Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal Dániel Marx Humboldt-Universität zu Berlin Joint work with Daniel Lokshtanov Saket Saurabh ACM-SIAM Symposium on Discrete Algorithms (SODA 2011) – Jan 24, 2011 Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.1/20
Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.2/20
Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.2/20
Treewidth Treewidth: A measure of how “tree-like” the graph is. (Introduced by Robertson and Seymour in the Graph Minors project.) Significance: Appears naturally in graph structure theory. Polynomial or even linear algorithms for NP-hard problems on bounded treewidth graphs. Crucial tool for planar approximation schemes. Useful for fixed-parameter tractability results. Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.3/20
Treewidth Tree decomposition: Vertices are arranged in a a tree structure satisfying the following properties: c b d 1. If u and v are neighbors, then there is a bag containing both of them. 2. For every vertex v , the bags containing v form a connected subtree. e g f h c , d , f d , f , g b , c , f g , h a , b , c b , e , f Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.4/20
Treewidth Tree decomposition: Vertices are arranged in a a tree structure satisfying the following properties: c b d 1. If u and v are neighbors, then there is a bag containing both of them. 2. For every vertex v , the bags containing v form a connected subtree. e g f h c , d , f d , f , g b , c , f g , h a , b , c b , e , f Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.4/20
Treewidth Tree decomposition: Vertices are arranged in a a tree structure satisfying the following properties: c b d 1. If u and v are neighbors, then there is a bag containing both of them. 2. For every vertex v , the bags containing v form a connected subtree. e g f h Width of decomposition: largest bag size − 1 . treewidth: width of the best decomposition. c , d , f Fact: treewidth = 1 ⇐ ⇒ graph is a forest d , f , g b , c , f g , h a , b , c b , e , f Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.4/20
Treewidth Tree decomposition: Vertices are arranged in a a tree structure satisfying the following properties: c b d 1. If u and v are neighbors, then there is a bag containing both of them. 2. For every vertex v , the bags containing v form a connected subtree. e g f h Width of decomposition: largest bag size − 1 . treewidth: width of the best decomposition. c , d , f Fact: treewidth = 1 ⇐ ⇒ graph is a forest d , f , g b , c , f g , h a , b , c b , e , f Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.4/20
Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.5/20
MAX INDEPENDENT SET and tree decompositions Fact: Given a tree decomposition of width w , MAX INDEPENDENT SET can be solved in time O (2 w · n ) . c , d , f B x : vertices appearing in node x . V x : vertices appearing in the subtree rooted b , c , f d , f , g at x . g , h a , b , c b , e , f Define table M [ x , S ] : the maximum weight of an independent set I ⊆ V x with ∅ =? bc =? I ∩ B x = S . b =? cf =? c =? bf =? Compute the tables in bottom-up order. f =? bcf =? Size of each table is 2 w +1 . Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.6/20
Algorithms Given a tree decomposition of width w , dynamic programming gives: O (2 w · n ) INDEPENDENT SET O (3 w · n ) DOMINATING SET O (2 w · n ) MAX CUT O (3 w · n ) ODD CYCLE TRANSVERSAL O ( q w · n ) q -COLORING ( q ≥ 3 ) O (2 w · n ) PARTITION INTO TRIANGLES [various authors] Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.7/20
Algorithms Given a tree decomposition of width w , dynamic programming gives: O (2 w · n ) INDEPENDENT SET O (3 w · n ) DOMINATING SET O (2 w · n ) MAX CUT O (3 w · n ) ODD CYCLE TRANSVERSAL O ( q w · n ) q -COLORING ( q ≥ 3 ) O (2 w · n ) PARTITION INTO TRIANGLES [various authors] Question: Can we improve the base in any of these algorithms? Supporting evidence: Running time matches the obvious DP table size. But ... Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.7/20
Some history DOMINATING SET Obvious approach: 9 w [Telle and Proskurowski ’93] More clever algorithm: 4 w [Alber et al. ’02] Even more clever algorithm: 3 w [Rooij et al. ’09] using fast subset convolution of [Björklund et al. ’07] Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.8/20
Some history DOMINATING SET Obvious approach: 9 w [Telle and Proskurowski ’93] More clever algorithm: 4 w [Alber et al. ’02] Even more clever algorithm: 3 w [Rooij et al. ’09] using fast subset convolution of [Björklund et al. ’07] HAMILTONIAN CYCLE 2 n time [Held and Karp ’62] 1.657 n (randomized) time [Björklund ’10] Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.8/20
Some history DOMINATING SET Obvious approach: 9 w [Telle and Proskurowski ’93] More clever algorithm: 4 w [Alber et al. ’02] Even more clever algorithm: 3 w [Rooij et al. ’09] using fast subset convolution of [Björklund et al. ’07] HAMILTONIAN CYCLE 2 n time [Held and Karp ’62] 1.657 n (randomized) time [Björklund ’10] DIRECTED FEEDBACK VERTEX SET Trivial 2 n algorithm. Nontrivial 1.9977 n algorithm [Razgon ’07] Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.8/20
Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.9/20
SETH Obviously, we need a hardness assumption. √ w · n ) algorithm seems to be P � = NP is not sufficiently strong: even a O (2 compatible with it. Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.10/20
SETH Obviously, we need a hardness assumption. √ w · n ) algorithm seems to be P � = NP is not sufficiently strong: even a O (2 compatible with it. Strong Exponential Time Hypothesis (SETH): s k = inf { δ | n -variable k -SAT can be solved in 2 δ n } Conjecture: [Impagliazzo-Paturi ’01] s k → 1 We can use a somewhat weaker assumption: No faster SAT: Conjecture: n -variable m -clause SAT (with arbitrary clause length) cannot be solved in time (2 − ǫ ) n · poly ( m ) for any ǫ > 0 . Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.10/20
Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.11/20
Results Main result: If the Strong Exponential Time Hypothesis (SETH) is true, then given a tree decomposition of width w , (2 − ǫ ) w · n O (1) INDEPENDENT SET (3 − ǫ ) w · n O (1) DOMINATING SET (2 − ǫ ) w · n O (1) MAX CUT cannot be (3 − ǫ ) w · n O (1) ODD CYCLE TRANSVERSAL solved in time ( q − ǫ ) w · n O (1) q -COLORING ( q ≥ 3 ) (2 − ǫ ) w · n O (1) PARTITION INTO TRIANGLES The lower bounds match the known algorithms (up to the ǫ in the base). Note: For some problems, we can obtain stronger results by proving the same lower bound with respect to pathwidth or feedback vertex number. Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.12/20
Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.13/20
Reductions Suppose we have a reduction: INDEPENDENT SET instance ⇒ n -variable SAT instance of treewidth w ≤ c · n . Then: (2 − ǫ ) w · n O (1) algorithm for (2 − ǫ ) c · n algorithm for SAT ⇐ INDEPENDENT SET To get a (2 − ǫ ) w lower bound, we need c ≤ 1 . Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.14/20
Reductions Suppose we have a reduction: INDEPENDENT SET instance ⇒ n -variable SAT instance of treewidth w ≤ c · n . Then: (2 − ǫ ) w · n O (1) algorithm for (2 − ǫ ) c · n algorithm for SAT ⇐ INDEPENDENT SET To get a (2 − ǫ ) w lower bound, we need c ≤ 1 . More generally: For any c , we get a (2 1 / c − ǫ ) w lower bound ⇒ To get a (3 − ǫ ) w lower bound (e.g., for DOMINATING SET), we need c ≤ log 3 2 ≈ 0.631 . Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.14/20
Textbook reduction How large is the treewidth in the textbook reduction from SAT to INDEPENDENT SET? x 1 x n ¯ ¯ x 1 x n C 1 C 2 C 3 Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.15/20
Textbook reduction How large is the treewidth in the textbook reduction from SAT to INDEPENDENT SET? x 1 x n ¯ ¯ x 1 x n C 1 C 2 C 3 2 − ǫ ) w ≈ 1.41 w lower bound. 1 Treewidth is about 2 n , which gives a (2 We need treewidth ≤ n for the (2 − ǫ ) w lower bound. Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal – p.15/20
Recommend
More recommend
