Minimal interval completions

1
Minimal interval completions

• Pinar Heggernes, Bergen, Norway
• Karol Suchan, Orléans, France
• Ioan Todinca, Orléans, France
• Yngve Villanger, Bergen, Norway

2
Minimal triangulations

• Input an arbitrary graph G(V,E)
• Output a chordal graph H(V,E F)
• treewidth s.t. ?(H) is minimum
• minimum fill-in s.t. F is minimum
• Both problems are NP-hard.
• Compute a minimal triangulation H.
• F is inclusion-minimal among all possibilities

3
Minimal interval completions

• Input an arbitrary graph G(V,E)
• Output an interval graph H(V,E F)
• pathwidth s.t. ?(H) is minimum
• profile s.t. F is minimum
• Both problems are NP-hard.
• Find a minimal interval completion H.
• F is inclusion-minimal among all possibilities

4
Minimal interval completions versus minimal
triangulations

• MinTriang
• O(nm) algorithms Lex-M by Rose, Tarjan, Lueker
  76
• o(n3) Kratsch, Spinrad 04
• O(n?log n) Heggernes, Telle, Villanger 05
• MinIntComp
• nothing before
• O(n5) in this talk even O(n4), perhaps O(n3 log
  n)

5
Naive approaches

• edge-by-edge removal
• H any int. completion (e.g. a clique)
• while 9e 2 E(H) E(G) s.t. H-e interval H
  H - e
• interval chordal Å AT-free
• minimal triangulation then destroy the ATs
• destroy the ATs then minimal triangulation
• None of them works

6
The vertex incremental approach

• Vv1, v2, , vn.
• Hi a min int comp of GiGv1, , vi.
• use Hi to compute Hi1 by adding an
  inclusion-minimal set of fill edges from vi1 to
  Hi.
• Hi1 is indeed a min int comp of Gi1 (easy).

Hi
vi1
7
The new problem

• G an interval graph G G x
• MinIntComp of G, fill edges incident to x

G
KL
KR

• fill a, c, b

8
Nice clique paths
G
KR
KL
G
KL
KR

• find a nice clique-path of G
• Spared(CP), the set of cliques left to KL or
  right to KR, must be maximal

9
Forced separators

• Theorem. Berry, Heggernes, Villanger 04
• u,v 2 N(x) non adjacent
• S a minimal u,v-separator in G
• S must be filled.
• S is the intersection of two consecutive cliques

G
Sa
10
Blocks associated to S

• Oi Ci NG(Ci)
• the cliques in Oi form a connected subpath in any
  clique path of G
• they partition the set of cliques of G
• Clique path permutation of the Ois.

C1
S
C2
C3
11
Pre-order on blocks

• Can Oj appear between Oi and the S-edge?
• Only if Oi Å S is contained in every clique of
  Oj.
• We write Oi ¹ Oj
• pre-order
• of width at most two

G
S
O1
O2
O3
O1
O2

O3
12
Dilworth decompositions and clique paths

• Theorem. Clique paths of G Dilworth
  decompositions (L,R) of (Blocks(S),¹).
• here L(O1, O2), R(O3)

L
R
G
S
O1
O2
O3
O1
O2

O3
13
Nice Dilworth decompositions

• OL (OR) the leftmost (rightmost) block containing
  a private neighbour of x
• Spared(L,R) outside the OL OR interval

L
R
S
OR
OL
Theorem. Spared(L,R) must be maximal. A nice
decomp. can be computed in O(n3).
14
Computing nice Dilworth decomposition

• topological sorting of the refined pre-order
• greedy decomposition, bottom-up
• equivalent tilt each component of the
  co-comparability graph
• the components do not interlace

pre-order
15
The algorithm MinIntComp

• At each intremental step
• choose S
• compute a nice Dilworth decomposition of
  (Blocks(S),¹).
• fill the blocks strictly between OL and OR
• call MinIntComp on OL then on OR

L
R
S
OR
OL
16
Summary

• Theorem A minimal interval completion of G can
  be computed in O(n5) time.
• vertex-incremental approach.
• optimized Dilworth decompositon of a pre-order of
  width two.
• O(n4) with a faster computation of the pre-order.

17
Further questions

• Pathwidth and profile
• find the local optimum when adding a new vertex
• choose a good ordering of the vertices
• An algorithm computing any minimal interval
  completion?
• Extracting a minimal interval completion from an
  arbitrary one?

18
Thank you!
?
19
Edge-by-edge removal counterexample
20
Minimal triangulations destroying ATs
counterexamples