS+ Let K be a clique of size k in G 0 ≤ S (x0 –xi) (xi + xj –x0) +((k –1)x0 –Sxi) 2 i≠jK iK Edge constraint SDP condition =S xi2 – (k –1) x02 Afterprojecting S xi ≥ k –1 iK

  14. Proving Lower Bounds in LS+ Hierarchies LP relaxation K for Gwith min VC ~n: xi + xj≥1 ij  E (½, ½,…) (½+g, ½+g, …) x K(r)if  matrix Ys.t. • diagonal is x • Y is PSD • “columns”  K(r 1) “Protection” matrix for x Lemma (LS’91): I.H. K(1) K(3) K(2) Int. gap of K is ≥ 2 – o(1)  Use inductive proof: find appropriate Y’s

  15. “Frankl-Rödl” graphs m-dimensional Hamming cube: n = 2m points parameter V = {1,1}m (i, j) E iff(i, j) = (1  )m } Theorem: [Frankl-Rödl’87] Max Ind.Set size  |B(v,n/2(1- ))| m2m(1  2/64)m (i, j) = (1  )m Cor: If  = (√log m/m) then max IS is o(2m) = o(n) Graphs used for int.gaps in [KK91, AK94, KG95, C02, HMM06]

  16. What’s so wonderful about them?... Start with a perfect matching n/2 o(n) • Vertex Cover= +O( ) • ``Geometric’’ vertex cover = n/2 Perturb : edges connect vertices of Ham. Dist. (1-)n

  17. Proof Outline (i, j) = (1  )m VC 1  o(n) x K(r) if  PSD matrix Ys.t. • diagonal is x • “columns”  K(r1) x = (½ + )1 In induction: need vectors vi to define matrix Yij = vi vj • Show vi exist whenever • x {0, 1, ½ + }n and •  > 6 • Ensure S {0, 1, ½ + }nwhere     O()  (/) round lower bound for x = (½ + )1 Constantand = (√log m/m)  Int. gap 2  o(1) after (√log n/log log n) rounds 2’.Show some set SK(r1) where “columns” conv(S)

  18. V = {1,1}m (i, j) = (1  )m Back to Frankl-Rödl graphs VC 1  o(n) 1 √m Natural set {ui} of unit vectors: {1,1}m Note:ui · uj = 1  2(i, j)/m 2 1 for (i, j)  E Hence (i, j)  E  uianduj nearly antipodal Nearly true for vi =ui (v0  vi) · (v0  vj) = 0, (i, j)  E ui · uj vi · vj linear function F of vi · vj Kleinberg-Goemans: Affine translation onuito obtain vi 1 1 1 F 0 21 F 1 1 1

  19. F(vi · vj) (i, j) = (1  )m Use Kleinberg-Goemans vi for LS+? Fact: One round of LS+ also requires following ineq: (v0  vi) · (v0  vj)  0 i,j equality whenever ij  E I.e, when ui · uj = 2 1 VC 1  o(n) ui · uj vi · vj F(wi · wj) F(vi · vj) 1 1 1 1 linear map • Idea (Charikar): Map ui to wi s.t. • F(wi · wj)  0 • F(wi · wj) = 0 if ij  E Desired mapping on dot-products [KG] affine map on ui 0 0 21 How? Use tensoring 1 1 1 1

  20. Tensoring u, v Rn Tensor product: uv Rn2 Value uivj at coordinate (i, j)  [n]2 Easy fact:(uv) · (uv) = (u · v)2 Let P(x) = c1xt1 + … + cqxtq Consider map TP(u) = (c1ut1,…, cqutq) Example:P(x) = x2 + 4x TP(u) = (uu, 2u)  Rn2+2n TP(u) · TP(v) = (u · v)2 + 4(u · v)2 = P(u · v) Fact: TP(u) · TP(v) = P(u · v) Positive coefficients 2 2 P determines dot-product of resulting vectors

  21. F(vi · vj) (i, j) = (1  )m (v0  vi) · (v0  vj)  0 i,j equality whenever (i, j)  E Back to finding solution for stronger SDP: Use TP VC 1  o(n) I.e, when ui · uj = 2 1 ui · uj Want wi = TP(ui)s.t. F(wi · wj) min at (i, j)  E 1 1 KG 21 0 ui · uj 1 0 1 C 21 F 1 1 Charikar exhibits appropriate P

  22. I.H. Charikar sol’n gives one round LS+ lower bound x = (½ + )1 x K(r) if  PSD matrix Ys.t. • diagonal is x • “columns”  K(r1) Charikar vectors define Yij = vi·vj that: • Diagonal is x = (½ + )1 • “Columns” K VC = 1  o(n) Must have seq of polynomials Can Charikar vectors show “columns” K(1)? Values distributed like polynomial of Gaussian • Problems: • (1) “Columns” not of form (½ + )1 • (2) Charikar’s vectors work only for one value

  23. Making non-uniform “columns” uniform values distributed like polynomial of Gaussian “Columns” we want to continue from not of form (½ + )1 Def [STT]:x  K is -saturated if for all ij  E so that xi, xj<1 there is surplus: xi + xj 1 + 2 Lemma [STT]:x is -saturated  there exists set of vectors x(i) {0, 1, ½ + }n in K s.t. x  conv({x(i) }).  Can convert “columns” to (essentially) (½ + )1 IF “columns” are -saturated Will be safe to “ignore” 0/1

  24. Is saturation good enough? Goal: matrix Y for x with “column” saturation  O() Recall P(x) definesTP(u) such that TP(u) · TP(v) = P(u · v) deg(PC) =O(1/) Fact: Yhas “columns” s.t. some edges never have surplus Problem: saturation of “close by” edges? = o(m) ~ P(1)P(1-1/m) ≤ P’(1)/m For all P P Saturation  Bad saturation zone Normal. Ham. Dist. from blue edge Necessary: deg(P) ≥  · m ! The blue edge

  25. Want column saturation   O() Precise technical property needed for P: For all vertices k and all edges ij : | P(ui · uk ) + P(uj · uk) |  O() y 1  [1, 1] 11/m Need | P(x) + P(y) |  O() over R 12 R But ui · uj = 2  1 for all edges ij, so 12 1 x • |ui·uk+uj·uk|  2 1 21 11/m • |ui·ukuj·uk|  2(1-) 21 Red points correspond to 0-1 edges  Ignored in saturation calculation 1 Domain of P(x) + P(y)

  26. Defining the sequence of tensoring polynomials arbitrary  > 6 So far: • There must be a seq of polys dep. on , , m. • Polynomials must have large degree. Let x {0, 1, ½ + }n Take P(x) = (-O()) x(x  1)m/ +  x 1/ + (1- -O()) x Properties: • Minimum at ui · uj, ij  E • P’(1) >m  • Works as long as  > 6 • The “Columns” of Y that is produced by using TP,m(ui) have saturation   O()  KG P ui · uj 1 21 0 1 C

  27. (i, j) = (1  )m Putting everything together VC 1  o(n) r = (/) x = (½ + )1 Induction: Have x {0, 1, ½ + }n where  > 6 • Define Y using TP,m(ui)  “Columns” have saturation     O() [STT]  Exists S K  {0, 1, ½ + }ns.t. “columns” conv(S) Induction Hypothesis  S K(r 1) Takeconstant and = (√log m/m) x K(r) if  PSD matrix Ys.t. • diagonal is x • “columns”  K(r1) 2’.Show some set SK(r1) where “columns” conv(S)  x K(r)

  28. Requiring that ||vi-vj||2 is l1? • As is, no l1 inequalities are not implied. • The results of [HMM] (showing that metric-cut ineqaities and pentagonal inequalities hold) suggest the examples are still good. • Need to • Give Sherali Adams LB • introduce dij = ||vi-vj||2 • Add more reqs the LS+ proof need to satisfy.

  29. Sherali-Adams [SA’90] Lift-and-Project Idea: Keep “lifting” but never project! Simulate third, fourth, etc, degree products with linear vars Only known integrality gap [FK’06]: W(log n)SA rounds  int. gap ≤2  e for MAX-CUT SA+ lower bound would inequalities for lifted variables  Triangle, pentagonal, etc., inequalities derivable E.g., x1x2x3 Y123 LP not SDP version

  30. Relations to Unique Games Conjecture (UGC) • LS+ lower bounds may provide evidence of inapproximability • UGC [Khot’02] implies optimal inapproximability results for Vertex Cover, MAX-CUT, etc  Strong LS+, SA+ lower bounds for VC, MAX-CUT

  31. Thanks

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