Region Growing for Multi-route Cuts€¦ · Multi-route Cuts and LP Formulation Region growing lemma Algorithms 2/49. Multi-route CutsLPRegion Growing LemmaAlgorithms Connectivity

Multi-route Cuts LP Region Growing Lemma Algorithms

Region Growing for Multi-route Cuts

Siddharth Barman

Joint work with Shuchi Chawla

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Multi-route Cuts LP Region Growing Lemma Algorithms

Outline

Multi-route Cuts and LP Formulation

Region growing lemma

Algorithms

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Multi-route Cuts LP Region Growing Lemma Algorithms

Connectivity

Connectivity between two terminals is defined to be thenumber of edge disjoint paths between them.

In classical cut problems the goal is to find a set of edgeswhose removal reduces the connectivity to zero.

Relaxing to higher connectivity requirements gives usmulti-route cuts.

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Multi-route Cuts LP Region Growing Lemma Algorithms

Connectivity

Connectivity between two terminals is defined to be thenumber of edge disjoint paths between them.

In classical cut problems the goal is to find a set of edgeswhose removal reduces the connectivity to zero.

Relaxing to higher connectivity requirements gives usmulti-route cuts.

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Multi-route Cuts LP Region Growing Lemma Algorithms

Connectivity

Connectivity between two terminals is defined to be thenumber of edge disjoint paths between them.

In classical cut problems the goal is to find a set of edgeswhose removal reduces the connectivity to zero.

Relaxing to higher connectivity requirements gives usmulti-route cuts.

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Multi-route Cuts LP Region Growing Lemma Algorithms

Road Runner in Konigsberg

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Multi-route Cuts LP Region Growing Lemma Algorithms

Multi-route Cut Instance

Objective: Find a low cost set of edges such that connectivitybetween s to t to reduces to 1.

t

s

1010

10

1

11

ce = 100

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Multi-route Cuts LP Region Growing Lemma Algorithms

Multi-route Cut Instance

Objective: Find a low cost set of edges such that connectivitybetween s to t to reduces to 1.

t

s

1010

10

1

11

ce = 100

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Multi-route Cuts LP Region Growing Lemma Algorithms

Multi-route Cut Instance

Objective: Find a low cost set of edges such that connectivitybetween s to t to reduces to 1.

t

s

1010

10

1

11

ce = 100

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Multi-route Cuts LP Region Growing Lemma Algorithms

Menger’s Theorem

Menger’s Theorem: Maximum number of pairwise edgedisjoint paths between any two terminals is equal to minimumedge cut between them.

Connectivity := Number of edge disjoint paths = 2.

Menger’s Theorem: ∃ cut C with exactly 2 edges.

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Multi-route Cuts LP Region Growing Lemma Algorithms

Menger’s Theorem

Menger’s Theorem: Maximum number of pairwise edgedisjoint paths between any two terminals is equal to minimumedge cut between them.

Connectivity := Number of edge disjoint paths = 2.

Menger’s Theorem: ∃ cut C with exactly 2 edges.

s t

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Multi-route Cuts LP Region Growing Lemma Algorithms

Spectrum of Cut Problems

s

t1t2

t3

Single Source MultipleSink

Multiway Cuts

t1

t2

th

Multicuts

s1

sh

t1

th

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Multi-route Cuts LP Region Growing Lemma Algorithms

Multicut version of multi-route cut

Given: A graph G with edge costs with terminal pairs,(s1, t1), (s2, t2), · · · , (sh, th), and connectivity threshold, k.

Goal: To produce a minimum cost set of edges E ′ ⊆ E , suchthat for each i , si and ti are at most (k − 1)-edge-connectedin the graph (V ,E \ E ′).

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Multi-route Cuts LP Region Growing Lemma Algorithms

Related Work

Bruhn, Cerny, Hall and Kolman [SODA07] gave a dualitytheorem for multi-route flows and cuts on uniform capacitygraphs.

Chekuri and Khanna [ICALP08], gave a randomizedO(log2 n log h) approximation algorithm for 2-route multicuts.

Classical Multicut via region growing by Garg, Vazirani andYannakakis [STOC03]

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Multi-route Cuts LP Region Growing Lemma Algorithms

Results

EDRC: Edge Disjoint Route Cut

NDRC: Node Disjoint Route Cut

SS: Single Source Multiple Sink, MW: Multiway Cut, MC:Multicut,

Problem Previous best result Our result

SS-2-EDRC, SS-2-NDRC O(log n) O(log h)

MW-2-EDRC, MW-2-NDRC O(log n log h) O(log2 h)

MC-2-EDRC O(log2 n log h) O(log2 h)

MC-2-NDRC – O(log2 h)

SS-k-EDRC – (6,O(√

h ln h))SS-k-EDRC-Uniform – (2, 4)SS-k-EDRC (constant h) – (4, 4)

0n = # of nodes, h = # of source-sink pairs15 / 49

Multi-route Cuts LP Region Growing Lemma Algorithms

Results

EDRC: Edge Disjoint Route Cut

NDRC: Node Disjoint Route Cut

SS: Single Source Multiple Sink, MW: Multiway Cut, MC:Multicut,

Problem Previous best result Our result

SS-2-EDRC, SS-2-NDRC O(log n) O(log h)

MW-2-EDRC, MW-2-NDRC O(log n log h) O(log2 h)

MC-2-EDRC O(log2 n log h) O(log2 h)

MC-2-NDRC – O(log2 h)

SS-k-EDRC – (6,O(√

h ln h))SS-k-EDRC-Uniform – (2, 4)SS-k-EDRC (constant h) – (4, 4)

0n = # of nodes, h = # of source-sink pairs16 / 49

Multi-route Cuts LP Region Growing Lemma Algorithms

LP Formulation for Multicut Version

z = min∑e∈E

xe ce

s.t∑e∈E

y ie ≤ k − 1 ∀i ∈ [h]

d i (u, v) = xe + y ie ∀i ∈ [h], e = (u, v) ∈ E

d i is a metric ∀i ∈ [h]

d i (si , ti ) ≥ 1 ∀i ∈ [h]

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Multi-route Cuts LP Region Growing Lemma Algorithms

Guessing Witness Edges

We can guess k − 1 witness edges and then find min-cut.

But does not scale up for:

Higher k values, as there are(

nk−1

)possibilities

Multiple terminals

t

s

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Multi-route Cuts LP Region Growing Lemma Algorithms

Guessing Witness Edges

We can guess k − 1 witness edges and then find min-cut.

But does not scale up for:

Higher k values, as there are(

nk−1

)possibilities

Multiple terminals

t

s

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Multi-route Cuts LP Region Growing Lemma Algorithms

Guessing Witness Edges

We can guess k − 1 witness edges and then find min-cut.

But does not scale up for:

Higher k values, as there are(

nk−1

)possibilities

Multiple terminals

t

s

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Multi-route Cuts LP Region Growing Lemma Algorithms

Guessing Witness Edges

We can guess k − 1 witness edges and then find min-cut.

But does not scale up for:

Higher k values, as there are(

nk−1

)possibilities

Multiple terminals

t

s

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Multi-route Cuts LP Region Growing Lemma Algorithms

Construction Highlight: k = 2

Let cost1(C ) be the total cost of the cut minus the mostexpensive edge

Say, for s and all ti , there exists a cut C such that cost1(C )≤ α× LP contribution contained inside C .

s

t1

t2

t3

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Multi-route Cuts LP Region Growing Lemma Algorithms

Construction Highlight: k = 2

Let cost1(C ) be the total cost of the cut minus the mostexpensive edge

Say, for s and all ti , there exists a cut C such that cost1(C )≤ α× LP contribution contained inside C .

s

t1

t2

t3

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Multi-route Cuts LP Region Growing Lemma Algorithms

Construction Highlight: k = 2

Let cost1(C ) be the total cost of the cut minus the mostexpensive edge

Say, for s and all ti , there exists a cut C such that cost1(C )≤ α× LP contribution contained inside C .

s

t1

t2

t3

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Multi-route Cuts LP Region Growing Lemma Algorithms

Construction Highlight: k = 2

Let cost1(C ) be the total cost of the cut minus the mostexpensive edge

Say, for s and all ti , there exists a cut C such that cost1(C )≤ α× LP contribution contained inside C .

s

t1

t2

t3

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Multi-route Cuts LP Region Growing Lemma Algorithms

Construction Highlight: k = 2

Let cost1(C ) be the total cost of the cut minus the mostexpensive edgeSay, for s and all ti , there exists a cut C such that cost1(C )≤ α× LP contribution contained inside C .

s

t1

t2

t3

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Multi-route Cuts LP Region Growing Lemma Algorithms

Construction Highlight: k = 2

Let cost1(C ) be the total cost of the cut minus the mostexpensive edgeSay, for s and all ti , there exists a cut C such that cost1(C )≤ α× LP contribution contained inside C .

s

t1t3

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Multi-route Cuts LP Region Growing Lemma Algorithms

Construction Highlight: k = 2

Let cost1(C ) be the total cost of the cut minus the mostexpensive edge

Say, for s and all ti , there exists a cut C such that cost1(C )≤ α× LP contribution contained inside C .

s

t1

t2

t3

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Multi-route Cuts LP Region Growing Lemma Algorithms

Construction Highlight: k = 2

Total cost of the generated solution is no more than α timesthe value obtained by the linear program.

We still need to consider whether proper connectivity isachieved in the residual graph.

s

t1

t2

t3

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Multi-route Cuts LP Region Growing Lemma Algorithms

Region Growing Lemma

LP solution assigns length xe to edges

Surface area ce

Volume xece

xe

ce

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Multi-route Cuts LP Region Growing Lemma Algorithms

Region Growing Lemma: Single Edge Length

For well separated terminals s and t, there exists a cut C suchthat cost(C ) ≤ log h× LP contribution contained inside C .

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Multi-route Cuts LP Region Growing Lemma Algorithms

Region Growing Lemma: Single Edge Length

For well separated terminals s and t, there exists a cut C suchthat cost(C ) ≤ log h× LP contribution contained inside C .

s t

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Multi-route Cuts LP Region Growing Lemma Algorithms

Region Growing Lemma: Single Edge Length

For well separated terminals s and t, there exists a cut C suchthat cost(C ) ≤ log h× LP contribution contained inside C .

s t

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Multi-route Cuts LP Region Growing Lemma Algorithms

Region Growing Lemma: Single Edge Length

∃r ∈ [0, 1] such that A(r) ≤ log h × V (r).

s tA(r)r

V (r)Ball B(r)

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Multi-route Cuts LP Region Growing Lemma Algorithms

Region Growing Lemma for Multi-route Case

Even with x and y lengths there exists a cut C such that # yedges in the cut small

And x-cost(C ) ≤ O(log h)× LP contribution (x’s) containedinside C .

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Multi-route Cuts LP Region Growing Lemma Algorithms

Region Growing Lemma for Multi-route Case

Even with x and y lengths there exists a cut C such that # yedges in the cut smallAnd x-cost(C ) ≤ O(log h)× LP contribution (x’s) containedinside C .

s t

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Multi-route Cuts LP Region Growing Lemma Algorithms

Region Growing Lemma for Multi-route Case

Even with x and y lengths there exists a cut C such that # yedges in the cut smallAnd x-cost(C ) ≤ O(log h)× LP contribution (x’s) containedinside C .

s t

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Multi-route Cuts LP Region Growing Lemma Algorithms

Region Growing Lemma for Multi-route Case

∃r ∈ [0, 1] such thatAx(r) ≤ 2 log h × V x(r)y(r) < 2(k − 1)

Markov’s Inequality

s tAx(r)

y edge

r

V x(r)Ball B(r)

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Multi-route Cuts LP Region Growing Lemma Algorithms

Region Growing Lemma for 2-route

∃r ∈ [0, 1] such that

Ax(r) ≤ 2 log h × V x(r)y(r) < 2(k − 1) = 2

s tAx(r)

y edge

r

V x(r)Ball B(r)

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Multi-route Cuts LP Region Growing Lemma Algorithms

2-Route Cut Algorithms: Single Source Multiple Sink

First we find a cut S that separates terminal, say t1, from thesource via region growing lemma.Then we then remove the cut from the graph and iterate overG [V \ S ].Overall this gives a O(log h) approximation.

s

t1

t2

t3

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Multi-route Cuts LP Region Growing Lemma Algorithms

2-Route Cut Algorithms: Multicut

For 2-route multicuts in order to guarantee separation, aftercut S is found by region growing, we need to recurse on G [S ]and G [V \ S ]. This results in an O(log2 h) approximation.

s1 t1

s2 t2s4 t4

s5 t5

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Multi-route Cuts LP Region Growing Lemma Algorithms

2-Route Cut Algorithms: Multicut

For 2-route multicuts in order to guarantee separation, aftercut S is found by region growing, we need to recurse on G [S ]and G [V \ S ]. This results in an O(log2 h) approximation.

s1 t1

s2 t2s4 t4

s5 t5

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Multi-route Cuts LP Region Growing Lemma Algorithms

2-Route Cut Algorithms: Multicut

For 2-route multicuts in order to guarantee separation, aftercut S is found by region growing, we need to recurse on G [S ]and G [V \ S ]. This results in an O(log2 h) approximation.

s1 t1

s2 t2s4 t4

s5 t5

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Multi-route Cuts LP Region Growing Lemma Algorithms

k-Route Cut Algorithms

Integrality gap of the LP is Ω(k) for k + 1 route cut

s t

u

ye = kk+1

xe = 1k+1

# = k + 1# = 2k

ce = 1

ce =∞

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Multi-route Cuts LP Region Growing Lemma Algorithms

k-Route Cut Algorithms

(α, β) approximation: si and ti get α× k separated with acost of β × OPT

s t

u

ye = kk+1

xe = 1k+1

# = k + 1# = 2k

ce = 1

ce =∞

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Multi-route Cuts LP Region Growing Lemma Algorithms

k-Route Cut Algorithms

Plainly applying the lemma we get (2h, 2) and (2, 2h)Single Source Multiple Sink k-route cut using a separationidea over a strengthened LP we get a (6,O(

√h log h))

approximation.

s

t1

t2

t3

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Multi-route Cuts LP Region Growing Lemma Algorithms

Open Problems

Poly-logarithmic (bicriteria) approximation for k-route cuts fork ≥ 3.

Hardness of approximation for k-route cut problems.

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Multi-route Cuts LP Region Growing Lemma Algorithms

Open Problems

Poly-logarithmic (bicriteria) approximation for k-route cuts fork ≥ 3.

Hardness of approximation for k-route cut problems.

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Multi-route Cuts LP Region Growing Lemma Algorithms

Questions?

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