Cuts & Metrics: Multiway Cut

Transcription

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

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

3 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.

4 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

5 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.

6 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.

7 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.

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

9 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.

10 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.

11 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

12 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

13 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

14 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

15 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)

16 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

17 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

18 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.

19 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)

20 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.

21 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

22 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

23 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 ) x u x v = 3 4 x u x v

24 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)

25 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.

26 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).

27 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= < 1 individual ver7ces v

28 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( 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( k 2 ) 2 k+1 4 d(u, v).

29 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.

30 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.

31 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

32 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

33 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

34 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.