Tree Spanners for Bipartite Graphs and Probe Interval Graphs Andreas Brandstädt 1, Feodor Dragan 2, Oanh Le 1, Van Bang Le 1, and Ryuhei Uehara 3 1 Universität Rostock 2 Kent State University 3 Komazawa University

Tree Spanners for Bipartite Graphs and Probe Interval Graphs Andreas Brandstädt 1, Feodor Dragan 2, Oanh Le 1, Van Bang Le 1, and Ryuhei Uehara 3 1 Universität Rostock 2 Kent State University 3 Komazawa University

Tree Spanner Spanning tree T is a tree t-spanner iff d T (x,y) ≦ t d G (x,y) for all x and y in V. G T x y x y

Tree Spanner Spanning tree T is a tree t-spanner iff G T d T (x,y) ≦ t d G (x,y) for all {x,y} in E.

Tree Spanner Spanning tree T is a tree 6-spanner. G T

Tree Spanner G admits a tree 4-spanner (which is optimal). Tree t-spanner problem asks if G admits a tree t-spanner for given t. G T

Applications in distributed systems and communication networks synchronizers in parallel systems topology for message routing there is a very good algorithm for routing in trees in biology evolutionary tree reconstruction in approximation algorithms approximating the bandwidth of graphs Any problem related to distances can be solved approximately on a complex graph if it admits a good tree spanner G 7-spanner for G

Known Results for tree t -spanner general graphs [Cai&Corneil’95] a linear time algorithm for t =2 (t=1 is trivial) tree t -spanner is NP-complete for any t ≧ 4 ( ⇒ NP-completeness of bipartite graphs for t ≧ 5) tree t -spanner is Open for t=3

Known Results for tree t -spanner chordal graphs [Brandst ä dt, Dragan, Le & Le ’02] tree t -spanner is NP-complete for any t ≧ 4 tree 3-spanner admissible graphs [a Number of Authors] cographs, complements of bipartite graphs, interval graphs, directed path graphs, split graphs, permutation graphs, convex bipartite graphs, regular bipartite graphs, distance-hereditary graphs tree 4-spanner admissible graphs AT-free graphs [PKLMW’99], strongly chordal graphs, dually chordal graphs [BCD’99] tree 3 -spanner is in P for planar graphs [FK’2001]

Known Results for tree t -spanner chordal graphs [Brandst ä dt, Dragan, Le & Le ’02] tree t -spanner is NP-complete for any t ≧ 4 tree 3-spanner admissible graphs [a Number of Authors] cographs, complements of bipartite graphs, interval graphs, directed path graphs, split graphs, permutation graphs, convex bipartite graphs, regular bipartite graphs, distance-hereditary graphs tree 4-spanner admissible graphs AT-free graphs [PKLMW’99], strongly chordal graphs, dually chordal graphs [BCD’99] tree 3 -spanner is in P for planar graphs [FK’2001] ⇒ Bipartite Graphs??

Known Results for tree t -spanner bipartite graphs [Cai&Corneil ’95] tree t -spanner is NP-complete for any t ≧ 5 chordal graphs [Brandst ä dt, Dragan, Le & Le ’02] tree t -spanner is NP-complete for any t ≧ 4 tree 3-spanner admissible graphs [a Number of Authors] cographs, complements of bipartite graphs, interval graphs, directed path graphs, split graphs, permutation graphs, convex bipartite graphs, regular bipartite graphs, distance-hereditary graphs convex bipartite ⊂ interval bigraphs ⊂ bipartite ATE-free graphs ⊂ chordal bipartite graphs ⊂ bipartite graphs

This Talk interval rooted directed path strongly chordal weakly chordal bipartite interval bigraph convex AT-free bipartite ATE-free bipartite NP-C 4-Adm. 3-Adm.

This Talk interval rooted directed path strongly chordal weakly chordal enhanced probe interval chordal bipartite probe interval bigraph convex STS-probe interval AT-free bipartite ATE-free bipartite NP-C 4-Adm. 3-Adm. =

This Talk interval rooted directed path strongly chordal weakly chordal enhanced probe interval chordal bipartite probe interval bigraph convex STS-probe interval AT-free bipartite ATE-free bipartite NP-C 4-Adm. 3-Adm. = 7-Adm.

NP-hardness for chordal bipartite graphs [Thm] For any t ≧ 5, the tree t-spanner problem is NP-complete for chordal bipartite graphs. Reduction from 3SAT Monotone … (x, y, z) or (x, y, z)

NP-hardness for chordal bipartite graphs Reduction from 3SAT Basic gadgets Monotone … (x, y, z) or (x, y,z) S 1 [a,b]S 2 [a,b]S 3 [a,b] a a’ b b’ ab a’b’ S 1 [a,a’] S 1 [a’,b’] S 1 [b,b’] S 2 [a,a’] S 2 [a’,b’] S 2 [b,b’] ab a’b’

NP-hardness for chordal bipartite graphs Reduction from 3SAT Basic gadget S k [a,b] and its spanning trees Monotone … (x, y, z) or (x, y,z) a a’ b b’ a a’ b b’ a a’ b b’ H with {a,b} (2k+1)-spanner without {a,b} h (2k+h)-spanner a a’ b b’ without {a,b} (2k-1)-spanner

NP-hardness for chordal bipartite graphs Reduction from 3SAT Gadget for x i Monotone … (x, y, z) or (x, y,z) qr sp xixi xixi xixi xixi xixi xixi 1 2 m 1 2 m … … S k-1 [] S k []× ２ ＝ ＝ Must be selected

NP-hardness for chordal bipartite graphs Reduction from 3SAT Gadget for C j Monotone … (x, y, z) or (x, y,z) cjcj cjcj djdj djdj S k []× ２＝

NP-hardness for chordal bipartite graphs Reduction from 3SAT Gadget for C 1 =(x 1,x 2,x 3 ) and C 2 =(x 1,x 2,x 4 ) Monotone … (x, y, z) or (x, y,z) qr sp x1x1 x1x1 x1x1 x1x S k-2 [] ＝ x2x2 x2x2 x2x2 x2x x3x3 x3x3 x3x3 x3x x4x4 x4x4 x4x4 x4x c1c1 c1c1 d1d1 d1d c2c2 c2c2 d2d2 d2d

Tree 3-spanner for a bipartite ATE-free graph An ATE(Asteroidal-Triple-Edge) e 1,e 2,e 3 [Mul97]: Any two of them there is a path from one to the other avoids the neighborhood of the third one. [Lamma] interval bigraphs ⊂ bipartite ATE-free graphs ⊂ chordal bipartite graphs. e1e1 e3e3 e2e2

Tree 3-spanner for a bipartite ATE-free graph A maximum neighbor w of u: N(N(u))=N(w) [Lamma] Any chordal bipartite graph has a vertex with a maximum neighbor. u w chordal bipartite graph ⇔ bipartite graph any cycle of length at least 6 has a chord

Tree 3-spanner for a bipartite ATE-free graph G; connected bipartite ATE-free graph u; a vertex with maximum neighbor For any connected component S induced by V ＼ D k-1 (u), there is w in N k-1 (u) s.t. N(w) ⊇ S∩N k (u) S u … w

Tree 3-spanner for a bipartite ATE-free graph Construction of a tree 3-spanner of G: u; a vertex with maximum neighbor u … w

Conclusion and open problems Many questions remain still open. Among them: Can Tree 3–Spanner be decided efficiently on general graphs??? on chordal graphs? on chordal bipartite graphs? Tree t – Spanner on (enhanced) probe interval graphs for t<7? Thank you!