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 𝑒2 = 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
• f 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
• f 𝑠 around the source 𝑡 with respect to the distances 𝑦+, on arcs

𝑣𝑤 ∈ 𝐵 𝐻

• Output 𝜀I 𝐶 𝑡, 𝑠

≝ 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢0 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 = {𝑡U, … , 𝑡W} , find minimum cost edge set blocking all 𝑡X − 𝑡

Z paths

𝑨[\]^ = min ) 𝑥+, 𝑦+,

• T_`T2 +,

s.t. ∑ 𝑦+,

• +,∈#

≥ 1 ∀ 𝑄 𝑡X − 𝑡

Z paths

𝑦+, ≥ 0 ∀𝑣𝑤 edges

Natural generalization of level cuts

• Pick random cutting radius 𝑠 ∈ 0,

U d

• Cut edges leaving the ball 𝐶(𝑡X, 𝑠) of nodes contained within a radius
• f 𝑠 around the each source 𝑡X with respect to the distances 𝑦+, on

edges 𝑣𝑤 ∈ 𝐹 𝐻

• Output all but the heaviest cut in ∪X 𝜀 𝐶 𝑡X, 𝑠

≝ 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢0

A 2-approximation algorithm

Claim: 𝑄 𝑣𝑤 ∈ 𝑑𝑣𝑢 ≤

fgh

i j

= 2𝑦+, Corollary: 𝐹 𝑥 𝑑𝑣𝑢 ≤ 2𝑨[\]^ min

ST,TS2 0 𝑥 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢0 ≤ 1 − 1

𝑙 𝐹 𝑥 𝑑𝑣𝑢 ≤ 2 1 −

U W 𝑨[\]^

≤ 2 1 −

U 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 = {𝑡U, … , 𝑡W} , find minimum cost arc set blocking all 𝑡X → 𝑡

Z paths

for all ordered pairs of sources 𝑗 ≠ 𝑘. Note: Min multiway cut for S = {𝑡U, 𝑡d} 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 = {𝑡U, … , 𝑡W} , find minimum cost arc set blocking all 𝑡X → 𝑡

Z paths for all

• rdered 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 = {𝑡U, … , 𝑡W} , and nonnegative costs on non-source nodes, find minimum cost node set blocking all 𝑡X − 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 = {𝑡U, … , 𝑡W} , find minimum cost arc set blocking all 𝑡X → 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)

Theorem: Level-cutting algorithm on the above LP gives a 2- approximation 𝑨[\]^ = min ) 𝑥+, 𝑦+,

• T_`T2 +,

s.t. ∑ 𝑦+,

• +,∈#

≥ 1 ∀ 𝑄 𝑡X → 𝑡

Z paths

𝑦+, ≥ 0 ∀𝑣𝑤 arcs

Level cuts – Attempt 1

• Pick random cutting radius 𝑠 ∈ 0,1
• Cut arcs leaving the ball 𝐶(𝑡X, 𝑠) of nodes contained within a radius
• f 𝑠 around the each source 𝑡X with respect to the distances 𝑦+, on

arcs 𝑣𝑤 ∈ 𝐵 𝐻

• Output ∪X 𝜀I 𝐶 𝑡X, 𝑠

≝ 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢0 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 𝑡U, 𝑣 ≤ 𝑒 𝑡d, 𝑣 ≤ ⋯ ≤ 𝑒 𝑡W, 𝑣 Note that 𝑒 𝑢d, 𝑣 = 𝑒 𝑢w, 𝑣 = ⋯ = 𝑒 𝑢W, 𝑣 = 𝑒 𝑡U, 𝑣 If one of these balls cut 𝑣𝑤 then all of them do Thus 𝑣𝑤 is either cut by the ball around 𝑢U or by the above set of balls. 𝑄 𝑣𝑤 ∈ 𝑑𝑣𝑢 ≤ 2𝑦+, min

ST,TS2 0 𝑥 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢0 ≤ 2𝑥 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)

Multicuts in undirected graphs

Given undirected graph G=(V,E), with nonnegative costs on edges, and source-sink pairs = {(𝑡U, 𝑢U), … , (𝑡W, 𝑢W)} , find minimum cost edge set blocking all 𝑡X − 𝑢X paths 𝑨[\^ = min ) 𝑥+, 𝑦+,

• T_`T2 +,

s.t. ∑ 𝑦+,

• +,∈#

≥ 1 ∀ 𝑄 𝑡X − 𝑢X paths 𝑦+, ≥ 0 ∀𝑣𝑤 edges

Level cut algorithm – attempt 1

• Pick random cutting radius 𝑠 ∈ 0,1
• Cut edges leaving the ball 𝐶(𝑡X, 𝑠) of nodes contained within a radius
• f 𝑠 around the each source 𝑡X with respect to the distances 𝑦+, on

edges 𝑣𝑤 ∈ 𝐹 𝐻

• Output ∪X 𝜀 𝐶 𝑡X, 𝑠

≝ 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢0

• 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(𝑡X, 𝑠) of nodes contained

within a radius of 𝑠 around the current source 𝑡X 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 𝑡X, 𝑠

≝ 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢0 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
• Pick random cutting radius 𝑠 ∈ 0,

U d

• In sorted order, cut edges leaving the ball 𝐶(𝑡X, 𝑠) of nodes contained

within a radius of 𝑠 around the current source 𝑡X 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 𝑡X, 𝑠

≝ 𝑚𝑓𝑤𝑓𝑚 𝑑𝑣𝑢0 for a random 𝑠

A 4 ln k approximation algorithm

Fix an edge 𝑣𝑤. Order sources so that 𝑒 𝑡U, 𝑣 ≤ 𝑒 𝑡d, 𝑣 ≤ ⋯ ≤ 𝑒 𝑡W, 𝑣 When does 𝑡X 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 𝑡X, 𝑣 , 𝑒 𝑡X, 𝑣 + 𝑦+,]

A 4 ln k approximation algorithm

When does 𝑡X cut the edge from the 𝑣 side?

• When no other 𝑡

Z for 𝑘 < 𝑗 occurs before it in the random order

probability ≤

U X

• And when 𝑣𝑤 lies in the correct range: r ∈ 𝑒 𝑡X, 𝑣 , 𝑒 𝑡X, 𝑣 + 𝑦+,

probability ≤

fgh

i j

𝑄 𝑣𝑤 ∈ 𝑑𝑣𝑢 = ) 𝑄(𝑣𝑤 𝑑𝑣𝑢 𝑐𝑧 𝑡X)

• X

≤ 2 ) 1 𝑗

• X

2𝑦+, ≤ 4 ln 𝑙

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[\^

Summary

• If you need to cut a graph, write a distance

based linear program to round

• 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)