Cuts & Metrics: Multiway Cut

Cuts & Metrics: Multiway Cut Barna Saha April 21, 2015

Cuts and Metrics Multiway Cut Probabilistic Approximation of Metrics by Tree Metric

Cuts & Metrics Metric Space: A metric (V, d) on a set of vertices V gives a distance d uv for each pair of vertices u, v V such that the following properties are obeyed. d u,v = 0 iff v = u (semimetric does not respect this property though d(u, u) = 0 always) d u,v = d v,u u, v V d u,v d u,w + d w,v u, v, w V May some time ignore the distinction of semimetric and metric Metrics are useful way of thinking about graph problems involving cuts.

Cut Semimetric For any cut S V, define d u,v = 1 if u S and v / S, else d(u, v) = 0 If we are trying to find a s-t cut which minimizes total weight of edges crossing the cut, we can write: min w e d u,v subject to d s,t = 1 e=(u,v) E d u,v d u,w + d w,v u, v, w V d u,v = {0, 1} u, v E

Cut Semimetric For any cut S V, define d u,v = 1 if u S and v / S, else d(u, v) = 0 If we are trying to find a s-t cut which minimizes total weight of edges crossing the cut, we can write: min w e d u,v subject to d s,t = 1 e=(u,v) E d u,v d u,w + d w,v u, v, w V d u,v [0, 1] u, v E (Metric Relaxation) We can solve s-t mincut via Metric Relaxation.

Multiway Cut Problem Given an undirected graph G = (V, E), costs c e 0 for all edges e E and k distinguished vertices s 1, s 2,..., s k, remove a minimum-cost collection of edges such that every pair of distinguished vertices s i, s j, i j, get separated.

Multiway Cut Problem Given an undirected graph G = (V, E), costs c e 0 for all edges e E and k distinguished vertices s 1, s 2,..., s k, remove a minimum-cost collection of edges such that every pair of distinguished vertices s i, s j, i j, get separated. Applications Many! Look for them.

Multiway Cut Isola(ng cut for s1 s3 Can compute isola(ng cut op(mally

Multiway Cut Isola(ng cut for s2 Isola(ng cut for s1 Isola(ng cut for s3 s3 Return union of the isola(ng cuts A 2-approximation.

3 2-approximation for Multiway Cut Problem Partition the set of vertices into C 1, C 2,..., C k such that s i C i, for i = 1, 2,.., k, to minimize the edges crossing them. Formulate it in terms of integer linear program. Obtain a linear programming relaxation. Round (Randomized Rounding) it utilizing connection to metrics.

3 2-approximation for Multiway Cut Problem k different variables x i u for a vertex u, x i u = 1 if u C i. k different variables z i e for an edge e, z i e = 1 if e δ(c i ) Each edge in exactly two C i, C j. Hence the objective is Constraints: min 1 2 xs i i = 1, i = 1, 2,.., k k xu i = 1, u V i=1 k c e ze i e E i=1 For all edge e = (u, v) z i e x i u x i v x i u = {0, 1}, z i e = {0, 1} i, u V, e E

3 2-approximation for Multiway Cut Problem k different variables x i u for a vertex u, x i u = 1 if u C i. k different variables z i e for an edge e, z i e = 1 if e δ(c i ) Each edge in exactly two C i, C j. Hence the objective is Constraints: min 1 2 xs i i = 1, i = 1, 2,.., k k xu i = 1, u V i=1 k c e ze i e E i=1 For all edge e = (u, v) z i e = x i u x i v x i u = {0, 1}, z i e = {0, 1} i, u V, e E

3 2-approximation for Multiway Cut Problem k different variables x i u for a vertex u, x i u = 1 if u C i. k different variables z i e for an edge e, z i e = 1 if e δ(c i ) Each edge in exactly two C i, C j. Hence the objective is Constraints: min 1 2 c e e=(u,v) E i=1 k xu i xv i xs i i = 1, i = 1, 2,.., k k xu i = 1, u V i=1 x i u = {0, 1}, i, u V

3 2-approximation for Multiway Cut Problem k different variables x i u for a vertex u, x i u = 1 if u C i. k different variables z i e for an edge e, z i e = 1 if e δ(c i ) Each edge in exactly two C i, C j. Hence the objective is Constraints: min 1 2 e=(u,v) E c e x u x v l1 xs i i = 1, i = 1, 2,.., k k xu i = 1, u V i=1 x i u = {0, 1}, i, u V

3 2-approximation for Multiway Cut Problem k different variables x i u for a vertex u, x i u = 1 if u C i. k different variables z i e for an edge e, z i e = 1 if e δ(c i ) Each edge in exactly two C i, C j. Hence the objective is Constraints: min 1 2 e=(u,v) E xs i i = 1, i = 1, 2,.., k k xu i = 1, u V i=1 c e x u x v l1 x i u [0, 1] i, u V (Relaxation)

3 2-approximation for Multiway Cut Problem k different variables x i u for a vertex u, x i u = 1 if u C i. k different variables z i e for an edge e, z i e = 1 if e δ(c i ) Each edge in exactly two C i, C j. Hence the objective is Constraints: min 1 2 e=(u,v) E c e x u x v l1 x si = e i, i = 1, 2,.., k x u k u V (Relaxation) Find location of U {s 1, s 2,.., s k } points in k-dimensional simplex such that their all-pair weighted sum of l 1 -distances is minimized

3 2-approximation for Multiway Cut Problem Intuition. Gives location to points in simplex. If a point is closest to e i assign it to C i Use randomization to define close so that each edge (u, v) is cut with probability 3 4 x u x v l1

3 2-approximation for Multiway Cut Problem For any r 0, define B(e i, r) = {u V : 1 2 e i x u l1 r} All vertices are within radius 1.

3 2-approximation for Multiway Cut Problem Algorithm. Let x be an optimum LP solution for all 1 i k do C i φ Pick r (0, 1) uniformly at random Pick a random permutation π of {1, 2,..., k} X φ // X keeps track of vertices already assigned for i = 1 to k 1 do C π(i) B(s π(i), r) X X X C π(i) C π(k) V X return F = k i=1 δ(c i)

3 2-approximation for Multiway Cut Problem Analysis Lemma x l u x l v 1 2 x u x v l1 Lemma u B(s i, r) if and only if 1 x i u r. Show that Prob((u, v)is cut) 3 4 x u x v l1.

3 2-approximation for Multiway Cut Problem Analysis Show that Prob((u, v) is cut) 3 4 x u x v l1. Define events S i : i is the first in the permutation s.t. one of u or v is in B(e i, r) X i : exactly one of u or v belongs to B(e i, r) For e to be cut, there must exists at least one i such that both S i and X i happens. Compute k Prob(S i X i ) i=1

3 2-approximation for Multiway Cut Problem Prob(X i ) = Prob(min (1 x i u, 1 x i v ) r max (1 x i u, 1 x i v )) = x i u x i v 1 2 x u x v l1 Let l be the vertex which minimizes min i min(1 x i u, 1 x i v ). If i l and l comes before i in π then S i is false. Prob(S i X i ) = Prob(S i X i π(l) < π(i))prob(π(l) < π(i)) + Prob(S i X i π(l) > π(i))prob(π(l) > π(i)) = Prob(S i X i π(l) > π(i))prob(π(l) > π(i)) = Prob(S i X i π(l) > π(i)) 1 2 Prob(X i π(l) > π(i)) 1 2 = Prob(X i) 1 2 = 1 2 x i u x i v

3 2-approximation for Multiway Cut Problem Prob(S i X i ) = Prob(X i ) 1 2 = 1 2 x i u x i v Prob(S i X i ) = 1 2 x u i xv i + Prob(S l X l ) i i,i l 1 2 ( x u x v ( x l u x l v )) + x l u x l v = 1 2 ( x u x v + x l u x l v ) 1 2.3 2 x u x v = 3 4 x u x v

Probabilistic Approximation of Metrics by Tree Metric Use tree metrics to approximate a given distance metric d on a set of vertices V A tree metric (V, T ) for a set of vertices V is a tree T on V V with nonnegative edge lengths. Distance T u,v is the unique shortest path length from u to v. We want for some small α 1 the following to hold for all vertices u, v V α: distortion of embedding. d(u, v) T (u, v) αd(u, v)

Importance of Low Distortion Embedding Many problems are easy to solve on trees. Use low distortion embedding to reduce problem on general metric space to tree metric. α dictates the approximation factor. See section 8.6 of Shmoys-Williamsons for examples.

Tree Embedding Theorem We Want to Prove Unfortunately no deterministic embedding can have distortion o(n). Embed into a distribution of trees. Show expected distortion is low. Theorem Given a distance metric (V, d), such that d(u, v) 1 for all u v, u, v V, there is a randomized, polynomial-time algorithm that produces a tree metric (V, T ), V V, such that for all u, v V, d(u, v) T (u, v) and E[T (u, v)] O(log n)d(u, v).

Hierarchical Cut Decomposition Let = 2 k such that 2 k 2 max u,v {d(u, v)} > 2 k 1. Level= log 2 (Δ) < Δ All ver7ces V Δ Δ Δ Level=log 2 (Δ)- 1 < Δ/2 < Δ/2 < Δ/2 log 2 (Δ)- 2 Δ/2 Δ/2 Δ/2 < Δ/4 < Δ/4 < Δ/4 2 i+1 2 i- 1 <=r< 2 i Level=i Level=1 < 2 2 i 2 i 2 i Level=0 2 2 2 < 1 individual ver7ces v

Hierarchical Cut Decomposition Lemma For all pairs of vertices u, v V, d(u, v) T (u, v). Proof. Let d(u, v) = 2 k + r, 2 k < r < 0. u and v cannot appear together in any set at level k 1 or lower. Hence the lowest level in which u and v may appear together is k. The total distance is 2 k i=1 2i = 4(1 + 2 + 2 2 +... + 2 k 1 ) 2 k+2 4 d(u, v). If d(u, v) = 2 k, then the lowest level in which u and v may appear together is k 1, k 2. The total distance is 2 k 1 i=1 2i = 4(1+2+2 2 +...+2 k 2 ) 2 k+1 4 d(u, v).

Hierarchical Cut Decomposition Lemma If the least common ancestor of vertices u, v V is at level i then T (u, v) 2 i+2.

Hierarchical Cut Decomposition Pick a random permutation π of V Set to the smallest power of 2 greater than 2max u,v d(u, v) Pick r 0 [ 1 2, 1) uniformly at random; set r i = 2 i r 0 for all 1 i log 2 // C(i) will be the sets corresponding to the nodes at level i, it creates a partition of V. C(log 2 ) = {V }: create tree node corresponding to V.

Hierarchical Cut Decomposition for i log 2 downto 1 do C(i 1) φ for all C C(i) do S C for j 1 to n do if B(π(j), r i 1 ) S φ then Add B(π(j), r i 1 ) S to C(i 1) Remove B(π(j), r i 1 ) S from S Create tree nodes corresponding to all sets in C(i 1), join them to C with edges of length 2 i

Hierarchical Cut Decomposition Look at level i and a pair of vertices u, v S iw : In the ith level w is the first vertex such that at least one of u and v is contained in its ball. X iw : In the ith level exactly one of u and v is contained in the ball of w T (u, v) max 1( w V : X i,w S i,w ).2 i+3 i=0,...,log 1 w V E[T (u, v)] w V log 1 i=0 1(X i,w S i,w ).2 i+3 log 1 i=0 Prob(X i,w S i,w ).2 i+3

Hierarchical Cut Decomposition E[T (u, v)] w V = w V log 1 i=0 w V log 1 i=0 Prob(X i,w S i,w ).2 i+3 Prob(S i,w X i,w )Prob(X i,w )2 i+3 log 1 b w Prob(X i,w )2 i+3 i=0

Hierarchical Cut Decomposition Show log i=1 Prob(X iw).2 i+3 16d u,v. Show if w is the jth closest vertex according to min (d(u, w), d(v, w)) then Prob(S i,w X i,w ) 1 j Set b w = 1 j.