Minimum cuts via Breadth-First search R. Ravi ravi@cmu.edu - PowerPoint PPT Presentation

Minimum cuts via Breadth-First search R. Ravi ravi@cmu.edu

Outline • Minimum s-t cut in digraphs (folklore) • Multiway-cuts in undirected graphs (folklore) • Multiway-cuts in digraphs (Chekuri & Madan) • Multicuts in undirected graphs (Calinescu, Karloff & Rabani)

� � Minimum s-t cut problem Given digraph G=(V,A), with nonnegative costs/capacities on arcs, a source s and a sink t, find minimum cost arc set blocking all s-t paths 𝑨 "# = min ) 𝑥 +, 𝑦 +, /012 +, s.t. ∑ 𝑦 +, ≥ 1 ∀ 𝑄 s-t paths +,∈# 𝑦 +, ≥ 0 ∀𝑣𝑤 arcs

� Compact LP formulation 𝑨 "# = min ) 𝑥 +, 𝑦 +, /012 +, s.t. 𝑒 , ≤ 𝑒 + + 𝑦 +, ∀𝑣𝑤 𝑒 2 = 0 𝑒 ? ≥ 1 𝑦 +, ≥ 0 ∀𝑣𝑤

Djikstra’s algorithm to generate level cuts • Pick random cutting radius 𝑠 ∈ 0,1 • Remove edges leaving the ball 𝐶(𝑡, 𝑠) of nodes contained within a radius of 𝑠 around the source 𝑡 with respect to the distances 𝑦 +, on arcs 𝑣𝑤 ∈ 𝐵 𝐻

Every level cut is an optimal cut! • Pick random cutting radius 𝑠 ∈ 0,1 • Cut edges leaving the ball 𝐶(𝑡, 𝑠) of nodes contained within a radius of 𝑠 around the source 𝑡 with respect to the distances 𝑦 +, on arcs 𝑣𝑤 ∈ 𝐵 𝐻 • Output 𝜀 I 𝐶 𝑡, 𝑠 ≝ 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢 0 Claim: 𝑄 𝑣𝑤 ∈ 𝑑𝑣𝑢 ≤ 𝑦 +, ≤ ∑ Corollary: 𝐹 𝑥 𝑑𝑣𝑢 𝑥 +, 𝑄 𝑣𝑤 ∈ 𝑑𝑣𝑢 ≤ 𝑨 "# �+, 𝑥 𝑛𝑗𝑜𝑑𝑣𝑢 ≤ min ST,TS2 𝑥 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢 ≤ 𝐹 𝑥 𝑑𝑣𝑢 ≤ 𝑨 "# ≤ 𝑥(𝑛𝑗𝑜𝑑𝑣𝑢)

Outline • Minimum s-t cut in digraphs (folklore) • Multiway-cuts in undirected graphs (folklore) • Multiway-cuts in digraphs (Chekuri & Madan) • Multicuts in undirected graphs (Calinescu, Karloff & Rabani)

� � Multiway cut in undirected graphs Given undirected graph G=(V,E), with nonnegative costs on edges, a source set S = { 𝑡 U , … , 𝑡 W } , find minimum cost edge set blocking all 𝑡 X − 𝑡 Z paths 𝑨 [\]^ = min ) 𝑥 +, 𝑦 +, T_`T2 +, s.t. ∑ 𝑦 +, ≥ 1 ∀ 𝑄 𝑡 X − 𝑡 Z paths +,∈# 𝑦 +, ≥ 0 ∀𝑣𝑤 edge s

Natural generalization of level cuts U • Pick random cutting radius 𝑠 ∈ 0, d • Cut edges leaving the ball 𝐶(𝑡 X , 𝑠) of nodes contained within a radius of 𝑠 around the each source 𝑡 X with respect to the distances 𝑦 +, on edges 𝑣𝑤 ∈ 𝐹 𝐻 • Output all but the heaviest cut in ∪ X 𝜀 𝐶 𝑡 X , 𝑠 ≝ 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢 0

A 2-approximation algorithm f gh Claim: 𝑄 𝑣𝑤 ∈ 𝑑𝑣𝑢 ≤ = 2𝑦 +, i j Corollary: 𝐹 𝑥 𝑑𝑣𝑢 ≤ 2𝑨 [\]^ ST,TS2 0 𝑥 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢 0 ≤ 1 − 1 min 𝑙 𝐹 𝑥 𝑑𝑣𝑢 U ≤ 2 1 − W 𝑨 [\]^ U ≤ 2 1 − W 𝑥 min 𝑛𝑣𝑚𝑢𝑗𝑥𝑏𝑧 𝑑𝑣𝑢

Outline • Minimum s-t cut in digraphs (folklore) • Multiway-cuts in undirected graphs (folklore) • Multiway-cuts in digraphs (Chekuri & Madan) • Multicuts in undirected graphs (Calinescu, Karloff & Rabani)

Multiway cuts in digraphs Given directed graph G=(V,A), with nonnegative costs on arcs, a source set S = { 𝑡 U , … , 𝑡 W } , find minimum cost arc set blocking all 𝑡 X → 𝑡 Z paths for all ordered pairs of sources 𝑗 ≠ 𝑘. Note : Min multiway cut for S = { 𝑡 U , 𝑡 d } is NP-hard so does not specialize to regular min-cut (Also need to cut all reverse paths)

Multiway cuts in digraphs Given directed graph G=(V,A), with nonnegative costs on arcs, a source set S = { 𝑡 U , … , 𝑡 W } , find minimum cost arc set blocking all 𝑡 X → 𝑡 Z paths for all ordered pairs of sources 𝑗 ≠ 𝑘. • In digraphs, node weights can be represented as arc weights by dividing nodes • Generalizes node-weighted multiway cut in undirected graphs: Given undirected graph G=(V,E), a source set S = { 𝑡 U , … , 𝑡 W } , and nonnegative costs on non-source nodes , find minimum cost node set blocking all 𝑡 X − 𝑡 Z paths • In undirected graphs, node weights generalize edge weights by subdividing

Multiway cuts in digraphs Given directed graph G=(V,A), with nonnegative costs on arcs, a source set S = { 𝑡 U , … , 𝑡 W } , find minimum cost arc set blocking all 𝑡 X → 𝑡 Z paths for all ordered pairs of sources 𝑗 ≠ 𝑘. Theorem (Naor-Zosin, FOCS’97): 2-approximation for multiway cuts in digraphs by exactly rounding a relaxed multiway flow relaxation which is within factor 2 of natural relaxation

� � Multiway cuts in digraphs (Chekuri & Madan, SODA ‘16) 𝑨 [\]^ = min ) 𝑥 +, 𝑦 +, T_`T2 +, s.t. ∑ 𝑦 +, ≥ 1 ∀ 𝑄 𝑡 X → 𝑡 Z paths +,∈# 𝑦 +, ≥ 0 ∀𝑣𝑤 arc s Theorem: Level-cutting algorithm on the above LP gives a 2- approximation

Level cuts – Attempt 1 • Pick random cutting radius 𝑠 ∈ 0,1 • Cut arcs leaving the ball 𝐶(𝑡 X , 𝑠) of nodes contained within a radius of 𝑠 around the each source 𝑡 X with respect to the distances 𝑦 +, on arcs 𝑣𝑤 ∈ 𝐵 𝐻 • Output ∪ X 𝜀 I 𝐶 𝑡 X , 𝑠 ≝ 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢 0 for a random 𝑠 • Argue no arc is overused by more than factor 2 in expectation?

Level cuts – Attempt 1 fails

Level cuts of Chekuri and Madan

A 2-approximation algorithm For arc 𝑣𝑤 order sources so that 𝑒 𝑡 U , 𝑣 ≤ 𝑒 𝑡 d , 𝑣 ≤ ⋯ ≤ 𝑒 𝑡 W , 𝑣 Note that 𝑒 𝑢 d , 𝑣 = 𝑒 𝑢 w , 𝑣 = ⋯ = 𝑒 𝑢 W , 𝑣 = 𝑒 𝑡 U , 𝑣 If one of these balls cut 𝑣𝑤 then all of them do Thus 𝑣𝑤 is either cut by the ball around 𝑢 U or by the above set of balls. 𝑄 𝑣𝑤 ∈ 𝑑𝑣𝑢 ≤ 2𝑦 +, ST,TS2 0 𝑥 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢 0 ≤ 2𝑥 min 𝑛𝑣𝑚𝑢𝑗𝑥𝑏𝑧 𝑑𝑣𝑢 min

Outline • Minimum s-t cut in digraphs (folklore) • Multiway-cuts in undirected graphs (folklore) • Multiway-cuts in digraphs (Chekuri & Madan) • Multicuts in undirected graphs (Calinescu, Karloff & Rabani)

� � Multicuts in undirected graphs Given undirected graph G=(V,E), with nonnegative costs on edges, and source-sink pairs = {( 𝑡 U , 𝑢 U ), … , (𝑡 W , 𝑢 W ) } , find minimum cost edge set blocking all 𝑡 X − 𝑢 X paths 𝑨 [\^ = min ) 𝑥 +, 𝑦 +, T_`T2 +, s.t. ∑ 𝑦 +, ≥ 1 ∀ 𝑄 𝑡 X − 𝑢 X paths +,∈# 𝑦 +, ≥ 0 ∀𝑣𝑤 edge s

Level cut algorithm – attempt 1 • Pick random cutting radius 𝑠 ∈ 0,1 • Cut edges leaving the ball 𝐶(𝑡 X , 𝑠) of nodes contained within a radius of 𝑠 around the each source 𝑡 X with respect to the distances 𝑦 +, on edges 𝑣𝑤 ∈ 𝐹 𝐻 • Output ∪ X 𝜀 𝐶 𝑡 X , 𝑠 ≝ 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢 0 • Argue no arc is overused by more than factor 2 in expectation? Caution : LP has a Ω log 𝑙 integrality gap

Level cuts of Calinescu, Karloff & Rabani • Sort source-sink pairs in random order • Pick random cutting radius 𝑠 ∈ 0,1 • In sorted order, cut edges leaving the ball 𝐶(𝑡 X , 𝑠) of nodes contained within a radius of 𝑠 around the current source 𝑡 X with respect to the distances 𝑦 +, on edges 𝑣𝑤 ∈ 𝐹 𝐻 • “Protect” edges both of whose ends are contained in earlier balls from being cut later • Output the unprotected parts ∪ X 𝜀 + 𝐶 𝑡 X , 𝑠 ≝ 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢 0 for a random 𝑠

Level cuts of Calinescu, Karloff & Rabani

Level cuts of Calinescu, Karloff & Rabani Is the solution feasible?

Level cuts of Calinescu, Karloff & Rabani Reduce cutting radius to half

Level cuts of Calinescu, Karloff & Rabani • Sort source-sink pairs in random order U • Pick random cutting radius 𝑠 ∈ 0, d • In sorted order, cut edges leaving the ball 𝐶(𝑡 X , 𝑠) of nodes contained within a radius of 𝑠 around the current source 𝑡 X with respect to the distances 𝑦 +, on edges 𝑣𝑤 ∈ 𝐹 𝐻 • “Protect” edges both of whose ends are contained in earlier balls from being cut later • Output the unprotected parts ∪ X 𝜀 + 𝐶 𝑡 X , 𝑠 ≝ 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢 0 for a random 𝑠

A 4 ln k approximation algorithm Fix an edge 𝑣𝑤. Order sources so that 𝑒 𝑡 U , 𝑣 ≤ 𝑒 𝑡 d , 𝑣 ≤ ⋯ ≤ 𝑒 𝑡 W , 𝑣 When does 𝑡 X cut the edge from the 𝑣 side? • When no other 𝑡 Z for 𝑘 < 𝑗 occurs before it in the random order • And when 𝑣𝑤 lies in the correct range: r ∈ [𝑒 𝑡 X , 𝑣 , 𝑒 𝑡 X , 𝑣 + 𝑦 +, ]

� � A 4 ln k approximation algorithm When does 𝑡 X cut the edge from the 𝑣 side? • When no other 𝑡 Z for 𝑘 < 𝑗 occurs before it in the random order U probability ≤ X • And when 𝑣𝑤 lies in the correct range: r ∈ 𝑒 𝑡 X , 𝑣 , 𝑒 𝑡 X , 𝑣 + 𝑦 +, f gh probability ≤ i j ≤ 2 ) 1 𝑄 𝑣𝑤 ∈ 𝑑𝑣𝑢 = ) 𝑄(𝑣𝑤 𝑑𝑣𝑢 𝑐𝑧 𝑡 X ) 2𝑦 +, ≤ 4 ln 𝑙 𝑗 X X

A 4 ln k approximation algorithm For the CKR cutting algorithm, 𝑄 𝑣𝑤 ∈ 𝑑𝑣𝑢 ≤ 4 ln 𝑙 Theorem (CKR): Expected cost of output multicut is 4 ln 𝑙 𝑨 [\^