Steiner-Star-Free Graphs and Equivalence of Steiner Tree Relaxations - - PowerPoint PPT Presentation
Steiner-Star-Free Graphs and Equivalence of Steiner Tree Relaxations Andreas Emil Feldmann 1 Jochen Knemann 1 Neil Olver 2 Laura Sanit 1 1 Combinatorics & Optimization, University of Waterloo 2 VU University & CWI, Amsterdam The
Andreas Emil Feldmann1 Jochen Könemann1 Neil Olver2 Laura Sanità1
1Combinatorics & Optimization, University of Waterloo 2VU University & CWI, Amsterdam
T erminals Steiner vertices
T erminals Steiner vertices
T erminals Steiner vertices
◮ applications:
network design, VLSI
◮ one of Karp’s original
21 NP-hard problems
◮ APX-hard
T erminals Steiner vertices
◮ applications:
network design, VLSI
◮ one of Karp’s original
21 NP-hard problems
◮ APX-hard ◮ (ln(4) + ε)-approximation:
[Byrka et al. 2012] ◮ iterative rounding of
hypergraphic (HYP) LP
◮ solving HYP is strongly NP-hard ◮ runtime bottleneck: PTAS for HYP
T erminals Steiner vertices
◮ applications:
network design, VLSI
◮ one of Karp’s original
21 NP-hard problems
◮ APX-hard ◮ (ln(4) + ε)-approximation:
[Byrka et al. 2012] ◮ iterative rounding of
hypergraphic (HYP) LP
◮ solving HYP is strongly NP-hard ◮ runtime bottleneck: PTAS for HYP
◮ hypergraphic (HYP) LP:
[Warme 1998]
– strongly NP-hard to solve → PTAS necessary + HYP gap ≤ ln(4) ≈ 1.39
[Goemans et al. 2012]
◮ hypergraphic (HYP) LP:
[Warme 1998]
– strongly NP-hard to solve → PTAS necessary + HYP gap ≤ ln(4) ≈ 1.39
[Goemans et al. 2012]
◮ bidirected cut (BCR) LP:
[Edmonds 1967]
+ compact formulation → efficiently solvable – BCR gap ≤ 2
[Folklore]
◮ hypergraphic (HYP) LP:
[Warme 1998]
– strongly NP-hard to solve → PTAS necessary + HYP gap ≤ ln(4) ≈ 1.39
[Goemans et al. 2012]
◮ bidirected cut (BCR) LP:
[Edmonds 1967]
+ compact formulation → efficiently solvable – BCR gap ≤ 2
[Folklore]
→ improve upper bound of BCR
→ loss: β
→ loss: ln(4)
→ loss: β
→ loss: ln(4)
+ efficient algorithm – total loss: β ln(4) but: if β < 2/ ln(4) then BCR gap < 2
◮ always: BCR gap ≥ HYP gap
◮ always: BCR gap ≥ HYP gap ◮ sometimes: BCR gap > HYP gap
HYP opt BCR opt = 12 11
◮ always: BCR gap ≥ HYP gap ◮ sometimes: BCR gap > HYP gap
HYP opt BCR opt = 12 11
◮ sometimes: BCR gap = HYP gap
quasi-bipartite
[Chakrabarty et al. 2010] [Fung et al. 2012] [Goemans et al. 2012]
Theorem
In every Steiner claw-free instance, BCR gap = HYP gap.
Theorem
In every Steiner claw-free instance, BCR gap = HYP gap. HYP opt BCR opt = 12 11
Theorem
In every Steiner claw-free instance, BCR gap = HYP gap.
Theorem
It is NP-hard to decide whether BCR opt = HYP opt (even on instances with only one Steiner star).
equivalent LP
[Goemans, Myung 1993] T erminals Steiner vertices
equivalent LP
[Goemans, Myung 1993]
notation:
◮ E(S): induced edges of S ◮ ymax(S) = maxv∈S yv
T erminals Steiner vertices
min
ze cost(e) s.t.
ze ≤
yv − ymax(S) ∀S ⊆ V (no cycles)
ze =
yv − 1 (connectedness) yt = 1 ∀t ∈ R (terminals in tree) yv, ze ≥ 0 ∀v ∈ V, e ∈ E
based on full components:
T erminals Steiner vertices
based on full components: notation:
◮ R(C): terminals in C ◮ (a)+ = max{0, a}
T erminals Steiner vertices
min
xC cost(C) s.t.
xC(|R(C) ∩ S| − 1)+ ≤ |S| − 1 ∀S ⊆ R (no cycles)
xC(|R(C)| − 1)+ = |R| − 1 (connectedness) xC ≥ 0 ∀C ∈ K
BCR
− →
HYP
BCR
− →
HYP
Constant cost:
ze cost(e) = xC cost(C)
Bottleneck: tight set S
ze =
yv − ymax(S) (no cycles)
◮ E(S): induced edges of S ◮ ymax(S) = maxv∈S yv
Bottleneck: tight set S
Lemma
An iteration succeeds if for every tight set S intersecting C,
Lemma
An iteration succeeds if for every tight set S intersecting C,
Identifying a component:
tight sets are connected
Lemma
An iteration succeeds if for every tight set S intersecting C,
Identifying a component:
tight sets are connected
s.t. all tight sets are connected
Lemma
An iteration succeeds if for every tight set S intersecting C,
If the iteration fails, ∃ demanding set: tight set intersecting C s.t. maximizer not in C.
A demanding set S has
◮ maximizers /
∈ C
◮ connected Steiners ⊆ C
Lemma
Every tight set is internally connected. A demanding set S has
◮ maximizers /
∈ C
◮ connected Steiners ⊆ C ◮ blocked edge ab:
a /
∈ C, b ∈ C
Lemma
Every tight set is internally connected. A demanding set S has
◮ maximizers /
∈ C
◮ connected Steiners ⊆ C ◮ blocked edge ab:
a /
∈ C, b ∈ C
◮ blocking set S′:
a ∈ S′, b /
∈ S′,
d ∈ S′ ∩ V(C)
Lemma
Demanding and blocking sets do not intersect in terminals. A demanding set S has
◮ maximizers /
∈ C
◮ connected Steiners ⊆ C ◮ blocked edge ab:
a /
∈ C, b ∈ C
a is Steiner
◮ blocking set S′:
a ∈ S′, b /
∈ S′,
d ∈ S′ ∩ V(C)
Lemma
Demanding and blocking sets do not intersect in terminals. Identifying a component:
all tight sets are connected
Steiners, s.t. all tight sets are connected
Lemma
Demanding and blocking sets do not intersect in terminals. A demanding set S has
◮ maximizers /
∈ C
◮ connected Steiners ⊆ C ◮ blocked edge ab:
a /
∈ C, b ∈ C
a is Steiner
◮ blocking set S′:
a ∈ S′, b /
∈ S′,
d ∈ S′ ∩ V(C) Steiner
Lemma
b is connected to a maximizer of S in S \ S′. A demanding set S has
◮ maximizers /
∈ C
◮ connected Steiners ⊆ C ◮ blocked edges a1b1, a2b2:
ai /
∈ C, bi ∈ C
a1 is Steiner
◮ blocking set S′:
a1 ∈ S′, b1 /
∈ S′,
d ∈ S′ ∩ V(C) Steiner
Lemma
b is connected to a maximizer of S in S \ S′. A demanding set S has
◮ maximizers /
∈ C
◮ connected Steiners ⊆ C ◮ blocked edges a1b1, a2b2:
ai /
∈ C, bi ∈ C
a1 = a2 are Steiners
◮ blocking sets S′, S′′:
a1 ∈ S′, b1 /
∈ S′,
a2 ∈ S′′, b2 /
∈ S′′,
d ∈ S′ ∩ V(C) Steiner
Lemma
b is connected to a maximizer of S in S \ S′. A demanding set S has
◮ maximizers /
∈ C
◮ connected Steiners ⊆ C ◮ blocked edges a1b1, a2b2:
ai /
∈ C, bi ∈ C
a1 = a2 are Steiners
◮ blocking sets S′, S′′:
a1 ∈ S′, b1 /
∈ S′,
a2 ∈ S′′, b2 /
∈ S′′,
d ∈ S′ ∩ V(C) Steiner
Theorem
In every Steiner claw-free instance, BCR gap = HYP gap.
Conjecture
If the following minor does not exist, then BCR gap = HYP gap.
Conjecture
If the following minor does not exist, then BCR gap = HYP gap.
Conjecture
If the following minor does not exist, then BCR gap = HYP gap. HYP opt BCR opt = 16 15