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Tree Spanners for Bipartite Graphs and Probe Interval Graphs

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.

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Tree Spanners for Bipartite Graphs and Probe Interval Graphs

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  1. Tree Spanners for Bipartite Graphs and Probe Interval Graphs Andreas Brandstädt1, Feodor Dragan2, Oanh Le1, Van Bang Le1, and Ryuhei Uehara3 1 Universität Rostock 2 Kent State University 3 Komazawa University

  2. Tree Spanners for Bipartite Graphs and Probe Interval Graphs Andreas Brandstädt1, Feodor Dragan2, Oanh Le1, Van Bang Le1, and Ryuhei Uehara3 1 Universität Rostock 2 Kent State University 3 Komazawa University

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

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

  5. Tree Spanner G T • Spanning tree T is a tree6-spanner.

  6. Tree Spanner G 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.

  7. 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

  8. 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

  9. Known Results for tree t -spanner • 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] • 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]

  10. Known Results for tree t -spanner • 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] • cographs, complements of bipartite graphs, intervalgraphs, directed pathgraphs, 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??

  11. Known Results for tree t -spanner • 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] • 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

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

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

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

  15. 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)

  16. NP-hardness for chordal bipartite graphs Reduction from 3SAT • Basic gadgets Monotone … (x, y, z) or (x, y ,z) S1[a,b] S2[a,b] S3[a,b] a b a b a b S2[a,a’] S2[b,b’] S1[a,a’] S1[b,b’] a’ b’ a’ b’ a’ b’ S2[a’,b’] S1[a’,b’]

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

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

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

  20. NP-hardness for chordal bipartite graphs Reduction from 3SAT • Gadget for C1=(x1,x2,x3) and C2=(x1,x2,x4) Monotone … (x, y, z) or (x, y ,z) 1 2 1 2 1 2 1 2 x2 x2 x4 x4 q r x1 x1 x3 x3 1 2 1 2 1 2 1 2 x2 x2 x4 x4 p s x1 x1 x3 x3 + - + - c1 c1 c2 c2 Sk-2[] + - + - ＝ d1 d1 d2 d2

  21. Tree 3-spanner for a bipartite ATE-free graph e1 • An ATE(Asteroidal-Triple-Edge)e1,e2,e3 [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. e2 e3

  22. Tree 3-spanner for a bipartite ATE-free graph • A maximum neighborw 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

  23. 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＼Dk-1(u), there is w in Nk-1(u) s.t. N(w)⊇S∩Nk(u) w … u S

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

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

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