An Approximation Algorithm for Requirement cut on graphs Viswanath Nagarajan Joint work with R. Ravi

Cut problems Undirected G=(V,E) Edge costs c : E → R + Remove minimum cost set of edges satisfying some requirement

Example : Multicut Pairs (s 1,t 1 ), (s 2,t 2 ), …, (s k,t k ) Separate each pair O(log k) approximation [GVY’93]

Example : Steiner multicut Groups X 1, X 2, …, X g µ V Separate each group O(log 3 (gt)) approximation [KPRT’97] (t = max i |X i |)

More examples k-cut : 2 [SV ’95] Multiway cut : 1.34 [KKSTY ’99] Steiner k-cut : 2 [CGN ’03] Multi-multiway cut : log k [AL’04]

Requirement cut Groups X 1, X 2, …, X g µ V Requirements r i (0 ≤ r i ≤ |X i |) Separate group X i into r i pieces Generalizes previous cut problems

Containment of cut problems Multicut Steiner multicut Multi-multiway cut k-cut Requirement cut Steiner k-cut

Our results O(log n ¢ log(gR)) approximation n = number of vertices g = number of groups R = maximum requirement O(log(gR)) approximation on trees

IP formulation min  e2 E c e d e s.t.  e2 T i d e ≥ r i -1 8 T i : Steiner tree on X i, 8 i=1, …, g d e 2 {0,1} 8 e2 E

Solving LP relaxation Minimum Steiner tree – NP hard! Relax to spanning trees Require that d is metric

LP relaxation min  e2 E c e d e s.t.  e2 T i d e ≥ r i – 18 T i : spanning tree on X i 8 i = 1, …, g d : metric d e 2 [0,1]8 e2 E

Requirement cut on trees Input graph G is a tree Reduction from set cover )  (log g) hard LP rounding yields O(log(gR)) approximation

Tree rounding 1) Solve LP to get d * (OPT lp =  e2 E c e d * e ) 2) Define d e = min{2¢ d * e, 1} 3) Repeat O(log(gR)) times: Pick each edge of tree G with probability d e

Tree rounding Theorem : Randomized rounding for requirement cut on trees yields a solution of cost at most O(log(gR)) ¢ OPT lp Cost in single phase ≤ 2 ¢ OPT lp Total cost of rounding ≤ O(log (gR)) ¢ OPT lp Need to satisfy all requirements

Single phase c i : current number of components of X i Residual requirement of X i = r i – c i Lemma : In each phase of randomized rounding, the total residual requirement reduces by a factor of 4/3, in expectation.

Bounding number of phases Initial requirement ≤ g¢R Expected requirement after phase k, E[R k ] ≤ (¾) k ¢g¢R k = 4¢ln(gR) ) E[R k ] ≤ ½

Rounding in single phase Current forest F F i : forest induced by X i H i : shortcut F i over degree 2 Steiner vertices

Single phase - analysis Removing edges of H i ≡ new components in F i Lemma : Expected number of “edges” of H i removed is at least (1-1/e)¢ d(H i ) Here, d(H i ) =  e2 H i d e

Single phase - analysis Claim : In any Steiner forest H i with each non terminal having degree ≥ 3, removing any m edges results in at least d(m+1)/2e new terminal components Expected number of new components containing X i ≥ (1- 1/e)¢ ½ ¢ d(H i ) ≥ ¼ ¢ d(H i )

Single phase - analysis Extend H i to Steiner tree on X i Add c i – 1 edges d(H i ) + c i – 1 ≥ r i – 1 (~ Sp. tree constraint) d(H i ) ≥ r i – c i = residual requirement of X i

Single phase - analysis Number of new components ≥ ¼ ¢ (r i -c i ) new requirement of group X i = old requirement – number of new components ≤ ¾ ¢ old requirement total new requirement ≤ ¾ ¢ total old requirement (All in expectation)

General graphs Solve LP on G to get metric d * Use FRT embedding to tree metric ( ,T) (edges of T ≡ cuts in G) Use tree rounding on T O(log n ¢ log(gR)) approximation

Conclusions Introduced requirement cut problem O(log n ¢ log(gR)) integrality gap Improvement – for planar graphs? Even on trees, gap between  (log g) lower bound, O(log(gR)) upper bound

Thank you!

Extra claim Claim : Minimum Steiner tree on X i w.r.t. d is at least r i – 1 d * costs on edges

Single phase - analysis Lemma : Expected number of “edges” of H i removed is at least (1-1/e)¢ d(H i ) (u,v) : edge of H i P : path connecting u and v in F i Pµ F i u v (u,v)2 H i

Lemma – contd. Pr [separating u and v] = 1 -  e2 P (1-d e ) ≥ 1 – e -d(P) A) d(P) ≤ 1 Pr ≥ 1 – e -d(P) ≥ (1-1/e) ¢ d(P) ≥ (1-1/e) d u,v B) d(P) ≥ 1 Pr ≥ 1 – e -d(P) ≥ 1 – 1/e ≥ (1-1/e) d u,v