covering problems in edge- and node-weighted graphs€¦ · covering problems in edge- and...

Covering problemsin edge- and node-weighted graphs

Takuro Fukunaga

National Institute of Informatics, JapanJST, ERATO, Kawarabayashi Large Graph Project

SWAT, July 2-4, 2014

Graph covering problem

Problem

Choose a minimum weight set of edges/nodessatisfying covering constraints

edge set• edge cover• edge dominating set• spanning tree• graph cut

this talk

node set• vertex cover• dominating set

Graph covering problem

Problem

Choose a minimum weight set of edges/nodessatisfying covering constraints

edge set• edge cover• edge dominating set• spanning tree• graph cut

this talk

node set• vertex cover• dominating set

Weights of an edge set FEdge weights: w(F) :=

∑e∈F w(e)

7 = 20

Node weights: w(F) :=∑

v∈V(F) w(v)

each node weight is counted once Ü subadditivity

∑e∈F w(e)

7 = 20

v∈V(F) w(v)

∑e∈F w(e)

7 = 20

v∈V(F) w(v)

Previous works on node-weight minimization• Steiner tree [Klein, Ravi, 95]

• Prize-collecting Steiner tree[Moss, Rabani, 07] [Konemann, Sadeghabad, Sanita, 13]

• Prize-collecting Steiner forest [Bateni, Hajiaghayi, Liaghat, 13]

• Survivable network design [Nutov 10,12]

• Prize-collecting survivable network design[Chekuri, Ene, Vakilian, 12]

• Online Steiner tree [Naor, Panigrahi, Singh, 11]

• Online Steiner forest [Hajiaghayi, Liaghat, Panigrahi, 13]

What about other covering problems?

Two questions

edge-weights

node-weights

1. How hard are problems?problems look hard...2. How to solve them?existing techniques look difficult to apply...

Our contribution1. Problem hardness:Many covering problems in bipartite graphs are set cover hard

0, 1-edge coveredge dominating setT -join

Üset cover

Ω(log n)-approx hardness

2. Extending exsiting LP rounding algorithms:• O(log n)-approx for the prize-collecting edge dominating set in

general graphs• exact algorithm for the prize-collecting edge dominating set in trees• 2-approx for the multicut in trees

Positive results matching approximation hardness

Our contribution1. Problem hardness:Many covering problems in bipartite graphs are set cover hard

0, 1-edge coveredge dominating setT -join

Üset cover

Ω(log n)-approx hardness

2. Extending exsiting LP rounding algorithms:• O(log n)-approx for the prize-collecting edge dominating set in

general graphs• exact algorithm for the prize-collecting edge dominating set in trees• 2-approx for the multicut in trees

Positive results matching approximation hardness6/17

Hardness

Edge dominating set (EDS)• δ(e) := set of edges sharing end nodes with e• F ∩ δ(e) 6= ∅ for ∀e ∈ E Ü F ⊆ E is an EDS

Reduction: Set cover Ü EDS

elements0 0 0 0 0 0 0

w(S1)w(S2)w(S3)w(S4)w(S5)w(S6)w(S7)w(S8)w(S9)

∞ ∞ ∞ ∞ ∞ ∞ ∞

w(S1) w(S4) w(S6)

Hardness

elements0 0 0 0 0 0 0

∞ ∞ ∞ ∞ ∞ ∞ ∞

w(S1) w(S4) w(S6)

Hardness

elements0 0 0 0 0 0 0

∞ ∞ ∞ ∞ ∞ ∞ ∞

w(S1) w(S4) w(S6)

Hardness

elements0 0 0 0 0 0 0

∞ ∞ ∞ ∞ ∞ ∞ ∞

w(S1) w(S4) w(S6)

Positive results

Edge-weights• 2-approximation algorithm for the EDS in general graphs [Fujito,

Nagamochi, 02]• An exact primal-dual algorithm for the prize-collecting EDS in trees

[Kamiyama, 10]

Edge- & node-weighted graph

• O(log n)-approximation algorithm for the prize-collecting EDS ingeneral graphs• An exact primal-dual algorithm for the prize-collecting EDS in trees

Main contributionA natural LP relaxation for node-weighted graphs has a large integralitygap in many covering problems Ü We present a new LP relaxation

Positive results

Edge-weights• 2-approximation algorithm for the EDS in general graphs [Fujito,

Nagamochi, 02]• An exact primal-dual algorithm for the prize-collecting EDS in trees

[Kamiyama, 10]

Edge- & node-weighted graph

• O(log n)-approximation algorithm for the prize-collecting EDS ingeneral graphs• An exact primal-dual algorithm for the prize-collecting EDS in trees

Main contributionA natural LP relaxation for node-weighted graphs has a large integralitygap in many covering problems Ü We present a new LP relaxation

Natural LP relaxation

• x(e) ∈ 0, 1: x(e) = 1 Ü e is chosenx(e) = 0 Ü e is NOT chosen

• x(v) ∈ 0, 1: x(v) = 1 Ü an edge incident to v is chosenx(v) = 0 Ü NO edge incident to v is chosen

Natural LP

min∑

e∈E w(e)x(e) +∑

v∈V w(v)x(v)

s.t.∑

e′∈δ(e) x(e′) ≥ 1, ∀e ∈ E

x(v) ≥ x(e), ∀v ∈ V , e ∈ δ(v)

x ≥ 0

A bad example for the natural LP

u1 u2 uk

w(v) = M

w(ui) = 0

w(e) = 0

Optimal solutionchooing a single edge Ü weight = M

LP solutionx(e) = 1/k for ∀e ∈ Ex(v) = 1/k for ∀v ∈ V

Ü weight = M/k

Gap = k

x(v) ≥ x(e),∀v ∈ V , e ∈ δ(v)

x(v) ≥∑

e∈δ(v) x(e),∀v ∈ V

u1 u2 uk

w(v) = M

w(ui) = 0

w(e) = 0

Ü weight = M/k

Gap = k

x(v) ≥ x(e),∀v ∈ V , e ∈ δ(v)

x(v) ≥∑

e∈δ(v) x(e),∀v ∈ V

u1 u2 uk

w(v) = M

w(ui) = 0

w(e) = 0

Ü weight = M/k

Gap = k

x(v) ≥ x(e),∀v ∈ V , e ∈ δ(v)

x(v) ≥∑

e∈δ(v) x(e),∀v ∈ V

u1 u2 uk

w(v) = M

w(ui) = 0

w(e) = 0

Ü weight = M/k

Gap = k

x(v) ≥ x(e),∀v ∈ V , e ∈ δ(v)

x(v) ≥∑

e∈δ(v) x(e), ∀v ∈ V

u1 u2 uk

w(v) = M

w(ui) = 0

w(e) = 0

Ü weight = M/k

Gap = k

x(v) ≥ x(e),∀v ∈ V , e ∈ δ(v)

x(v) ≥∑

e∈δ(v) x(e), ∀v ∈ V

u1 u2 uk

w(v) = M

w(ui) = 0

w(e) = 0

Ü weight = M/k

Gap = k

x(v) ≥ x(e),∀v ∈ V , e ∈ δ(v)

x(v) ≥∑

e∈δ(v) x(e), ∀v ∈ V

u1 u2 uk

w(v) = M

w(ui) = 0

w(e) = 0

Ü weight = M/k

Gap = k

x(v) ≥ x(e),∀v ∈ V , e ∈ δ(v)

x(v) ≥∑

e∈δ(v) x(e), ∀v ∈ V

New LP

· · ·

y(e, e′) ∈ 0, 1 : y(e, e′) = 1⇔ e is chosen for covering e′

Constraints

• covering costraints∑

e∈E y(e, e′) ≥ 1 ∀e′ ∈ E

• e is chosen for some covering costraints Ü x(e) = 1x(e) ≥ y(e, e′) ∀e, e′ ∈ E

• ∃edges in δ(v) is chosen for some covering costraints Ü x(v) = 1x(v) ≥

∑e∈δ(v) y(e, e′) ∀v ∈ V , e′ ∈ E

New LP

· · ·

y(e, e′) ∈ 0, 1 : y(e, e′) = 1⇔ e is chosen for covering e′

Constraints

• covering costraints∑

e∈E y(e, e′) ≥ 1 ∀e′ ∈ E

• e is chosen for some covering costraints Ü x(e) = 1x(e) ≥ y(e, e′) ∀e, e′ ∈ E

• ∃edges in δ(v) is chosen for some covering costraints Ü x(v) = 1x(v) ≥

∑e∈δ(v) y(e, e′) ∀v ∈ V , e′ ∈ E

New LP

min∑

e∈E w(e)x(e) +∑

v∈V w(v)x(v)

s.t.∑

e∈E y(e, e′) ≥ 1 ∀e′ ∈ E ,

x(e) ≥ y(e, e′) ∀e, e′ ∈ E ,

x(v) ≥∑

e∈δ(v) y(e, e′) ∀v ∈ V , e′ ∈ E ,

x , y ≥ 0

Integrality gap of the new LP

Natural LP• Integrality gap≤ 2.1 for edge-weighted graphs

• Integrality gap = 1 for edge-weighted trees

• Integrality gap = Ω(n) for node-weighted trees

New LP• Integrality gap = O(log n) for node-weighted graphs

• Integrality gap = 1 for node-weighted trees

Dual of the new LPDual of New LP

max∑

e∈E ξ(e)

s.t.∑

e∈E ν(e, e′) ≤ w(e′) ∀e′ ∈ E ,∑

e∈E µ(e, v) ≤ w(v) ∀v ∈ V ,

ξ(e) ≤ µ(e, u) + µ(e, v) + ν(e, e′) ∀e ∈ E , e′ = uv ∈ δ(e),

ξ, ν, µ ≥ 0

ve′e

µ(e, v)

µ(e, u)

ν(e, e′)

Dual of the new LPDual of New LP

max∑

e∈E ξ(e)

s.t.∑

e∈E ν(e, e′) ≤ w(e′) ∀e′ ∈ E ,∑

e∈E µ(e, v) ≤ w(v) ∀v ∈ V ,

ξ(e) ≤ µ(e, u) + µ(e, v) + ν(e, e′) ∀e ∈ E , e′ = uv ∈ δ(e),

ξ, ν, µ ≥ 0

ve′e

µ(e, v)

µ(e, u)

ν(e, e′)

Primal-dual algorithm for trees

Base case: G is a star

i∗ = arg min0≤i≤kw(u) + w(vi) + w(ei)α = w(u) + w(vi∗) + w(ei∗)

Primal solution F := ei∗Dual solution ξ(ei∗) = α, ξ(ei) = 0 for i 6= i∗

w(F) =∑

e ξ(e)

Induction

v1 v2 vk

w ′(u) = 0

w ′(e0) = maxw(e0)− α, 0

w ′(v0) = w(v0)−maxα− w(e0), 0

Algorithm

• α ≥ w(u) + w(e0) & an edge in δ(v0) is chosen in the inducedinstance Ü choose e0

• otherwise Ü choose ei∗

Induction

v1 v2 vk

w ′(u) = 0

w ′(e0) = maxw(e0)− α, 0

w ′(v0) = w(v0)−maxα− w(e0), 0

Algorithm

Induction

v1 v2 vk

w ′(u) = 0

w ′(e0) = maxw(e0)− α, 0

w ′(v0) = w(v0)−maxα− w(e0), 0

Algorithm

Induction

v1 v2 vk

w ′(u) = 0

w ′(e0) = maxw(e0)− α, 0

w ′(v0) = w(v0)−maxα− w(e0), 0

Algorithm

Conclusion

• Set cover hardness for the edge dominating set, edge cover, andT -join problems

• O(log n)-approx for the prize-collecting edge dominating setproblem in general graphs

• exact algorithm for the prize-collecting edge dominating setproblem in trees

• 2-approx for the multicut problem in trees

New LP relaxation for covering problemsin edge- & node-weighted graphs

covering problems in edge- and node-weighted graphs€¦ · covering problems in edge- and...

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