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

1 / 49


Outline

Multi-route Cuts and LP Formulation

Region growing lemma

Algorithms

2 / 49


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.

3 / 49


Connectivity




4 / 49


Connectivity




5 / 49


Road Runner in Konigsberg

6 / 49


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

7 / 49




t

s

1010

10

1

11

ce = 100

8 / 49




t

s

1010

10

1

11

ce = 100

9 / 49


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.

10 / 49


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

11 / 49


Spectrum of Cut Problems

s

t1t2

t3

Single Source MultipleSink

Multiway Cuts

t1

t2

th

Multicuts

s1

sh

t1

th

12 / 49


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 ′).

13 / 49


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]

14 / 49


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


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


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]

17 / 49


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

18 / 49






nk−1

)possibilities

Multiple terminals

t

s

19 / 49






nk−1

)possibilities

Multiple terminals

t

s

20 / 49






nk−1

)possibilities

Multiple terminals

t

s

21 / 49


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

22 / 49





s

t1

t2

t3

23 / 49





s

t1

t2

t3

24 / 49





s

t1

t2

t3

25 / 49



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

26 / 49



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

27 / 49





s

t1

t2

t3

28 / 49



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

29 / 49


Region Growing Lemma

LP solution assigns length xe to edges

Surface area ce

Volume xece

xe

ce

30 / 49


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 .

31 / 49




s t

32 / 49




s t

33 / 49



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

s tA(r)r

V (r)Ball B(r)

34 / 49


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 .

35 / 49



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

36 / 49



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

37 / 49



∃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)

38 / 49


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)

39 / 49


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

40 / 49


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

41 / 49




s1 t1

s2 t2s4 t4

s5 t5

42 / 49




s1 t1

s2 t2s4 t4

s5 t5

43 / 49


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 =∞

44 / 49



(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 =∞

45 / 49



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

46 / 49


Open Problems

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

Hardness of approximation for k-route cut problems.

47 / 49


Open Problems

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

Hardness of approximation for k-route cut problems.

48 / 49


Questions?

49 / 49

Documents

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