Stable Sets in Graphs with Bounded Odd Cycle Packing Number Tony - - PowerPoint PPT Presentation

▶
stable sets in graphs with bounded odd cycle packing
SMART_READER_LITE
LIVE PREVIEW

Stable Sets in Graphs with Bounded Odd Cycle Packing Number Tony - - PowerPoint PPT Presentation

Aug 16, 2023 24 likes •802 views

Stable Sets in Graphs with Bounded Odd Cycle Packing Number Tony Huynh (Monash) joint with Michele Conforti, Samuel Fiorini, Gwena el Joret, and Stefan Weltge Maximum Weight Stable Set Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle

slide-1
SLIDE 1

Stable Sets in Graphs with Bounded Odd Cycle Packing Number

Tony Huynh (Monash) joint with Michele Conforti, Samuel Fiorini, Gwena¨ el Joret, and Stefan Weltge

slide-2
SLIDE 2

Maximum Weight Stable Set

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 2 / 32

slide-3
SLIDE 3

Maximum Weight Stable Set

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 2 / 32

slide-4
SLIDE 4

Maximum Weight Stable Set

Problem Given a graph G and w : V (G) → R0, compute a maximum weight stable set (MWSS) of G.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 2 / 32

slide-5
SLIDE 5

Maximum Weight Stable Set

Problem Given a graph G and w : V (G) → R0, compute a maximum weight stable set (MWSS) of G. Theorem For every ǫ > 0, it is NP-hard to approximate maximum stable set within a factor of n1−ǫ.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 2 / 32

slide-6
SLIDE 6

Bipartite Graphs

Theorem MWSS can be solved on bipartite graphs in polynomial time.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 3 / 32

slide-7
SLIDE 7

Bipartite Graphs

Theorem MWSS can be solved on bipartite graphs in polynomial time. max

  • v∈V (G)

w(v)xv s.t. xu + xv 1 ∀uv ∈ E(G) x ∈ {0, 1}V (G) ≡ max

  • v∈V (G)

w(v)xv s.t. Mx 1 x ∈ {0, 1}V (G)

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 3 / 32

slide-8
SLIDE 8

Bipartite Graphs

Theorem MWSS can be solved on bipartite graphs in polynomial time. max

  • v∈V (G)

w(v)xv s.t. xu + xv 1 ∀uv ∈ E(G) x ∈ [0, 1]V (G) ≡ max

  • v∈V (G)

w(v)xv s.t. Mx 1 x ∈ [0, 1]V (G)

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 3 / 32

slide-9
SLIDE 9

Bipartite Graphs

Theorem MWSS can be solved on bipartite graphs in polynomial time. max

  • v∈V (G)

w(v)xv s.t. xu + xv 1 ∀uv ∈ E(G) x ∈ [0, 1]V (G) ≡ max

  • v∈V (G)

w(v)xv s.t. Mx 1 x ∈ [0, 1]V (G) If G is bipartite, then M is a totally unimodular matrix.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 3 / 32

slide-10
SLIDE 10

Integer Programming

Conjecture Fix k ∈ N. Integer Linear Programming can be solved in strongly polynomial time when all subdeterminants of the constraint matrix are in {−k, . . . , k}.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 4 / 32

slide-11
SLIDE 11

Integer Programming

Conjecture Fix k ∈ N. Integer Linear Programming can be solved in strongly polynomial time when all subdeterminants of the constraint matrix are in {−k, . . . , k}. Theorem (Artmann, Weismantel, Zenklusen ’17) True for k = 2. Bimodular Integer Programming can be solved in strongly polynomial time.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 4 / 32

slide-12
SLIDE 12

Integer Programming

Conjecture Fix k ∈ N. Integer Linear Programming can be solved in strongly polynomial time when all subdeterminants of the constraint matrix are in {−k, . . . , k}. Theorem (Artmann, Weismantel, Zenklusen ’17) True for k = 2. Bimodular Integer Programming can be solved in strongly polynomial time. Open for k 3.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 4 / 32

slide-13
SLIDE 13

Odd Cycle Packing Number

M = M(G) edge-vertex incidence matrix of graph G

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 5 / 32

slide-14
SLIDE 14

Odd Cycle Packing Number

M = M(G) edge-vertex incidence matrix of graph G M =                           

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

                          

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 5 / 32

slide-15
SLIDE 15

Odd Cycle Packing Number

Observation max |sub-determinant of M(G)| = 2OCP(G)

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 6 / 32

slide-16
SLIDE 16

Odd Cycle Packing Number

Observation max |sub-determinant of M(G)| = 2OCP(G) Corollary MWSS can be solved in polynomial time in graphs without two vertex-disjoint odd cycles.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 6 / 32

slide-17
SLIDE 17

Odd Cycle Packing Number

Observation max |sub-determinant of M(G)| = 2OCP(G) Corollary MWSS can be solved in polynomial time in graphs without two vertex-disjoint odd cycles. Conjecture Fix k ∈ N. MWSS can be solved in polynomial time in graphs without k vertex-disjoint odd cycles.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 6 / 32

slide-18
SLIDE 18

Polynomial Time Approximation Schemes

Theorem (Bock, Faenza, Moldenhauer, Ruiz-Vargas ’14) For every fixed k ∈ N, MWSS on graphs with OCP(G) k has a PTAS.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 7 / 32

slide-19
SLIDE 19

Polynomial Time Approximation Schemes

Theorem (Bock, Faenza, Moldenhauer, Ruiz-Vargas ’14) For every fixed k ∈ N, MWSS on graphs with OCP(G) k has a PTAS. Theorem (Tazari ’10) For every fixed k ∈ N, MWSS and Minimum Vertex Cover on graphs with OCP(G) k has a PTAS.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 7 / 32

slide-20
SLIDE 20

Extension Complexity

Definition A polytope Q ⊆ Rp is an extension of a polytope P ⊆ Rd if there exists an affine map π : Rp → Rd with π(Q) = P. The extension complexity of P, denoted xc(P), is the minimum number of facets of any extension of P.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 8 / 32

slide-21
SLIDE 21

Extension Complexity

Definition A polytope Q ⊆ Rp is an extension of a polytope P ⊆ Rd if there exists an affine map π : Rp → Rd with π(Q) = P. The extension complexity of P, denoted xc(P), is the minimum number of facets of any extension of P.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 8 / 32

slide-22
SLIDE 22

Spanning Tree Polytope

Theorem (Edmonds ’71) Let G = (V , E) be a graph. Then x ∈ T(G) if and only if

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 9 / 32

slide-23
SLIDE 23

Spanning Tree Polytope

Theorem (Edmonds ’71) Let G = (V , E) be a graph. Then x ∈ T(G) if and only if

  • x 0,

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 9 / 32

slide-24
SLIDE 24

Spanning Tree Polytope

Theorem (Edmonds ’71) Let G = (V , E) be a graph. Then x ∈ T(G) if and only if

  • x 0,
  • x(E) = |V | − 1,

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 9 / 32

slide-25
SLIDE 25

Spanning Tree Polytope

Theorem (Edmonds ’71) Let G = (V , E) be a graph. Then x ∈ T(G) if and only if

  • x 0,
  • x(E) = |V | − 1,
  • x(E[U]) |U| − 1, for all non-empty U ⊆ V .

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 9 / 32

slide-26
SLIDE 26

Spanning Tree Polytope

Theorem (Wong ’80 and Martin ’91) For every connected graph G = (V , E), xc(T(G)) = O(|V | · |E|).

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 10 / 32

slide-27
SLIDE 27

Lower bounds

Theorem (Fiorini, Massar, Pokutta, Tiwary, and de Wolf ’12) There is no extended formulation of TSPn of polynomial size.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 11 / 32

slide-28
SLIDE 28

Lower bounds

Theorem (Fiorini, Massar, Pokutta, Tiwary, and de Wolf ’12) There is no extended formulation of TSPn of polynomial size. Theorem (Rothvoß ’14) The extension complexity of M(Kn) is exponential in n.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 11 / 32

slide-29
SLIDE 29

Surfaces

Classification of Surfaces:

  • orientable ∼

= sphere with h handles = Sh

  • non-orientable

∼ = sphere with c cross-caps = Nc Euler genus:

  • g(Sh) = 2h
  • g(Nc) = c

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 12 / 32

slide-30
SLIDE 30

Our Main Results

Theorem (Conforti, Fiorini, H, Weltge ’19) If OCP(G) 1 then STAB(G) has a size-O(n2) extended formulation.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 13 / 32

slide-31
SLIDE 31

Our Main Results

Theorem (Conforti, Fiorini, H, Weltge ’19) If OCP(G) 1 then STAB(G) has a size-O(n2) extended formulation. Theorem (Conforti, Fiorini, H, Joret, Weltge ’19) Fix k, g ∈ N. Then for every graph G with OCP(G) k and Euler genus g, MWSS can be solved in polynomial time and STAB(G) has a polynomial-size extended formulation.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 13 / 32

slide-32
SLIDE 32

OCP = 1 Graphs

Theorem (Lov´ asz) Let G be a 4-connected graph. Then OCP(G) 1 iff

  • G − X is bipartite for some X ⊆ V (G) with |X| 3
  • G has a nice embedding in the projective plane

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 14 / 32

slide-33
SLIDE 33

Nicely Embedded Graphs

Definition Let G be a graph embedded in a surface S. A cycle of G is 1-sided if it has a neighborhood that is a M¨

  • bius strip, and

2-sided if it has a neighborhood that is a cylinder.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 15 / 32

slide-34
SLIDE 34

Nicely Embedded Graphs

Definition Let G be a graph embedded in a surface S. A cycle of G is 1-sided if it has a neighborhood that is a M¨

  • bius strip, and

2-sided if it has a neighborhood that is a cylinder.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 15 / 32

slide-35
SLIDE 35

Nicely Embedded Graphs

Definition A graph G is nicely embedded in a surface S if every odd cycle in G is 1-sided.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 16 / 32

slide-36
SLIDE 36

Nicely Embedded Graphs

Definition A graph G is nicely embedded in a surface S if every odd cycle in G is 1-sided. Lemma If G is nicely embedded on a surface of Euler genus k, then OCP(G) k.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 16 / 32

slide-37
SLIDE 37

Nicely Embedded Graphs

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 17 / 32

slide-38
SLIDE 38

The Erd˝

  • s-P´
  • sa Theorem

Theorem (Erd˝

  • s and P´
  • sa, ’65)

Every graph has one of the following:

  • k vertex-disjoint cycles;
  • a feedback vertex set of size O(k log k).

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 18 / 32

slide-39
SLIDE 39

Escher Walls

Theorem (Thomassen ’88) The Erd˝

  • s-P´
  • sa Property does not hold for odd cycles.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 19 / 32

slide-40
SLIDE 40

Escher Walls

Theorem (Thomassen ’88) The Erd˝

  • s-P´
  • sa Property does not hold for odd cycles.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 19 / 32

slide-41
SLIDE 41

Escher Walls

Theorem (Thomassen ’88) The Erd˝

  • s-P´
  • sa Property does not hold for odd cycles.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 19 / 32

slide-42
SLIDE 42

Escher Walls

Theorem (Thomassen ’88) The Erd˝

  • s-P´
  • sa Property does not hold for odd cycles.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 19 / 32

slide-43
SLIDE 43

Escher Walls

Theorem (Thomassen ’88) The Erd˝

  • s-P´
  • sa Property does not hold for odd cycles.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 19 / 32

slide-44
SLIDE 44

Escher Walls

Theorem (Thomassen ’88) The Erd˝

  • s-P´
  • sa Property does not hold for odd cycles.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 19 / 32

slide-45
SLIDE 45

An Erd˝

  • s-P´
  • sa Theorem for 2-sided Odd Cycles

Theorem (CFHJW) There exists a computable function f (g, k) such that for all graphs G embedded in a surface with Euler genus g and with no k + 1 node-disjoint 2-sided odd cycles, there exists X ⊆ V (G) with |X| f (g, k) such that G − X does not contain a 2-sided

  • dd cycle. Furthermore, there is such a set X of size at most

19g+1 · k if the surface is orientable.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 20 / 32

slide-46
SLIDE 46

An Erd˝

  • s-P´
  • sa Theorem for 2-sided Odd Cycles

Theorem (CFHJW) There exists a computable function f (g, k) such that for all graphs G embedded in a surface with Euler genus g and with no k + 1 node-disjoint 2-sided odd cycles, there exists X ⊆ V (G) with |X| f (g, k) such that G − X does not contain a 2-sided

  • dd cycle. Furthermore, there is such a set X of size at most

19g+1 · k if the surface is orientable. Theorem (Kawarabayashi and Nakamoto ’07) Odd cycles satisfy the Erd˝

  • s-P´
  • sa property in graphs embedded

in a fixed orientable surface

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 20 / 32

slide-47
SLIDE 47

Dropping Nonnegativity Constraints

Let P(G) = conv{x ∈ ZV (G) | Mx ≤ 1}.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 21 / 32

slide-48
SLIDE 48

Dropping Nonnegativity Constraints

Let P(G) = conv{x ∈ ZV (G) | Mx ≤ 1}. Theorem For every graph G we have STAB(G) = P(G) ∩ [0, 1]V (G).

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 21 / 32

slide-49
SLIDE 49

Slack Space

  • Node space:

xv =    1 if v ∈ S

  • therwise

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 22 / 32

slide-50
SLIDE 50

Slack Space

  • Node space:

xv =    1 if v ∈ S

  • therwise
  • Slack space:

yuv =    1 if u, v / ∈ S

  • therwise

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 22 / 32

slide-51
SLIDE 51

The Dual Graph

Theorem If G is a nicely embedded, then the edges of the dual graph G ∗ can be oriented such in the local cyclic order of the edges incident to each dual node f , the edges alternatively leave and enter f . We call such an orietation alternating and denote it by D.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 23 / 32

slide-52
SLIDE 52

Slack Space

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 24 / 32

slide-53
SLIDE 53

Slack Space

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 24 / 32

slide-54
SLIDE 54

Slack Space

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 24 / 32

slide-55
SLIDE 55

Slack Space

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 24 / 32

slide-56
SLIDE 56

Slack Space

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 24 / 32

slide-57
SLIDE 57

Slack Space

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 24 / 32

slide-58
SLIDE 58

Slack Space

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 24 / 32

slide-59
SLIDE 59

Slack Space

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 24 / 32

slide-60
SLIDE 60

The Slack Map

  • For x ∈ RV (G), let y := 1 − Mx ∈ RE(G)

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 25 / 32

slide-61
SLIDE 61

The Slack Map

  • For x ∈ RV (G), let y := 1 − Mx ∈ RE(G)
  • So yuv = 1 − xu − xv for all edges uv ∈ E(G)

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 25 / 32

slide-62
SLIDE 62

The Slack Map

  • For x ∈ RV (G), let y := 1 − Mx ∈ RE(G)
  • So yuv = 1 − xu − xv for all edges uv ∈ E(G)
  • Slack map σ : RV (G) → RE(G) : x → y = σ(x) := 1 − Mx

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 25 / 32

slide-63
SLIDE 63

The Slack Map

Lemma The image σ(RV (G)) of the slack map is the linear subspace of RE(G) defined by

2k

  • i=1

(−1)i−1yei = 0 ∀even cycles C = (e1, e2, . . . , e2k)

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 26 / 32

slide-64
SLIDE 64

The Slack Map

Lemma The image σ(RV (G)) of the slack map is the linear subspace of RE(G) defined by

2k

  • i=1

(−1)i−1yei = 0 ∀even cycles C = (e1, e2, . . . , e2k)

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 26 / 32

slide-65
SLIDE 65

The Dual Graph

Observation By Euler’s formula, in S ∼ = Ng, |E(G)| − |V (G)| = (|V (G ∗)| − 1) + (g − 1)

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 27 / 32

slide-66
SLIDE 66

The Dual Graph

Observation By Euler’s formula, in S ∼ = Ng, |E(G)| − |V (G)| = (|V (G ∗)| − 1) + (g − 1) Observation If G is nicely embedded in S then, σ(RV (G)) = {circulations in G ∗ subject to g −1 extra constraints}

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 27 / 32

slide-67
SLIDE 67

Homology

Definition Two integer circulations y, y′ in G ∗ are homologous if y − y′ is a sum of facial circulations.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 28 / 32

slide-68
SLIDE 68

Homology

Definition Two integer circulations y, y′ in G ∗ are homologous if y − y′ is a sum of facial circulations.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 28 / 32

slide-69
SLIDE 69

Homology

Definition The first homology group H1(S; Z) is the additive group of all integer circulations in G ∗ quotiented by the zero-homologous circulations.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 29 / 32

slide-70
SLIDE 70

Homology

Definition The first homology group H1(S; Z) is the additive group of all integer circulations in G ∗ quotiented by the zero-homologous circulations. Fact H1(Ng; Z) ∼ = Z2 × Zg−1

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 29 / 32

slide-71
SLIDE 71

Minimum Cost Homology Flow

max

  • v∈V (G)

w(v)xv s.t. Mx 1 x 0 x ∈ ZV (G)

  • min
  • e∈E(G)

c(e)ye s.t. y circulation in G ∗ [y] = (1, 0, . . . , 0) y 0 y ∈ ZE(G)

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 30 / 32

slide-72
SLIDE 72

Minimum Cost Homology Flow

max

  • v∈V (G)

w(v)xv s.t. Mx 1 x 0 x ∈ ZV (G)

  • min
  • e∈E(G)

c(e)ye s.t. y circulation in G ∗ [y] = (1, 0, . . . , 0) y 0 y ∈ ZE(G) where c ∈ RE(G)

+

is such that c(δ(v)) = w(v) for all v ∈ V (G)

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 30 / 32

slide-73
SLIDE 73

Minimum Cost Homology Flow

Theorem (Chambers, Erickson, Nayyeri ’10) Given a graph G embedded on a surface of Euler genus g, a cost function c : E(G) → R, and a circulation θ : E(G) → R, a minimum-cost circulation homologous to θ can be computed in time gO(g)n3/2.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 31 / 32

slide-74
SLIDE 74

Minimum Cost Homology Flow

Theorem (Chambers, Erickson, Nayyeri ’10) Given a graph G embedded on a surface of Euler genus g, a cost function c : E(G) → R, and a circulation θ : E(G) → R, a minimum-cost circulation homologous to θ can be computed in time gO(g)n3/2. Theorem (Malniˇ c and Mohar ’92) Suppose G is embedded in a surface S with Euler genus g 1. If C1, . . . , Cℓ are vertex-disjoint directed cycles in G whose homology classes are mutually distinct, then ℓ 6g.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 31 / 32

slide-75
SLIDE 75

Summary

Theorem (Conforti, Fiorini, H, Weltge ’19) If OCP(G) 1 then STAB(G) has a size-O(n2) extended formulation.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 32 / 32

slide-76
SLIDE 76

Summary

Theorem (Conforti, Fiorini, H, Weltge ’19) If OCP(G) 1 then STAB(G) has a size-O(n2) extended formulation. Theorem (Conforti, Fiorini, H, Joret, Weltge ’19) Fix k, g ∈ N. Then for every graph G with OCP(G) k and Euler genus g, MWSS can be solved in polynomial time and STAB(G) has a polynomial-size extended formulation.

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 32 / 32

slide-77
SLIDE 77

Summary

Theorem (Conforti, Fiorini, H, Weltge ’19) If OCP(G) 1 then STAB(G) has a size-O(n2) extended formulation. Theorem (Conforti, Fiorini, H, Joret, Weltge ’19) Fix k, g ∈ N. Then for every graph G with OCP(G) k and Euler genus g, MWSS can be solved in polynomial time and STAB(G) has a polynomial-size extended formulation.

Thank you!

Tony Huynh Stable Sets in Graphs with Bounded Odd Cycle Packing Number 32 / 32