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
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.
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.
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.
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.
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.
Seminario Matemticas Discretas, DIM 7/20 A unifying approach through tree-decompositions • Tree-decomposition ⇐ ⇒ T = ( T G , W ) s.t. T G is a tree ∀ t ∈ V ( T G ) , W t ⊆ V ( G ) ( W t is called a bag)
Seminario Matemticas Discretas, DIM 7/20 A unifying approach through tree-decompositions • Three constraints to satisfy: S t W t = V ( G ); ∀ e = { u , v } ∈ E ( G ), there is W t ⊇ { u , v } ; All bags containing u ∈ V ( G ) induce a subtree of T G .
Seminario Matemticas Discretas, DIM 8/20 The topological side treewidth: minimize the size of bags • Examples: tw ( G ) = 1 ⇐ ⇒ G is a tree;
Seminario Matemticas Discretas, DIM 8/20 The topological side treewidth: minimize the size of bags • Examples: tw ( G ) = 1 ⇐ ⇒ G is a tree; cycle C n : tw ( C n ) = 2; 0 , 1 , 5 0 1 , 4 , 5 5 1 4 2 1 2 4 , , 3 2 3 4 , ,
Seminario Matemticas Discretas, DIM 8/20 The topological side treewidth: minimize the size of bags • Examples: tw ( G ) = 1 ⇐ ⇒ G is a tree; cycle C n : tw ( C n ) = 2; complete graph K n : tw ( K n ) = n − 1;
Seminario Matemticas Discretas, DIM 8/20 The topological side treewidth: minimize the size of bags • Examples: tw ( G ) = 1 ⇐ ⇒ G is a tree; cycle C n : tw ( C n ) = 2; complete graph K n : tw ( K n ) = n − 1; square grid G n , n : tw ( G n , n ) = n .
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);
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); ˚ n ˇ cycle C n : tl ( C n ) = ; 3
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); ˚ n ˇ cycle C n : tl ( C n ) = ; 3 complete graph K n : tl ( K n ) = 1;
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); ˚ n ˇ cycle C n : tl ( C n ) = ; 3 complete graph K n : tl ( K n ) = 1; square grid G n , n : tl ( G n , n ) = n − 1.
Seminario Matemticas Discretas, DIM 10/20 Observations • tl ( C n ) / tw ( C n ) → ∞ ; • tw ( K n ) / tl ( K n ) → ∞ ; • tw ( G n , n ) ≈ tl ( G n , n ); − → no relations in general − → need to introduce additional properties/parameters
Seminario Matemticas Discretas, DIM 11/20 Problems • When are treewidth and treelength comparable ? • Upper-bound or lower-bound on tl ( G ) / tw ( G ) ?
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 j k ch ( G ) • [Wu2011] tl ( G ) ≤ 2 with ch ( G ) the chordality.
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) j k ℓ ( G ) tl ( G ) / tw ( G ) ≤ 2 − 1 2
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) j k ℓ ( G ) tl ( G ) / tw ( G ) ≤ 2 − 1 2 Theorem Apex-minor free graphs G = ⇒ tl ( G ) = Ω( tw ( G )) (comprise planar, bounded-genus graphs) p tl ( G ) / tw ( G ) ≥ Ω(1 / g ( G ) · g ( G ))
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 .
Seminario Matemticas Discretas, DIM 14/20 Method • upper-bounding the diameter of minimal separators • Why? tree-decomposition ∼ pairwise // minimal separators [ParraScheffler1997] − → diam G ( S ) ≤ c · | S | = ⇒ tl ( G ) ≤ c · tw ( G ).
Seminario Matemticas Discretas, DIM 15/20 Upper-bounds: using cycle space • Cycles between nodes in S A S B
Seminario Matemticas Discretas, DIM 15/20 Upper-bounds: using cycle space • Cycles between nodes in S ⇒ “sum” of triangles in G ⌊ l 2 ⌋ . • if “sum” of cycles of small length ≤ l = 2 2 2 2 2 2
Seminario Matemticas Discretas, DIM 15/20 Upper-bounds: using cycle space • Cycles between nodes in S ⇒ “sum” of triangles in G ⌊ l 2 ⌋ . • if “sum” of cycles of small length ≤ l = • “sum of triangles” = ⇒ connectivity properties ¨ l ˝ diam G ( S ) ≤ (2 − 1)( | S | − 1). 2
Seminario Matemticas Discretas, DIM 16/20 Applications • Graphs with distance-preserving ordering: cycle base with C 3 , C 4 tl ( G ) ≤ 2( tw ( G ) − 1) (cop-win graphs, weakly modular graphs, etc . . . ) • General graphs: isometric cycles j k ℓ ( G ) tl ( G ) ≤ (2 − 1)( tw ( G ) − 1) 2
Seminario Matemticas Discretas, DIM 17/20 Lower-bounds: using surface embedding • graph genus ∼ number of holes in the surface (to avoid crossings)
Seminario Matemticas Discretas, DIM 17/20 Lower-bounds: using surface embedding • graph genus ∼ number of holes in the surface (to avoid crossings)
Seminario Matemticas Discretas, DIM 17/20 Lower-bounds: using surface embedding • graph genus ∼ number of holes in the surface (to avoid crossings)
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
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)
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 ).
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.
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 ?
Seminario Matemticas Discretas, DIM 20/20
Recommend
More recommend
