Sherali-Adams Relaxations of Graph Isomorphism Polytopes. Peter N. Malkin* and Mohammed Omar + UC Davis

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1 Sherali-Adams Relaxations of Graph Isomorphism Polytopes Peter N. Malkin* and Mohammed Omar + UC Davis University of Washington May 18th 2010 *Partly funded by an IBM OCR grant and the NSF. + Partly funded by NSERC Postgraduate Scholarship.

2 Overview There has recently been much interest in project & lift (L&P) approaches that construct hierarchies of LP or SDP relaxations of 0-1 problems: Techniques: Sherali-Adams (1990), Lovász and Schrijver (1991) and Lasserre (2001) among others. Problems: Max-Cut, Stable Set and Vertex Cover for example. We are interested in applying the Sherali-Adams (S-A) technique to the Graph Isomorphism (ISO) and Graph Automorphism (AUT) problems. It is not known whether ISO and AUT are in P or NPC. Our Goal! Find complexity results about L&P applied to ISO & AUT that will hopefully shed light on L&P and the complexity of ISO and AUT.

3 Talk Outline Overview: 1 Combinatorial approaches to ISO and AUT. 2 Polyhedral approaches (Sherali-Adams). 3 Comparison of the two approaches. Notation: We will always use V := {1,.., n} as the vertex set and use simple undirected graphs G = (V, E G ) and H = (V, E H ). An isomorphism from G to H is a bijection ψ : V V s.t. {u, v} E G iff {ψ(u), ψ(v)} E H. The set of all isomorphisms from G to H is ISO(G, H). The set of all automorphisms is AUT (G) = ISO(G, G). We say G is asymmetric if AUT (G) = 1.

4 Vertex Classification (V-C) Algorithm The V-C algorithm for AUT (G) partitions V into equivalence classes {V 1, V 2,..., V m } s.t. ψ(v i ) = V i for all ψ AUT (G). If the partition is complete ( V i = 1), then G is asymmetric. Start from the partition {V } and iteratively refine it using the number of neighbors of a vertex w.r.t. each equivalence class.

5 Vertex Classification (V-C) Algorithm The V-C algorithm for AUT (G) partitions V into equivalence classes {V 1, V 2,..., V m } s.t. ψ(v i ) = V i for all ψ AUT (G). If the partition is complete ( V i = 1), then G is asymmetric. Start from the partition {V } and iteratively refine it using the number of neighbors of a vertex w.r.t. each equivalence class.

6 Vertex Classification (V-C) Algorithm The V-C algorithm for AUT (G) partitions V into equivalence classes {V 1, V 2,..., V m } s.t. ψ(v i ) = V i for all ψ AUT (G). If the partition is complete ( V i = 1), then G is asymmetric. Start from the partition {V } and iteratively refine it using the number of neighbors of a vertex w.r.t. each equivalence class.

7 Vertex Classification (V-C) Algorithm The V-C algorithm for AUT (G) partitions V into equivalence classes {V 1, V 2,..., V m } s.t. ψ(v i ) = V i for all ψ AUT (G). If the partition is complete ( V i = 1), then G is asymmetric. Start from the partition {V } and iteratively refine it using the number of neighbors of a vertex w.r.t. each equivalence class.

8 Vertex Classification (V-C) Algorithm The V-C algorithm for AUT (G) partitions V into equivalence classes {V 1, V 2,..., V m } s.t. ψ(v i ) = V i for all ψ AUT (G). If the partition is complete ( V i = 1), then G is asymmetric. Start from the partition {V } and iteratively refine it using the number of neighbors of a vertex w.r.t. each equivalence class.

9 Vertex Classification (V-C) Algorithm for AUT Define δ G (u) := {v V : {u, v} E G }. V-C algorithm for AUT (G) Start from {V }. Given {V 1, V 2,..., V m }, the vertices u, v V i are in the same refined equivalence class in the next iteration iff for all j, δ G (u) V j = δ G (v) V j. Repeat this process until the partition stabilizes. Let π(δ G ) denote this stable partition. If π(δ G ) is complete, then return YES (asymmetric), otherwise return MAYBE and output π(δ G ).

10 Vertex Classification (V-C) Algorithm for ISO The V-C method for ISO is based upon the V-C method for AUT. First, we modify the V-C method for AUT (G) to return an ordered partition π(δ G ) s.t. the order of the sets in the partition is invariant under isomorphism. Let π(δ G ) = (V 1,..., V mg ) and π(δ H ) = (W 1,..., W mh ). If G H, then m G = m H and ψ(v i ) = W i ψ ISO(G, H). We say π(δ G ) matches π(δ H ) if m G = m H, V i = W i and δ G (u) V j = δ H (v) W j for all i, j and u V i, v W i. The V-C algorithm for ISO(G, H) Compute π(δ G ) := (V 1,..., V mg ) and π(δ H ) := (W 1,..., W mh ). If π(δ G ) does not match π(δ H ) return NO, otherwise return MAYBE and output π(δ G ) and π(δ H ).

11 Vertex Classification (V-C) Algorithm V-C is well-known (by Corneil and Gotleib 1970??). V-C is the basis of many implementations of AUT and ISO including the well-known nauty package of McKay. The V-C results after only two refinement steps in a canonical form (CF ) of random graphs with probability 1 e cn (Babai & Kucera 1979). CF (G) G s.t. CF (G) = CF (H) iff G H. E.g., relabel vertices in G in the order they appear in π(δ G ). The partitions π(δ G ) and π(δ H ) can be used with branching. An additional depth-first search yields a CF of all graphs in linear average time (Babai & Kucera 1979). The V-C method is useless for regular graphs.

12 Polyhedral Approach for ISO and AUT Tinhofer (1986) examined the following polyhedron where A G and A H are the adjacency matrices of G and H respectively: P G,H := {X [0, 1] n n : XA G = A H X, Xe = X T e = e}. The integer points in P G,H are permutation matrices in bijection with ISO(G, H), that is, X P G,H {0, 1} n n iff ψ ISO(G, H) s.t. if ψ(u) = v, then X uv = 1, else X uv = 0. Tinhofer 1986 Let π(δ G ) = (V 1,..., V m ) and π(δ H ) = (W 1,..., W m ). 1 P G,H = iff π(δ G ) doesn t match π(δ H ) 2 P G = {I } iff π(δ G ) is complete. 3 If π(δ G ) matches π(δ H ), then X uv = 0 for all X P G,H iff u V i, v W j where i j.

13 k-dim Vertex Classification (V-C) Algorithm Define the set of k-tuples V k := {(u 1,..., u k ) : u 1,..., u k V }. The k-dim V-C algorithm for AUT (G) partitions V k into equivalence classes {V1 k, V 2 k,..., V m} k s.t. ψ(vi k ) = Vi k for all ψ AUT (G) where ψ(u) = (ψ(u 1 ), ψ(u 2 ),..., ψ(u k )). If the partition is complete ( V k i = 1), then G is asymmetric. For the graph below, the partition returned by the 2-dim V-C is {(1, 1), (3, 3)}, {(2, 2), (4, 4)}, {(1, 3), (3, 1)}, {(2, 4), (4, 2)}, {(1, 2), (1, 4),(3, 2), (3, 4)}, {(2, 1), (4, 1),(2, 3), (4, 3)}

14 k-dim Vertex Classification (V-C) Algorithm For the k-dim V-C method, we start from the subgraph partition of V k w.r.t. G and iteratively refine it using the number of neighbors of a k-tuple w.r.t. each equivalence class. The subgraph partition w.r.t. G is πg k where u v iff u s = u t v s = v t and {u s, u t } E G {v s, v t } E G. E.g., the subgraph partition of V 2 w.r.t. G is {V1 2, V 2 2, V 3 2}: 1 V1 2 = {(u, u) : u V } (vertices), 2 V2 2 = {(u 1, u 2 ) : {u 1, u 2 } E G } (edges) and 3 V3 2 = {(u 1, u 2 ) : {u 1, u 2 } E G } (non-edges). 1 2 {(1, 1), (3, 3), (2, 2), (4, 4)}, {(1, 3), (3, 1), (1, 2), (2, 1), (1, 4), 4 3 (4, 1), (2, 3), (3, 2), (3, 4), (4, 3)}, {(2, 4), (4, 2)}

15 k-dim Vertex Classification (V-C) Algorithm We need to extend the concept of neighbor for k-tuples. Define δ i G (u) := {(u 1,..., u i 1, v, u i+1,..., u k ) : v δ G (u i )} and δ i G (u) := {(u 1,..., u i 1, v, u i+1,..., u k ) : v δ G (u i )} For example, δg 2 ((1, 1)) = {(1, 2), (1, 3), (1, 4)}, δg 2 ((2, 2)) = {(2, 1), (2, 3)}, δg 2 ((1, 2)) = {(1, 1), (1, 3)}, δg 2 ((1, 3)) = {(1, 1), (1, 2), (1, 4)}.

16 k-dim Vertex Classification (V-C) Algorithm k-dim V-C algorithm Start from the subgraph partition. Given {V1 k,..., V m}, k the tuples u, v Vi k are in the same refined equivalence class in the next iteration if for all i, j, δ i G (u) V k j = δ i G (v) V k j and δ i G (u) V k j = δ i G (v) V k j. This process is repeated until the partition stabilizes. Let π k (δ G ) denote this stable partition. If π k (δ G ) is complete, then return YES (asymmetric), otherwise return MAYBE and output π k (δ G ). For k = 1, this is the same as the V-C algorithm. For k = n, the algorithm is trivially exact. For ISO(G, H), we must check if π k (δ G ) matches π k (δ H ).

17 k-dim Weisfeiler-Lehman (W-L) Algorithm The k-dim W-L method (k 2) also partitions V k (1968). Define i (u) := {(u 1,..., u i 1, v, u i+1,..., u k ) : v V }. k-dim W-L algorithm for AUT (G) Start from the subgraph partition. Given {V1 k,..., V m}, k the vertices u, v Vs k are in the same refined equivalence class in the next iteration if for all i, j, i (u) V k j = i (v) V k j. This process is repeated until the partition stabilizes. Let π k ( G ) denote this stable partition. If π k ( G ) is complete, then return YES (asymmetric), otherwise return MAYBE and output π k ( G ). For ISO(G, H), we must check if π k ( G ) matches π k ( H ).

18 The k-dim V-C method vs the k-dim W-L method Theorem: Cai, Fürer and Immerman 1992 There is a class of pairs of non-isomorphic graphs (G n, H n ) having O(n) vertices such that the Ω(n)-dim W-L algorithm is needed to distinguish G n and H n. k-dim V-C implies k-dim W-L since i (u) = δ i G (u) δ i G (u). Also, (k + 1)-dim W-L implies k-dim V-C. The projection of the partition π k ( G ) onto the first k 1 components gives the partition π k (δ G ). Corollary: M- & Omar There is a class of pairs of non-isomorphic graphs (G n, H n ) having O(n) vertices such that the Ω(n)-dim V-C algorithm is needed to distinguish G n and H n.

19 Sherali-Adams Relaxations The S-A relaxations of a semi-algebraic set P = {x [0, 1] n f 1 (x) = 0,..., f s (x) = 0} [0, 1] n are a hierarchy of polyhedra P 1 P n such that P n = conv(p {0, 1} n ). 1 First, we generate some valid constraints on P: x i f j (x) = 0 j, I, I k 1, i I x i i I (1 x i ) 0 I J, J k. i J\I 2 Next, we replace x 2 i with x i since x 2 i = x i for x {0, 1} n. 3 Then, we linearize the system by replacing each monomial i I x i with a new variable y I giving the polyhedron ˆP k in y. 4 Lastly, we project ˆP k giving P k := {x : x i = y {i}, y ˆP k }.

20 Sherali-Adams Relaxations of P G,H We examine the S-A relaxations of P G,H (XA G = A H X ) as follows: X uw X wv = 0 u, v V, w δ G (v) w δ H (u) X uw 1 = 0 u V, w V X wv 1 = 0 v V, w V 0 X uv 1 u, v V.

21 Sherali-Adams Relaxations of P G,H We examine the S-A relaxations of P G,H (XA G = A H X ) as follows: 1 Multiply each equation by (r,s) I X rs for all I V 2. X uw X wv = 0 u, v V, w δ G (v) w δ H (u) X uw 1 = 0 u V, w V X wv 1 = 0 v V, w V 0 X uv 1 u, v V.

22 Sherali-Adams Relaxations of P G,H We examine the S-A relaxations of P G,H (XA G = A H X ) as follows: 1 Multiply each equation by (r,s) I X rs for all I V 2. X rs X uw (r,s) I w δ H (v) w δ G (u) ( ) X rs X uw (r,s) I w V ) X wv ( X rs (r,s) I w V X uv (u,v) I (u,v) J\I X wv (r,s) I (r,s) I = 0 I V 2, I k 1, u, v V X rs = 0 I V 2, I k 1, u V, X rs = 0 I V 2, I k 1, v V, (1 X uv ) 0 I J V 2, J k.

23 Sherali-Adams Relaxations of P G,H We examine the S-A relaxations of P G,H (XA G = A H X ) as follows: 1 Multiply each equation by (r,s) I X rs for all I V 2. 2 Replace X 2 uv with X uv and replace (u,v) I X uv with Y I. X rs X uw (r,s) I w δ H (v) w δ G (u) ( ) X rs X uw (r,s) I w V ) X wv ( X rs (r,s) I w V X uv (u,v) I (u,v) J\I X wv (r,s) I (r,s) I = 0 I V 2, I k 1, u, v V X rs = 0 I V 2, I k 1, u V, X rs = 0 I V 2, I k 1, v V, (1 X uv ) 0 I J V 2, J k.

24 Sherali-Adams Relaxations of P G,H We examine the S-A relaxations of P G,H (XA G = A H X ) as follows: 1 Multiply each equation by (r,s) I X rs for all I V 2. 2 Replace X 2 uv with X uv and replace (u,v) I X uv with Y I. Y I {(u,w)} Y I {(w,v)} = 0 I V 2, I k 1, u, v V, w δ H (v) w δ G (u) Y I {(u,w)} Y I = 0 I V 2, I k 1, u V, w V Y I {(w,v)} Y I = 0 I V 2, I k 1, v V, w V ( 1) K\I Y K 0 I J V 2, J k, Y = 1. I K J

25 Sherali-Adams Relaxations of P G,H We examine the S-A relaxations of P G,H (XA G = A H X ) as follows: 1 Multiply each equation by (r,s) I X rs for all I V 2. 2 Replace X 2 uv with X uv and replace (u,v) I X uv with Y I. Y I {(u,w)} Y I {(w,v)} = 0 I V 2, I k 1, u, v V, w δ H (v) w δ G (u) Y I {(u,w)} Y I = 0 I V 2, I k 1, u V, w V Y I {(w,v)} Y I = 0 I V 2, I k 1, v V, w V Y I 0 I V 2, I k, Y = 1.

26 Sherali-Adams Relaxations of P G,H We examine the S-A relaxations of P G,H (XA G = A H X ) as follows: 1 Multiply each equation by (r,s) I X rs for all I V 2. 2 Replace Xuv 2 with X uv and replace (u,v) I X uv with Y I. 3 Rewrite Y I as Y u,v where I = u, v := {(u i, v i ) : 1 i k}. Y I {(u,w)} Y I {(w,v)} = 0 I V 2, I k 1, u, v V, w δ H (v) w δ G (u) Y I {(u,w)} Y I = 0 I V 2, I k 1, u V, w V Y I {(w,v)} Y I = 0 I V 2, I k 1, v V, w V Y I 0 I V 2, I k, Y = 1.

27 Sherali-Adams Relaxations of P G,H We examine the S-A relaxations of P G,H (XA G = A H X ) as follows: 1 Multiply each equation by (r,s) I X rs for all I V 2. 2 Replace Xuv 2 with X uv and replace (u,v) I X uv with Y I. 3 Rewrite Y I as Y u,v where I = u, v := {(u i, v i ) : 1 i k}. Y u,w Y w,v = 0 u, v V k, 1 i k, w δh i (v) w δg i (u) Y u,w Y u,v \(ui,v i ) = 0 u, v V k, 1 i k, w i (v) Y w,v Y u,v \(ui,v i ) = 0 u, v V k, 1 i k, w i (u) Y u,v 0 u, v V k, Y = 1.

28 Comparison of S-A and V-C Note Y ˆP k G,H {0, 1}n n iff there exists ψ ISO(G, H) s.t. if ψ(u) = v, then Y u,v = 1, else Y u,v = 0. Theorem: M- and Omar 2010 Let π k (δ G ) = (V k 1,..., V k m) and π k (δ H ) = (W k 1,..., W k m). 1 P k G = {I } iff the partition πk (δ G ) is complete. 2 P k G,H = iff πk (δ G ) doesn t match π k (δ H ). 3 If π k (δ G ) matches π k (δ H ), then Y u,v = 0 for all Y ˆP G,H k iff u Vi k, v Wj k where i j. Corollary: M- & Omar 2010 There exists a class of pairs of non-isomorphic graphs (G n, H n ) with O(n) vertices s.t. the S-A relaxations of P G,H need Ω(n) iterations to converge to the convex hull proving non-isomorphism.

29 Sherali-Adams Relaxations of Q G,H Question: Does the W-L method correspond to the S-A relaxations of some polytope? Consider the following semi-algebraic set Q G,H : X u1 v 1 X u2 v 2 = 0 u 1, v 1 V, u 2 δ G (u 1 ), v 2 δ H (v 1 ) X u1 v 1 X u2 v 2 = 0 u 1, v 1 V, u 2 δ G (u 1 ), v 2 δ H (v 1 ) X uw 1 = 0 u V, w V X wv 1 = 0 v V, w V 0 X uv 1 u, v V. The integer points in Q G,H are in bijection with ISO(G, H).

30 Sherali-Adams Relaxations of Q G,H The kth S-A extended relaxation of Q G,H, written ˆQ k G,H, is Y u,v = 0 u, v V k, u i δ G (u j ), v i δ H (v j ) Y u,v = 0 u, v V k, u i δ G (u j ), v i δ H (v j ) Y u,w Y u,v \(ui,v i ) = 0 u, v V k, 1 i k, w i (v) w i (u) Y w,v Y u,v \(ui,v i ) = 0 u, v V k, 1 i k, Y u,v 0 u, v V k, Y = 1.

31 Comparison of S-A and W-L Theorem: M- & Omar 2010 Let π k ( G ) = (V k 1,..., V k m) and π k ( H ) = (W k 1,..., W k m). 1 Q k G = {I } iff πk ( G ) is complete. 2 Q k G,H = iff πk ( G ) doesn t match π k ( H ). 3 If π k ( G ) matches π k ( H ), then Y u,v = 0 for all Y ˆQ G,H k iff u Vi k, v Wj k where i j. Corollary: M- & Omar 2010 There exists a class of pairs of non-isomorphic graphs (G n, H n ) with O(n) vertices s.t. the S-A relaxations of Q G,H need Ω(n) iterations to converge to the convex hull proving non-isomorphism.

32 Conclusion We have shown the correspondence of the combinatorial algorithms V-C and W-L with the Sherali-Adams relaxations of P G,H and Q G,H respectively. We showed negative results about the convergence of the S-A relaxations of the polytopes P G,H and Q G,H.

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