Julia Chuzhoy (TTI-C)Yury Makarychev (TTI-C)

Aravindan Vijayaraghavan (Princeton)Yuan Zhou (CMU)

Cut minimization• Min st-cut: delete the min #edges to disconnect s, t

tt

ssDuality: Maxflow(s, t) = Mincut(s, t) = 2

Multicut• Given r pairs (si, ti), delete min #edges to disconnect

all (si, ti) pairst1t1

s1s1

s2s2

s3s3

t3t3

•Upper bound on max multicommodity flow

•Identifies bottlenecks in the graph

•O(log r) approximation algorithm [GVY95]

t2t2

Min k-route cuts• Unweighted version. Given r pairs (si, ti), delete min

#edges to k-disconnect all (si, ti) pairs

– i.e. for all i, (si, ti)-edge-connectivity < k

• General version. Given a weighted graph and r pairs, delete min wt. of edges to k-disconnect all (si, ti)

pairs

For example, when k = 2, OPT = 1.

t1t1

s1s1

s2s2

s3s3

t3t3

t2t2

Min k-route cuts: variants and specal cases

• EC-kRC: edge connectivity version, remove min. wt. of edges so that for each i, (si, ti)-edge-connectivity

< k– Unweighted case: all edge weights = 1– k = 1: Minimum multicut– s-t EC-kRC: single source-sink pair version

• VC-kRC: vertex connectivity version, remove min. wt. of edges so that for each i, (si, ti)-vertex-

connectivity < k

Motivation• Multiroute generalization

• st-k-route flow: a fractional combination of elementary k-route st-flows [Kis96, KT93, AO02]– Flow is resilient to (k-1) failures

Maxflow/Mincut

multiroute generalization st-k-route

flow

multicut k-route cut

multicommodityflow

k-routemulticommodity

flow

: a fault tolerant setting

Motivation (cont'd)

• Multiroute generalization: a fault tolerant setting

• As standard multicut, k-route cut also reveals network bottleneck, and in particular measures resilience of the network

multicut k-route cut

Approximation algorithms

• α-approximation: delete edges of wt. αOPT such that all the pairs are k-disconnected

• (β,α)-bicriteria approximation: delete edges of wt. αOPT such that all the pairs are βk-disconnected

Previous work• [Chekuri-Khanna'08]

– O(log2n log r)-approximation for k=2 (both EC-2RC and VC-2RC)

• [Barman-Chawla'10]

– O(log2r)-approximation for k=2 (both EC-2RC and VC-2RC)

– NP-Hardness for s-t EC-kRC

• [Kolman-Scheideler'11]

– O(log3r)-approximation for k=3 (EC-2RC)

• No sub-polynomial approx. algorithm known for k > 3

Our results : algorithms for EC-kRC• Unweighted EC-kRC

– O(k log1.5 r)-approximation

– (1+ε, (1/ε)log1.5 r)-bicriteria approximation

• General EC-kRC

– O(log1.5 r)-approximation for k = 2

– (2, log2.5 r loglog r)-bicriteria approx. in nO(k) time

– (log r, log3 r)-bicriteria approx. in poly(n, k) time

Our results : VC-kRC• Algorithms

– O(log1.5 r)-approximation for k = 2

– (2, d k log2.5 r loglog r)-bicriteria approx. in nO(k) time, where each node belongs to at most d source-sink pairs

• Harndess for VC-kRC

– NP-Hard to approximate VC-kRC within Ω(kε) for some specific ε > 0

• Hardness for st-VC-kRC– Superconstant hardness assuming random k-AND

hypothesis of [Feige'02]

– Ω(ρ0.5) hardness assuming ρ-inapproximability of Densest k-Subgraph

A comparison : EC-kRC

Previous results

Our results

k = 2 O(log2 r) [BC10] O(log1.5 r)

k = 3 O(log3 r) [KS11]

arbitrary k,unweighted

O(k log1.5 r)

(1+ε, (1/ε)log1.5 r)

arbitrary k,general

(2, log2.5 r loglog r)

in time nO(k)

(log r, log3 r)

in poly(n, k) time

A comparison : VC-kRC

Previous results Our results

k = 2 O(log2 r) [BC10] O(log1.5 r)

arbitrary k

multicut hardness:

APX-hard [DJP+94]

superconstant assuming UGC

[KV05, CKK+06]

(2, dklog2.5r loglog r)

alg in time nO(k)

Ω(kε)-hardness

Why only bicriteria algorithm for large k?

• The problem might be hard even for single s-t version– st-VC-kRC: no sub-polynomial approx. if assuming

no sub-polynomail approx. for Densest k-Subgraph– st-EC-kRC: no sub-polynomial approx. known (there is a (2, 2)-bicriteria approx. alg.)

• Embarrassing situation for st-EC-kRC: even APX-hardness is not known

• Recall the problem:– A weighted graph G, – source s, sink t– Goal: remove k edges in G to minimize min s-t cut

The rest of this talk...

• O(k log1.5 r)-approximation algorithm for unweighted EC-kRC

• O(log1.5 r)-approximation algorithm for general EC-2RC

• (2, log2.5 r loglog r)-bicriteria approx. algorithm for general EC-kRC (sketch)

The difficulty for large k (> 2)• Simple recursion (used in [BC10]) for k = 2

– Find a balanced cut (by region growing)– Remove all the cut edges but the most expensive one– recurse into both sides

• Key observation. the red edge cannot provide extra connectivity

for s1, t1

graph G

s1

t1

The difficulty for large k (> 2)• Simple recursion (used in [BC10]) for k = 2

– Find a balanced cut (by region growing)– Remove all the cut edges but the most expensive one– recurse into both sides

• Key observation. the red edge cannot provide extra connectivity

for s1, t1

• No longer true for k = 3 (or more)

graph G

s1

t1

a bad example for k = 3

Algorithms for k > 2• [Kolman-Scheideler'11] O(log3r)-approximation for

k=3, by multi-level region growing (based on the same LP used in [BC10])

• Our method– Idea 1. Relate k-route cut to the value of sparest

cut

– Idea 2. Solve the problem iteratively rather than recursively

O(k log1.5 r)-approximation algorithm for unweighted EC-kRC

Cut sparsity, and unweighted EC-kRC

• Let d(v) = #source-sink pairs that v participates in d(S) =

• Define uniform sparsity to be

• Intuition. Given a cut , is small when– the cut size is small– the cut separates many terminals (or, the cut is

balanced in terms of d)

Sv

vd )(

,)}(),(min{

),()(

SdSd

SSedgesS )(min)( SG

S

),( SS )(S

Cut sparsity, and unweighted EC-kRC

• Let d(v) = #source-sink pairs that v participates in d(S) =

• Define uniform sparsity to be

• Theorem.[ARV04] O(log0.5 r)-approx. for Φ(G).

Sv

vd )(

,)}(),(min{

),()(

SdSd

SSedgesS )(min)( SG

S

r

OPTkG

)(• Lemma.

Algorithm for unweighted EC-kRC• Step 0. Assume source-sink pairs are not k-

disconnected• Step 1. Use the algorithm in [ARV04] to find an

approximate sparse cut • Step 2. Delete all the edges across the cut• Step 3. Recurse into the subinstances defined by

each side of the cut

• Fact. #cut edges deleted in Step 2 is at most

),( SS

)}(),(min{)(log),( SdSdGrSSedges

OPTr

SdSdkr

)}(),(min{log

r

OPTkG

)(• Lemma.

A standard charging argument• Fact. #cut edges deleted in Step 2 is at most

• Charge this cost to the smaller part among – At one step, each terminal is charged by

• Each terminal can be in "small parts"– In total, each terminal is charged by

• Since there are r terminals, total cost:

),( SSedges OPTr

SdSdkr

)}(),(min{log

)(),( SdSd

OPTr

kr log

rlogOPT

r

kr 5.1log

OPTrk 5.1log

Proof of • Consider H = G \ OPT

• For every (si, ti) pair,

mincutH(si, ti) = |edges(Si, Ti)| < k

(a witness cut)

• Claim. The witness cuts are laminar

r

OPTkG

)(• Lemma.

si

ti

Si

Ti

Proof of Claim: witness cuts are laminar

• Gomory-Hu Tree. (exists for every graph) A weighted tree that consists of edges

representing all pairs minimum s-t cuts in the graph. mincutH(s, t) = mincutT(s, t)

• All s-t mincuts in the tree are laminar ==> All mincuts in H are laminar ==> All witness cuts are laminar

H:H:

Gomory-Hu tree TGomory-Hu tree T

Proof of • Consider H = G \ OPT

• For every (si, ti) pair,

mincutH(si, ti) = |edges(Si, Ti)| < k

(a witness cut)

• Claim. The witness cuts are laminar

• Let S1, S2, ..., Sm be the maximal

witness cuts (the smaller parts)

r

OPTkG

)(• Lemma.

S1

S2

S3

Proof of

• Let S1, S2, ..., Sm be the maximal witness cuts in H=G\

OPT

1. d(S1) + d(S2) + ... + d(Sm) >= r

2.

therefore

r

OPTkG

)(• Lemma.

S1

S2

S3

1),( iiOPT SSedges

1),( kSSedges iiH

),(),( iiOPTiiG SSedgeskSSedges

Proof of

• Let S1, S2, ..., Sm be the maximal witness cuts in H=G\

OPT

1. d(S1) + d(S2) + ... + d(Sm) >= r

2.

r

OPTkG

)(• Lemma.

S1

S2

S3

),(),( iiOPTiiG SSedgeskSSedges

OPTkSSedgesm

iiiG

2),(1

(since each edge is shared by at most 2

maximal cuts)

Proof of • Let S1, S2, ..., Sm be the maximal witness cuts in H=G\OPT

1. d(S1) + d(S2) + ... + d(Sm) >= r

2.

3. by expansion

In all:

r

OPTkG

)(• Lemma.

S1

S2

S3

),(),( iiOPTiiG SSedgeskSSedges

)()(),(11

i

m

i

m

iiiG SdGSSedges

)(Gr

OPTkGr 2)(

OPTkSSedgesm

iiiG

2),(1

O(log1.5 r)-approx. algorithm for EC-2RC

Another definition of sparsity• 2-route uniform sparsity

• Corollary of [ARV04]. 2-route uniform sparisty can

be efficiently approximated within O(log0.5 r) factor– Proof. Guess the red edge, remove it, and run

ARV.

,)}(),(min{

)}({max),()( ),()2(

SdSd

ewtSSwtS SSe

)(min)( )2()2( SGS

S S

the most expensive

edge

Another definition of sparsity• 2-route uniform sparsity

• Corollary of [ARV04]. 2-route uniform sparisty can

be efficiently approximated within O(log0.5 r) factor

,)}(),(min{

)}({max),()( ),()2(

SdSd

ewtSSwtS SSe

)(min)( )2()2( SGS

S S

the most expensive

edge

r

OPTG )()2(• Lemma.

Algorithm for EC-2RC• Step 1. Find an approximate 2-route sparse cut • Step 2. Delete all but the most expensive edge

across the cut• Step 3. Recurse into the subinstances defined by

each side of the cut

• Claim. The algorithm outputs a valid 2-route cut.• Proof. By the key observation we made before.

),( SS

Algorithm for EC-2RC• Step 1. Find an approximate 2-route sparse cut • Step 2. Delete all but the most expensive edge

across the cut• Step 3. Recurse into the subinstances defined by

each side of the cut

• Fact. wt of edges deleted in Step 2 is at most

• Corollary. wt of edges removed in total

),( SS

)}(),(min{)(log),( )2( SdSdGrSSedges

OPTr

SdSdr

)}(),(min{log

r

OPTG )()2(• Lemma.

OPTr 5.1log

Proof of • Consider H = G \ OPT

• For every (si, ti) pair,

mincutH(si, ti) = |edges(Si, Ti)| < 2

(a witness cut)

• Claim. The witness cuts are laminar

• Let S1, S2, ..., Sm be the maximal

witness cuts (the smaller parts)

r

OPTG )()2(• Lemma.

S1

S2

S3

Proof of

• Let S1, S2, ..., Sm be the maximal witness cuts in H=G\

OPT

1. d(S1) + d(S2) + ... + d(Sm) >= r

2.

),(

),(),(

)}({max),(),(

iiOPT

iiHiiG

SSeiiG

SSwt

SSwtSSwt

ewtSSwtii

OPTSSwt

ewtSSwt

m

iiiOPT

m

iSSe

iiGii

2),(

)}({max),(

1

1),(

, therefore,

r

OPTG )()2(• Lemma.

S1

S2

S3

Proof of

• Let S1, S2, ..., Sm be the maximal witness cuts in H=G\

OPT

1. d(S1) + d(S2) + ... + d(Sm) >= r

2.

In all,

thus, there exists i:

OPTewtSSwtm

iSSe

iiGii

2)}({max),(

1),(

r

OPT

Sd

ewtSSwt

m

ii

m

iSSe

iiGii

2

)(

)}({max),(

1

1),(

r

OPTG )()2(• Lemma.

r

OPT

Sd

ewtSSwtG

i

SSeiiG

ii

2

)(

)}({max),()( ),()2(

(2, log2.5 r loglog r)-bicriteria approx. for general EC-kRC (sketch)

(2, log2.5 r loglog r)-bicriteria approx. for general EC-kRC (sketch)• k-route non-uniform sparsity

where

• Corollary. (of [ALN05]) O(log0.5 r loglog r) approx. in nO(k) time

,),(

),(wt)()(

SSD

SSS

(k)k )(min)( )()( SG k

S

k

),(wt SS(k) : total wt of all the edges across the cut but the most expensive (k-1) ones

),(D SS : #source-sink pairs across the cut

r

OPTrGk log)()12(• Lemma.

(2, log2.5 r loglog r)-bicriteria approx. for general EC-kRC (sketch) (cont'd)

• The iterative algorithm. (Applying Idea 2)• Step 1. Use the algorithm in [ALN05] to find an

approximate sparse cut • Step 2. Delete all the edges across the cut but the (2k-2)

most expensive ones• Step 3. Remove all the source-sink pairs that are (2k-1)-

disconnected• Step 4. Repeat Step 1~3 until no source-sink pair

remains

• Theorem. Wt. of removed edges <= log2.5 r loglog r OPT

r

OPTrGk log)()12(• Lemma.

Open questions

• Algorithm side.– Better true approximation algorithm for general

EC-kRC (and VC-kRC)

• Hardness side.– Is EC-kRC (for large k) strictly harder than

multicut?

– Understand the simplest case: st-EC-kRC.

Thank you!