David Coudert Guillaume Duco e Nicolas Nisse · Seminario Matemticas Discretas, DIM 1/20 On the diameter of minimal separators in a graph David Coudert 1;2Guillaume Duco e Nicolas

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


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.


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.


Our focus in this work

Relating structural and metric tree-likeness

(treewidth with treelength)

−→ Algorithmic advantages from both sides.


Complexity

treewidth:

NP-hard;

FPT;

O(1)-approx for planar, bounded-genus graphs.

treelength:

NP-hard;

not FPT;

O(1)-approx for all graphs.


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.


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)


A unifying approach through tree-decompositions

• Three constraints to satisfy:St Wt = V (G);

∀e = u, v ∈ E(G), there is Wt ⊇ u, v;

All bags containing u ∈ V (G) induce a subtree of TG .


The topological side

treewidth: minimize the size of bags

• Examples:

tw(G) = 1⇐⇒ G is a tree;




• Examples:


cycle Cn: tw(Cn) = 2;

0

1

2

3

4

5

0 51, ,

1 54

1 2 4

2 3 4

, ,

, ,

, ,




• Examples:



complete graph Kn: tw(Kn) = n − 1;




• Examples:



complete graph Kn: tw(Kn) = n − 1;

square grid Gn,n: tw(Gn,n) = n.


The metric side

treelength: minimize the diameter of bags

• Examples:

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


The metric side


• Examples:


cycle Cn: tl(Cn) =˚

n3

ˇ;


The metric side


• Examples:



n3

ˇ;

complete graph Kn: tl(Kn) = 1;


The metric side


• Examples:



n3

ˇ;

complete graph Kn: tl(Kn) = 1;

square grid Gn,n: tl(Gn,n) = n − 1.


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


Problems

• When are treewidth and treelength comparable ?

• Upper-bound or lower-bound on tl(G)/tw(G) ?


Related work

• [Dieng2009] tw(G) < 12 · tl(G)

if G is planar

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

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

• [Wu2011] tl(G) ≤j

ch(G)2

kwith ch(G) the chordality.


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) ≤ 2j

`(G)2

k− 1


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) ≤ 2j

`(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))


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 .


Method

• upper-bounding the diameter of minimal separators

• Why?

tree-decomposition ∼ pairwise // minimal separators [ParraScheffler1997]

−→ diamG (S) ≤ c · |S | =⇒ tl(G) ≤ c · tw(G).


Upper-bounds: using cycle space

• Cycles between nodes in SA S B



• Cycles between nodes in S

• if “sum” of cycles of small length ≤ l =⇒ “sum” of triangles in Gbl2c.

2 2

2 2

2 2



• Cycles between nodes in S

• if “sum” of cycles of small length ≤ l =⇒ “sum” of triangles in Gbl2c.

• “sum of triangles” =⇒ connectivity properties

diamG (S) ≤ (2¨

l2

˝− 1)(|S | − 1).


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) ≤ (2j

`(G)2

k− 1)(tw(G)− 1)


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)





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

tl(G) = Ω(tw(G)/g(G)3/2).


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.


Main open questions

• Find a tree-decomposition with “good” tradeoff treewidth/treelength

• Complexity of graphs admitting a distance-preserving elimination ordering ?


Documents

David Coudert Guillaume Duco e Nicolas Nisse · Seminario Matemticas Discretas, DIM 1/20 On the diameter of minimal separators in a graph David Coudert 1;2Guillaume Duco e Nicolas