Tree Spanners for Bipartite Graphs and Probe Interval

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 x x G T y l 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. y

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

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

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

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

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

Known Results for tree t -spanner l l chordal graphs [Brandstädt, Dragan, Le & Le ’ 02] l tree t -spanner is NP-complete for any t ≧ 4 tree 3 -spanner admissible graphs [a Number of Authors] l l tree 4 -spanner admissible graphs l l l cographs, complements of bipartite graphs, interval graphs, directed path graphs, split graphs, permutation graphs, convex bipartite graphs, regular bipartite graphs, distance-hereditary 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 l l chordal graphs [Brandstädt, Dragan, Le & Le ’ 02] l tree t -spanner is NP-complete for any t ≧ 4 tree 3 -spanner admissible graphs [a Number of Authors] l l tree 4 -spanner admissible graphs l l l cographs, complements of bipartite graphs, interval graphs, directed path graphs, split graphs, permutation graphs, convex bipartite graphs, regular bipartite graphs, distance-hereditary 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 l l l bipartite graphs [Cai&Corneil ’ 95] tree t -spanner is NP-complete for any t ≧ 5 chordal graphs [Brandstädt, Dragan, Le & Le ’ 02] tree t -spanner is NP-complete for any t ≧ 4 tree 3 -spanner admissible graphs [a Number of Authors] l 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 chordal strongly chordal AT-free weakly chordal bipartite NP-C bipartite ATE-free 4 -Adm. rooted directed path interval bigraph interval convex 3 -Adm.

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

This Talk weakly chordal bipartite chordal strongly chordal AT-free enhanced probe interval rooted directed path interval probe interval STS-probe interval NP-C bipartite ATE-free 7 -Adm. 4 -Adm. interval bigraph = convex 3 -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. Monotone 3 SAT … (x, y, z) or (x, y, z)

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

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

NP-hardness for chordal bipartite graphs Monotone Reduction from 3 SAT… (x, y, z) or (x, y , z) l Gadget for xi xi 2 … xi m q r xi 1 p s xi 1 xi 2 … xi m ＝＝ Sk-1[] Sk[]×２ Must be selected

NP-hardness for chordal bipartite graphs Monotone Reduction from 3 SAT… (x, y, z) or (x, y , z) l Gadget for Cj cj+ cj - ＝ dj+ - dj Sk[]×２

NP-hardness for chordal bipartite graphs Monotone Reduction from 3 SAT… (x, y, z) or (x, y , z) l Gadget for C 1=(x 1, x 2, x 3) and C 2=(x 1, x 2, x 4) q r x 11 x 12 x 21 x 22 x 31 x 32 x 41 x 42 p s x 11 x 12 x 21 x 22 + c 1 + d 1 c 1 d 1 + c 2 + d 2 x 31 x 32 x 41 x 42 c 2 d 2 ＝ Sk-2[]

Tree 3 -spanner for a bipartite ATE-free graph l An ATE(Asteroidal-Triple-Edge) e 1, e 2, e 3 [Mul 97]: Any two of them there is a path from one to the other avoids the e 2 neighborhood of the third one. [Lamma] interval bigraphs ⊂ bipartite ATE-free graphs ⊂ chordal bipartite graphs. e 1 e 3

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

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

Tree 3 -spanner for a bipartite ATE-free graph Construction of a tree 3 -spanner of G: l 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!