On the diameter of minimal separators in a graph David Coudert 1 , 2 - - PowerPoint PPT Presentation

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On the diameter of minimal separators in a graph David Coudert 1 , 2 - - PowerPoint PPT Presentation

Mar 04, 2024 461 likes •845 views

Seminario Matemticas Discretas, DIM 1/20 On the diameter of minimal separators in a graph David Coudert 1 , 2 Guillaume Ducoffe 1 , 2 Nicolas Nisse 1 , 2 1Inria, France 2Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, 06900 Sophia Antipolis,

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Seminario Matemticas Discretas, DIM 1/20

On the diameter of minimal separators in a graph

David Coudert 1,2 Guillaume Ducoffe 1,2 Nicolas Nisse 1,2

1Inria, France

  • 2Univ. Nice Sophia Antipolis, CNRS, I3S, UMR 7271, 06900 Sophia Antipolis, France
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Seminario Matemticas Discretas, DIM 2/20

Solving hard problems on graphs

  • Motivation: many problems are “easy” to solve on trees
  • A classical graph parameter for “tree-likeness”:

treewidth = how close is the structure of the graph from a tree ? − → Several NP-hard problems solvable in polynomial-time on bounded-treewidth graphs.

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Seminario Matemticas Discretas, DIM 3/20

Treewidth in real-life networks

  • Problem: AS graphs of the Internet have large treewidth [MSV2011]
  • A complementary approach:

treelength = how close is the metric of the graph from a tree ? − → Introduced for: routing and distance schemes, comparison of phylogenetic networks, design of approximation algorithms.

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Seminario Matemticas Discretas, DIM 4/20

Our focus in this work

Relating structural and metric tree-likeness (treewidth with treelength)

− → Algorithmic advantages from both sides.

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Seminario Matemticas Discretas, DIM 5/20

Complexity

treewidth: NP-hard; FPT; O(1)-approx for planar, bounded-genus graphs. treelength: NP-hard; not FPT; O(1)-approx for all graphs.

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Seminario Matemticas Discretas, DIM 6/20

Overview of our contributions

  • characterization of graph classes s.t. treelength = θ(treewidth).

(including cop-win, bounded-genus graphs)

  • general bounds on the gap between treewidth and treelength.
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Seminario Matemticas Discretas, DIM 7/20

A unifying approach through tree-decompositions

  • Tree-decomposition ⇐

⇒ T = (TG, W ) s.t. TG is a tree ∀t ∈ V (TG), Wt ⊆ V (G) (Wt is called a bag)

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Seminario Matemticas Discretas, DIM 7/20

A unifying approach through tree-decompositions

  • Three constraints to satisfy:

S

t Wt = V (G);

∀e = {u, v} ∈ E(G), there is Wt ⊇ {u, v}; All bags containing u ∈ V (G) induce a subtree of TG.

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Seminario Matemticas Discretas, DIM 8/20

The topological side

treewidth: minimize the size of bags

  • Examples:

tw(G) = 1 ⇐ ⇒ G is a tree;

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Seminario Matemticas Discretas, DIM 8/20

The topological side

treewidth: minimize the size of bags

  • Examples:

tw(G) = 1 ⇐ ⇒ G is a tree; cycle Cn: tw(Cn) = 2;

1 2 3 4 5 5 1 , , 1 5 4 1 2 4 2 3 4 , , , , , ,

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Seminario Matemticas Discretas, DIM 8/20

The topological side

treewidth: minimize the size of bags

  • Examples:

tw(G) = 1 ⇐ ⇒ G is a tree; cycle Cn: tw(Cn) = 2; complete graph Kn: tw(Kn) = n − 1;

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Seminario Matemticas Discretas, DIM 8/20

The topological side

treewidth: minimize the size of bags

  • Examples:

tw(G) = 1 ⇐ ⇒ G is a tree; cycle Cn: tw(Cn) = 2; complete graph Kn: tw(Kn) = n − 1; square grid Gn,n: tw(Gn,n) = n.

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Seminario Matemticas Discretas, DIM 9/20

The metric side

treelength: minimize the diameter of bags

  • Examples:

tl(G) = 1 ⇐ ⇒ G is chordal (superclass of trees);

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Seminario Matemticas Discretas, DIM 9/20

The metric side

treelength: minimize the diameter of bags

  • Examples:

tl(G) = 1 ⇐ ⇒ G is chordal (superclass of trees); cycle Cn: tl(Cn) = ˚ n

3

ˇ ;

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Seminario Matemticas Discretas, DIM 9/20

The metric side

treelength: minimize the diameter of bags

  • Examples:

tl(G) = 1 ⇐ ⇒ G is chordal (superclass of trees); cycle Cn: tl(Cn) = ˚ n

3

ˇ ; complete graph Kn: tl(Kn) = 1;

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Seminario Matemticas Discretas, DIM 9/20

The metric side

treelength: minimize the diameter of bags

  • Examples:

tl(G) = 1 ⇐ ⇒ G is chordal (superclass of trees); cycle Cn: tl(Cn) = ˚ n

3

ˇ ; complete graph Kn: tl(Kn) = 1; square grid Gn,n: tl(Gn,n) = n − 1.

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Seminario Matemticas Discretas, DIM 10/20

Observations

  • tl(Cn)/tw(Cn) → ∞;
  • tw(Kn)/tl(Kn) → ∞;
  • tw(Gn,n) ≈ tl(Gn,n);

− → no relations in general − → need to introduce additional properties/parameters

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Seminario Matemticas Discretas, DIM 11/20

Problems

  • When are treewidth and treelength comparable ?
  • Upper-bound or lower-bound on tl(G)/tw(G) ?
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Seminario Matemticas Discretas, DIM 12/20

Related work

  • [Dieng2009] tw(G) < 12 · tl(G)

if G is planar

  • [Diestel2014] tl(G) ≤ ℓ(G) · (tw(G) − 1)

with ℓ(G) the length of a longest isometric cycle

  • [Wu2011] tl(G) ≤

j

ch(G) 2

k with ch(G) the chordality.

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Seminario Matemticas Discretas, DIM 13/20

Our contributions

Theorem Graphs G with bounded-length cycle base = ⇒ tl(G) = O(tw(G)) (comprise graphs with a distance-preserving elimination ordering) tl(G)/tw(G) ≤ 2 j

ℓ(G) 2

k − 1

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Seminario Matemticas Discretas, DIM 13/20

Our contributions

Theorem Graphs G with bounded-length cycle base = ⇒ tl(G) = O(tw(G)) (comprise graphs with a distance-preserving elimination ordering) tl(G)/tw(G) ≤ 2 j

ℓ(G) 2

k − 1 Theorem Apex-minor free graphs G = ⇒ tl(G) = Ω(tw(G)) (comprise planar, bounded-genus graphs) tl(G)/tw(G) ≥ Ω(1/g(G) · p g(G))

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Seminario Matemticas Discretas, DIM 14/20

Method

  • upper-bounding the diameter of minimal separators

S a separator ⇐ ⇒ G \ S disconnected. S a minimal separator ⇐ ⇒ ∃ A, B c.c. of G \ S s.t. N(A) = N(B) = S.

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Seminario Matemticas Discretas, DIM 14/20

Method

  • upper-bounding the diameter of minimal separators
  • Why?

tree-decomposition ∼ pairwise // minimal separators [ParraScheffler1997] − → diamG(S) ≤ c · |S| = ⇒ tl(G) ≤ c · tw(G).

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Seminario Matemticas Discretas, DIM 15/20

Upper-bounds: using cycle space

  • Cycles between nodes in S

A S B

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Seminario Matemticas Discretas, DIM 15/20

Upper-bounds: using cycle space

  • Cycles between nodes in S
  • if “sum” of cycles of small length ≤ l =

⇒ “sum” of triangles in G⌊ l

2⌋. 2 2 2 2 2 2

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Seminario Matemticas Discretas, DIM 15/20

Upper-bounds: using cycle space

  • Cycles between nodes in S
  • if “sum” of cycles of small length ≤ l =

⇒ “sum” of triangles in G⌊ l

2⌋.

  • “sum of triangles” =

⇒ connectivity properties diamG(S) ≤ (2 ¨ l

2

˝ − 1)(|S| − 1).

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Seminario Matemticas Discretas, DIM 16/20

Applications

  • Graphs with distance-preserving ordering: cycle base with C3, C4

tl(G) ≤ 2(tw(G) − 1) (cop-win graphs, weakly modular graphs, etc . . . )

  • General graphs: isometric cycles

tl(G) ≤ (2 j

ℓ(G) 2

k − 1)(tw(G) − 1)

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Seminario Matemticas Discretas, DIM 17/20

Lower-bounds: using surface embedding

  • graph genus ∼ number of holes in the surface (to avoid crossings)
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Seminario Matemticas Discretas, DIM 17/20

Lower-bounds: using surface embedding

  • graph genus ∼ number of holes in the surface (to avoid crossings)
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Seminario Matemticas Discretas, DIM 17/20

Lower-bounds: using surface embedding

  • graph genus ∼ number of holes in the surface (to avoid crossings)
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Seminario Matemticas Discretas, DIM 17/20

Lower-bounds: using surface embedding

  • graph genus ∼ number of holes in the surface (to avoid crossings)
  • bounded genus + large treewidth =

⇒ contractible to large “grid-like” graph

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Seminario Matemticas Discretas, DIM 17/20

Lower-bounds: using surface embedding

  • graph genus ∼ number of holes in the surface (to avoid crossings)
  • bounded genus + large treewidth =

⇒ contractible to large “grid-like” graph

  • Grid-like graphs have large treelength (like grids)
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Seminario Matemticas Discretas, DIM 17/20

Lower-bounds: using surface embedding

  • graph genus ∼ number of holes in the surface (to avoid crossings)
  • bounded genus + large treewidth =

⇒ contractible to large “grid-like” graph

  • Grid-like graphs have large treelength (like grids)

tl(G) = Ω(tw(G)/g(G)3/2).

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Seminario Matemticas Discretas, DIM 18/20

Conclusion

  • A general bridge between structural and metric graph invariants.
  • New bounds and approximation algorithms for treewidth
  • New algorithms for bounded-treewidth and bounded-treelength graphs.
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Seminario Matemticas Discretas, DIM 19/20

Main open questions

  • Find a tree-decomposition with “good” tradeoff treewidth/treelength
  • Complexity of graphs admitting a distance-preserving elimination ordering ?
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Seminario Matemticas Discretas, DIM 20/20