Tree Spanners on Chordal Graphs: Complexity, Algorithms, Open Problems A. Brandstaedt, F.F. Dragan, H.-O. Le and V.B. Le University of Rostock, Germany Kent State University, Ohio, USA

G tree 4-spanner T of G tree t -spanner Problem Given unweighted undirected graph G=(V,E) Does G admit a spanning tree T =(V,U) such that 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 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 –tree t -spanner is Open for t=3 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]

Chordal Graphs G is chordal if it has no chordless cycles of length >3 There is no constant t [McKee, H.-O.Le] no tree 1-spanner

Chordal Graphs G is chordal if it has no chordless cycles of length >3 There is no constant t [McKee, H.-O.Le] no tree 2-spanner

Chordal Graphs G is chordal if it has no chordless cycles of length >3 There is no constant t [McKee, H.-O.Le] no tree 3-spanner

Chordal Graphs G is chordal if it has no chordless cycles of length >3 There is no constant t [McKee, H.-O.Le] From far away they look like trees there is a tree T=(V,U) (not necessarily spanning) such that [BCD’99] there is a sparse (2n-2 edges) 5-spanner [PS’89] no tree 3-spanner These graphs are not only chordal but also are planar and 2-trees

Chordal Graphs G is chordal if it has no chordless cycles of length >3 There is no constant t [McKee, H.-O.Le] From “far away” they look like trees no tree 3-spanner These graphs are not only chordal but also are planar and 2-trees Q: What is the complexity of tree t - spanner in chordal graphs?

This Talk NP-completeness results for t>3 –to the best of our knowledge, this is the first hardness result for the problem on a restricted, well-understood graph class. [FK01] Tree t - Spanner, t>3, is NP-complete on planar graphs, if the integer t is part of the input. Some easy solvable cases Many open problems

NP-completeness results For any t 4, Tree t - Spanner is NP-complete on chordal graphs of diameter at most –t+1 (if t is even), –t+2 (if t is odd). Proof is by reduction from 3SAT. –Let F be a 3CNF formula with m clauses and n variables

The graph obtained from H and by identifying the edge e=xy Gadgets The graph obtained from and by identifying b=d and joining x=a with y=c. If k=t-1, then any tree t-spanner must connect x and y in the part

Vertex Gadgets Let For each variable create the graph as follows. –Consider a clique on l+2 vertices –For each edge create a –Take a chordless path and connect both to each vertex of this path –For each edge create a

The graph G(F) For each clause create the graph as follows. –If t is even, is simply a single vertex. –If t is odd, is the graph. Finally, the graph G=G(F) is obtained from all and by identifying all vertices to a single vertex s, and adding the following additional edges: –connect every vertex in with every vertex in –for each literal, if then connect with respectively, with according to the parity of t. t=4

NP-completeness G(F) is chordal, and diam(G(F)) is at most t+1 if t is even, and at most t+2 if t is odd. G(F) admits a tree t-spanner if and only if F is satisfiable. Diameter at mostComplexity t + 2, t 5 oddNP-C t + 1, t 4 evenNP-C

Efficient solvable cases For any even integer t, every chordal graph of diameter at most t-1 admits a tree t-spanner, and such a tree spanner can be constructed in linear time. For any odd integer t, every chordal graph of diameter at most t-2 admits a tree t-spanner, and such a tree spanner can be constructed in linear time. Diameter at mostComplexity t + 2, t 5 oddNP-C t + 1, t 4 evenNP-C t - 1, t 2 evenlinear t - 2, t 3 oddlinear Any BFS-tree started at a central vertex is such a spanner. A central vertex of a chordal graph can be found in linear time [CD’94]. For any chordal graph, Q: can we improve this to t-1?

t-1 question (t is odd) chordal graphs of diameter at most t-1 (t is odd) admit tree t-spanners if and only if chordal graphs of diameter 2 admit tree 3-spanners. –First we reduce this problem to the existence of a tree (2rad(G)-1)- spanner in a chordal graph of diameter 2rad(G)-2. –Then, any such graph has an m-convex two-set M such that –Therefore…. G M

Tree 3-spanners in chordal graphs of diam=2. Unfortunately, the reduction above (from arbitrary odd t to t=3) is of no direct use for general chordal graphs because not every chordal graph of diameter at most 2 admits a tree 3-spanner. –We used this reduction to obtain some results for planar chordal graphs and k-trees (k<4). A chordal graph G of diameter at most 2 admits a tree 3-spanner if and only if there is a vertex v in G such that any connected component of the second neighborhood of v has a dominating vertex in N(v).

Tree 3-spanners in chordal graphs of diam=2. Unfortunately, the reduction above (from arbitrary odd t to t=3) is of no direct use for general chordal graphs because not every chordal graph of diameter at most 2 admits a tree 3-spanner. –We used this reduction to obtain some results for planar chordal graphs and k-trees (k<4). A chordal graph G of diameter at most 2 admits a tree 3-spanner if and only if there is a vertex v in G such that any connected component of the second neighborhood of v has a dominating vertex in N(v). For a given chordal graph G=(V,E) of diameter at most 2, the Tree 3– Spanner can be decided in O(|V| |E|) time. Moreover, a tree 3- spanner of G, if it exists, can be constructed within the same time bound. OQ: Can that reduction and the result above be combined to solve the “t-1 question”?

Conclusion and open problems Many questions remain still open. Among them: Can Tree 3–Spanner be decided efficiently on chordal graphs? Can Tree (2 r (G)-1)-- Spanner be decided efficiently on chordal graphs of diameter 2r(G)-2? What is the complexity of Tree t –Spanner for chordal graphs of diameter at most t ?. Diameter at mostComplexity t + 2, t 5 oddNP-C t + 1, t 4 evenNP-C t + 1, t 3 odd? t, t 3? t - 1, t 5 odd? t - 1, t = 3polynomial t - 1, t 2 evenlinear t - 2, t 3 oddlinear

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