SDP Relaxations for MAXCUT
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1 SDP Relaxations for MAXCUT from Random Hyperplanes to Sum-of-Squares Certificates UMD March 3, 2017 Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
2 Overview 1 MAXCUT, Hardness and UGC 2 LP Shortcomings and SDP 3 Goemans and Williamson 4 SOS Recap 5 SOS Certificates (Some time later?) Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
3 MAXCUT, Hardness and UGC Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
4 The MAXCUT Problem Problem (MAXCUT) For an n-vertex graph G, find a bipartition (S, S) that maximizes { {u, v} E(G) u S, v S }. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
5 The MAXCUT Problem Problem (MAXCUT) For an n-vertex graph G, find a bipartition (S, S) that maximizes { {u, v} E(G) u S, v S }. NP-Complete by a straightforward reduction from MAX-2-SAT: {u, v} E(G) u S, v S (x u x v ) ( x u x v ). Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
6 The MAXCUT Problem Problem (MAXCUT) For an n-vertex graph G, find a bipartition (S, S) that maximizes { {u, v} E(G) u S, v S }. NP-Complete by a straightforward reduction from MAX-2-SAT: {u, v} E(G) u S, v S (x u x v ) ( x u x v ). Theorem (Håstad 2001, Trevisan et al. 2000) It is NP-hard to approximate MAXCUT better than Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
7 Unique Games and Label Covers In pictures: Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
8 Unique Games and Label Covers In pictures: Problem ((c, s)-gap Label Cover with Unique Constraints) The promise problem (L yes, L no ): - L yes = {G some assignment satisfies c-fraction of the constraints}. - L no = {G every assignment satisfies s-fraction of the constraints}. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
9 Unique Games and Label Covers In pictures: Problem ((c, s)-gap Label Cover with Unique Constraints) The promise problem (L yes, L no ): - L yes = {G some assignment satisfies c-fraction of the constraints}. - L no = {G every assignment satisfies s-fraction of the constraints}. Conjecture (Unique Games Conjecture (UCG)) For every sufficiently small pair of constants ɛ, δ > 0, k s.t. the (1 δ, ɛ)-gap label-cover problem with k colors is NP-hard. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
10 LP Shortcomings and SDP Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
11 LP Rounding for MAXCUT? A standard approach to design an approximation algorithm is to formulate an Integer Linear Program (ILP), then solve a relaxed LP yielding a (super)optimal solution in real variables, then round the solution back to integers trying to preserve optimality. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
12 LP Rounding for MAXCUT? A standard approach to design an approximation algorithm is to formulate an Integer Linear Program (ILP), then solve a relaxed LP yielding a (super)optimal solution in real variables, then round the solution back to integers trying to preserve optimality. (LP1) s.t. max {u,v} E(G) z uv z uv x u + x v z uv (1 x u ) + (1 x v ) z uv, x u {0, 1} Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
13 LP Rounding for MAXCUT? A standard approach to design an approximation algorithm is to formulate an Integer Linear Program (ILP), then solve a relaxed LP yielding a (super)optimal solution in real variables, then round the solution back to integers trying to preserve optimality. Claim (LP1) s.t. max {u,v} E(G) z uv z uv x u + x v z uv (1 x u ) + (1 x v ) z uv, x u {0, 1} Allowing x u [0, 1], we can set x u = 1 2 u and get z uv = 1 {u, v}. Hence, the optimal value of LP1 is E. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
14 LP Rounding for MAXCUT? A standard approach to design an approximation algorithm is to formulate an Integer Linear Program (ILP), then solve a relaxed LP yielding a (super)optimal solution in real variables, then round the solution back to integers trying to preserve optimality. Claim (LP1) s.t. max {u,v} E(G) z uv z uv x u + x v z uv (1 x u ) + (1 x v ) z uv, x u {0, 1} Allowing x u [0, 1], we can set x u = 1 2 u and get z uv = 1 {u, v}. Hence, the optimal value of LP1 is E. Bad: For K n MAXCUT E /2. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
15 0.5-approximation by Random Guessing Each vertex u E(G) is added independently to S with probability 1 2. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
16 0.5-approximation by Random Guessing Each vertex u E(G) is added independently to S with probability 1 2. Pr[{u, v} is cut] = Pr[(u S) (v S)] + Pr[(u S) (v S)] = Pr[u S] Pr[v S] + Pr[u S] Pr[v S] = = 1 2. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
17 0.5-approximation by Random Guessing Each vertex u E(G) is added independently to S with probability 1 2. Pr[{u, v} is cut] = Pr[(u S) (v S)] + Pr[(u S) (v S)] = Pr[u S] Pr[v S] + Pr[u S] Pr[v S] = = 1 2. Remark: Edge-by-edge analysis. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
18 0.5-approximation by Random Guessing Each vertex u E(G) is added independently to S with probability 1 2. Pr[{u, v} is cut] = Pr[(u S) (v S)] + Pr[(u S) (v S)] = Pr[u S] Pr[v S] + Pr[u S] Pr[v S] = = 1 2. Remark: Edge-by-edge analysis. Remark(2): Sub-exponential lower bounds on the size of LP relaxations beating 1 2 [P. Kothari et al., CATS@UMD February 17, 2017]. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
19 High Level Goals for SDP Generalize LP. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
20 High Level Goals for SDP Generalize LP. Simulate nonlinear programming, in some linear way. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
21 High Level Goals for SDP Generalize LP. Simulate nonlinear programming, in some linear way. Provide a systematic way to expand the search space by introducing more and more variables to obtain tighter approximations to the problem at hand. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
22 Features and Analogies to LP? Variables: real-valued = vector-valued. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
23 Features and Analogies to LP? Variables: real-valued = vector-valued. Linear constraints: on variables = on inner products. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
24 Features and Analogies to LP? Variables: real-valued = vector-valued. Linear constraints: on variables = on inner products. Recover vectors defining a solution by decomposing resulting matrix. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
25 Features and Analogies to LP? Variables: real-valued = vector-valued. Linear constraints: on variables = on inner products. Recover vectors defining a solution by decomposing resulting matrix. Convex, well-behaved, solvable in polynomial time. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
26 Features and Analogies to LP? Variables: real-valued = vector-valued. Linear constraints: on variables = on inner products. Recover vectors defining a solution by decomposing resulting matrix. Convex, well-behaved, solvable in polynomial time. Analysis involves statements about high dimensional geometric graphs (e.g., size of indep. set and expansion properties). Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
27 Positive Semidefinite Matrices Definition A symmetric matrix X R n n is positive semidefinite (PSD), or X 0, if any of the following equivalent conditions holds: 1 a T Xa 0, a R n. 2 X admits a Cholseky decomposition X = LL T. 3 All eigenvalues of X are non-negative. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
28 Positive Semidefinite Matrices Definition A symmetric matrix X R n n is positive semidefinite (PSD), or X 0, if any of the following equivalent conditions holds: 1 a T Xa 0, a R n. 2 X admits a Cholseky decomposition X = LL T. 3 All eigenvalues of X are non-negative. Remark: Condition(1) encodes an (uncountably) infinite set of linear constraints! Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
29 Positive Semidefinite Matrices Definition A symmetric matrix X R n n is positive semidefinite (PSD), or X 0, if any of the following equivalent conditions holds: 1 a T Xa 0, a R n. 2 X admits a Cholseky decomposition X = LL T. 3 All eigenvalues of X are non-negative. Remark: Condition(1) encodes an (uncountably) infinite set of linear constraints! Remark(2): Condition(2) is equivalent to X ij = v i, v j for vectors {v 1,..., v n } corresponding to the columns of L. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
30 Standard Form (Primal) (P) min C, X := ij C ij X ij s.t. A i, X = b i i {1,..., m}, X 0 Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
31 Standard Form (Primal) (P) min C, X := ij C ij X ij s.t. A i, X = b i i {1,..., m}, X 0 Remark: LP is a special case of SDP for which X is a diagonal matrix. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
32 Dual Form (D) s.t. max b T y yi A i + S = C, S 0 Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
33 Dual Form (D) s.t. max b T y yi A i + S = C, S 0 Remark: Weak duality holds. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
34 Powerful Technical Lemma Lemma ( Condition(4) for PSD matrices) For a symmetric matrix X R n n, X 0 iff A, X 0, A 0. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
35 Powerful Technical Lemma Lemma ( Condition(4) for PSD matrices) For a symmetric matrix X R n n, X 0 iff A, X 0, A 0. Fact(1): A, X = Trace(A T X ). Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
36 Powerful Technical Lemma Lemma ( Condition(4) for PSD matrices) For a symmetric matrix X R n n, X 0 iff A, X 0, A 0. Fact(1): A, X = Trace(A T X ). Fact(2): Trace(AB) = Trace(BA) (cyclic permutations). Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
37 Powerful Technical Lemma Lemma ( Condition(4) for PSD matrices) For a symmetric matrix X R n n, X 0 iff A, X 0, A 0. Fact(1): A, X = Trace(A T X ). Fact(2): Trace(AB) = Trace(BA) (cyclic permutations). Proof (Do on board). ( ) Suppose X is not PSD and obtain a witness a R n s.t. a T Xa < 0. ( ) Suppose A 0 and obtain the Cholseky decomposition A = LL T. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
38 Weak Duality Lemma If X, y are feasible for (P) and (D), respectively, then b T y C, X. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
39 Weak Duality Lemma If X, y are feasible for (P) and (D), respectively, then b T y C, X. Recall: In (P), X 0 and in (D), i y ia i + S = C, with S 0. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
40 Weak Duality Lemma If X, y are feasible for (P) and (D), respectively, then b T y C, X. Recall: In (P), X 0 and in (D), i y ia i + S = C, with S 0. Proof. C, X = i y i A i + S, X = i y i A i, X + S, X = i y i A i, X + S, X = i y i b i + S, X 0 (X is feasible) b T y (By the technical lemma) Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
41 Strong Duality? When the value of (P) coincides with that of (D). Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
42 Strong Duality? When the value of (P) coincides with that of (D). Usually holds under some condition. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
43 Strong Duality? When the value of (P) coincides with that of (D). Usually holds under some condition. Definition (Slater s Condition) Feasible region has an interior point. In other words, feasible X 0. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
44 Goemans and Williamson Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
45 SDP Formulation of MAXCUT max ( X uv) {u,v} E(G) s.t. X uu = 1, u V (G) X 0 Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
46 SDP Formulation of MAXCUT max {u,v} E(G) ( X uv) s.t. X uu = 1, u V (G) X 0 Intuition: Set X = aa T for a R n is defined as { 1, if u S, a u = 1, o.w. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
47 SDP Formulation of MAXCUT max {u,v} E(G) ( X uv) s.t. X uu = 1, u V (G) X 0 Intuition: Set X = aa T for a R n is defined as { 1, if u S, a u = 1, o.w. Cut edges contribute 1, uncut edges contribute 0. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
48 Rounding by a Random Hyperplane Theorem (Goemans and Williamson) There exists an α GW -approximation algorithm for MAXCUT where 2 α GW = min 0 θ π π θ 1 cos θ Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
49 Rounding by a Random Hyperplane Theorem (Goemans and Williamson) There exists an α GW -approximation algorithm for MAXCUT where 2 α GW = min 0 θ π π θ 1 cos θ Solve the SDP to get the solution X. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
50 Rounding by a Random Hyperplane Theorem (Goemans and Williamson) There exists an α GW -approximation algorithm for MAXCUT where 2 α GW = min 0 θ π π θ 1 cos θ Solve the SDP to get the solution X. - Recover vector variables y u for which X uv = y u, y v. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
51 Rounding by a Random Hyperplane Theorem (Goemans and Williamson) There exists an α GW -approximation algorithm for MAXCUT where 2 α GW = min 0 θ π π θ 1 cos θ Solve the SDP to get the solution X. - Recover vector variables y u for which X uv = y u, y v. - Pick a uniformly random vector a R n. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
52 Rounding by a Random Hyperplane Theorem (Goemans and Williamson) There exists an α GW -approximation algorithm for MAXCUT where 2 α GW = min 0 θ π π θ 1 cos θ Solve the SDP to get the solution X. - Recover vector variables y u for which X uv = y u, y v. - Pick a uniformly random vector a R n. - Set x u = sgn( a, y u ). Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
53 Rounding by a Random Hyperplane Theorem (Goemans and Williamson) There exists an α GW -approximation algorithm for MAXCUT where 2 α GW = min 0 θ π π θ 1 cos θ Solve the SDP to get the solution X. - Recover vector variables y u for which X uv = y u, y v. - Pick a uniformly random vector a R n. - Set x u = sgn( a, y u ). - Edge-by-edge analysis: Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
54 Rounding by a Random Hyperplane Theorem (Goemans and Williamson) There exists an α GW -approximation algorithm for MAXCUT where 2 α GW = min 0 θ π π θ 1 cos θ Solve the SDP to get the solution X. - Recover vector variables y u for which X uv = y u, y v. - Pick a uniformly random vector a R n. - Set x u = sgn( a, y u ). - Edge-by-edge analysis: Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
55 Second Generation of SDP Rounding Algorithms Beyond random hyperplanes. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
56 Second Generation of SDP Rounding Algorithms Beyond random hyperplanes. Initiated by Arora, Rao and Vazirani for c-balanced cuts. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
57 Second Generation of SDP Rounding Algorithms Beyond random hyperplanes. Initiated by Arora, Rao and Vazirani for c-balanced cuts. More global analysis. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
58 Second Generation of SDP Rounding Algorithms Beyond random hyperplanes. Initiated by Arora, Rao and Vazirani for c-balanced cuts. More global analysis. Different formulation + triangle inequality. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
59 Second Generation of SDP Rounding Algorithms Beyond random hyperplanes. Initiated by Arora, Rao and Vazirani for c-balanced cuts. More global analysis. Different formulation + triangle inequality. Breadth-first search on weighted geometric graph. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
60 SOS Recap Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
61 Certifying non-negativity over the hypercube Problem (Non-negativity) Given a low-degree polynomial f : {0, 1} n R, decide if f 0 over the hypercube or if there exists a point x {0, 1} n such that f (x) < 0. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
62 Certifying non-negativity over the hypercube Problem (Non-negativity) Given a low-degree polynomial f : {0, 1} n R, decide if f 0 over the hypercube or if there exists a point x {0, 1} n such that f (x) < 0. Example: For an n-vertex graph G, we encode a bipartition of the vertex set by a vector x {0, 1} n and we let f G (x) be the number of edges cut by the bipartition x. We get the degree-2 polynomial f G (x) = (x i x j ) 2. {i,j} E(G) Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
63 Certifying non-negativity over the hypercube Problem (Non-negativity) Given a low-degree polynomial f : {0, 1} n R, decide if f 0 over the hypercube or if there exists a point x {0, 1} n such that f (x) < 0. Example: For an n-vertex graph G, we encode a bipartition of the vertex set by a vector x {0, 1} n and we let f G (x) be the number of edges cut by the bipartition x. We get the degree-2 polynomial f G (x) = (x i x j ) 2. {i,j} E(G) Deciding if the polynomial c f G (x) is non-negative over the hypercube is the same as deciding if MAXCUT (G) c. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
64 Sum-of-Squares Certificates Definition A degree-d sos certificate (of non-negativity) for f : {0, 1} n R consists of polynomials g 1,..., g r : {0, 1} n R of degree at most d/2 for some r N such that f (x) = g 2 1 (x) + + g 2 r (x), for every x {0, 1} n. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
65 Sum-of-Squares Certificates Definition A degree-d sos certificate (of non-negativity) for f : {0, 1} n R consists of polynomials g 1,..., g r : {0, 1} n R of degree at most d/2 for some r N such that f (x) = g 2 1 (x) + + g 2 r (x), for every x {0, 1} n. A.k.a degree-d sos proof of the inequality f 0. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
66 Sum-of-Squares Certificates Definition A degree-d sos certificate (of non-negativity) for f : {0, 1} n R consists of polynomials g 1,..., g r : {0, 1} n R of degree at most d/2 for some r N such that f (x) = g 2 1 (x) + + g 2 r (x), for every x {0, 1} n. A.k.a degree-d sos proof of the inequality f 0. We can assume r = n O(d), thus verifying a certificate takes n O(d) time. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
67 Sum-of-Squares Certificates Definition A degree-d sos certificate (of non-negativity) for f : {0, 1} n R consists of polynomials g 1,..., g r : {0, 1} n R of degree at most d/2 for some r N such that f (x) = g 2 1 (x) + + g 2 r (x), for every x {0, 1} n. A.k.a degree-d sos proof of the inequality f 0. We can assume r = n O(d), thus verifying a certificate takes n O(d) time. Just check that f (g g 2 r ) vanishes for every x {0, 1} n. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
68 Finding SOS Certificates Theorem There exists an algorithm that given a polynomial f : {0, 1} n R and a number k N, outputs a degree-k sos certificate for f + 2 n in time n O(k) if f has a degree-k sos certificate. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
69 Finding SOS Certificates Theorem There exists an algorithm that given a polynomial f : {0, 1} n R and a number k N, outputs a degree-k sos certificate for f + 2 n in time n O(k) if f has a degree-k sos certificate. Intuition Such polynomials having a degree-d sos certificate form a convex cone, which admits a small semidefinite programming formulation. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
70 PSD Matrices and SOS Certificates Theorem A polynomial f has a degree-d sos certificate iff there exists a PSD matrix A such that for all x {0, 1} n, f (x) = (1, x) d/2, A(1, x) d/2. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
71 PSD Matrices and SOS Certificates Theorem A polynomial f has a degree-d sos certificate iff there exists a PSD matrix A such that for all x {0, 1} n, Proof (Do on board). f (x) = (1, x) d/2, A(1, x) d/2. ( ) For a PSD A, extract a degree-d certificate {g 1,... g r }. ( ) For a degree-d sos certificate f = r i=1 g i 2 form a PSD matrix A. Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
72 SOS Certificates (Some time later?) Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
73 The End.. often the sign of scientific success is when we eliminate the need for creativity and make boring what was once exciting.. is it just a matter of time until algorithm design will become as boring as solving a single polynomial equation? Questions? akader@cs.umd.edu Ahmed Abdelkader MAXCUT SDP SOS March 3, / 27
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