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The Square Root Phenomenon in Planar Graphs

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The Square Root Phenomenon in Planar Graphs The square root phenomenon in planar graphs Dániel Marx1 1Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI) Budapest, Hungary CS Theory Seminar The Hebrew University of Jerusalem May 22, 2013 Jerusalem, Israel 1 Main message Are NP-hard problems easier on planar graphs? Yes, usually. By how much? Often by exactly a square root factor. 2 Overview Chapter 1: Subexponential algorithms using treewidth. Chapter 2: Grid minors and bidimensionality. Chapter 3: Finding bounded-treewidth solutions. 3 The running time is still exponential, but significantly smaller: p 2O(n) ) 2O( n) p nO(k) ) nO( k) p 2O(k) · nO(1) ) 2O( k) · nO(1) Better exponential algorithms Most NP-hard problems (e.g., 3-Coloring, Independent Set, Hamiltonian Cycle, Steiner Tree, etc.) remain NP-hard on planar graphs,1 so what do we mean by “easier”? 1Notable exception: Max Cut is in P for planar graphs. 4 Better exponential algorithms Most NP-hard problems (e.g., 3-Coloring, Independent Set, Hamiltonian Cycle, Steiner Tree, etc.) remain NP-hard on planar graphs,1 so what do we mean by “easier”? The running time is still exponential, but significantly smaller: p 2O(n) ) 2O( n) p nO(k) ) nO( k) p 2O(k) · nO(1) ) 2O( k) · nO(1) 1Notable exception: Max Cut is in P for planar graphs. 4 Chapter 1: Subexponential algorithms using treewidth Treewidth is a measure of “how treelike the graph is.” We need only the following basic facts: 1 If a graph G has treewidth k, then many classical NP-hard problems can be solved in time2 O(k) · nO(1) or 2O(k log k) · nO(1) on G. p 2 A planar graph on n vertices has treewidth O( n). 3 Excluded Grid Theorem: a planar graph of treewidth k contains a Ω(k) × Ω(k) grid minor. 5 Width of the decomposition: largest bag size −1. treewidth: width of the best decomposition. Treewidth — a measure of “tree-likeness” Tree decomposition: Vertices are arranged in a tree structure satisfying the following properties: 1 If u and v are neighbors, then there is a bag containing both of them. 2 For every v, the bags containing v form a connected subtree. a c; d; f dcb ; c; f d; f ; gb ; b; c ; e; fa g; hb efgh 6 Width of the decomposition: largest bag size −1. treewidth: width of the best decomposition. Treewidth — a measure of “tree-likeness” Tree decomposition: Vertices are arranged in a tree structure satisfying the following properties: 1 If u and v are neighbors, then there is a bag containing both of them. 2 For every v, the bags containing v form a connected subtree. a c; d; f dcb b; c; f d; f ; g a; b; c b; e; f g; h efgh 6 Width of the decomposition: largest bag size −1. treewidth: width of the best decomposition. Treewidth — a measure of “tree-likeness” Tree decomposition: Vertices are arranged in a tree structure satisfying the following properties: 1 If u and v are neighbors, then there is a bag containing both of them. 2 For every v, the bags containing v form a connected subtree. a c; d; f dcb b; c; f d; f ; g ; b; c b; e; f g; ha efgh 6 Treewidth — a measure of “tree-likeness” Tree decomposition: Vertices are arranged in a tree structure satisfying the following properties: 1 If u and v are neighbors, then there is a bag containing both of them. 2 For every v, the bags containing v form a connected subtree. Width of the decomposition: largest bag size −1. treewidth: width of the best decomposition. a c; d; f dcb b; c; f d; f ; g ; b; c b; e; f g; ha efgh 6 Treewidth — a measure of “tree-likeness” Tree decomposition: Vertices are arranged in a tree structure satisfying the following properties: 1 If u and v are neighbors, then there is a bag containing both of them. 2 For every v, the bags containing v form a connected subtree. Width of the decomposition: largest bag size −1. treewidth: width of the best decomposition. a c; d; f dcb ; c; f d; f ; gb ; b; c ; e; fa g; hb efgh Each bag is a separator. 6 Treewidth — a measure of “tree-likeness” Tree decomposition: Vertices are arranged in a tree structure satisfying the following properties: 1 If u and v are neighbors, then there is a bag containing both of them. 2 For every v, the bags containing v form a connected subtree. Width of the decomposition: largest bag size −1. treewidth: width of the best decomposition. a c; d; f bcd ; c; f d; f ; gb ; b; c ; e; fa g; hb hgfe A subtree communicates with the outside world only via the root of the subtree. 6 Finding tree decompositions Various algorithms for finding optimal or approximate tree decompositions if treewidth is w: 3 optimal decomposition in time2 O(w ) · n [Bodlaender 1996]. 4-approximate decomposition in time2 O(w) · n2 [Robertson and Seymour]. 5-approximate decomposition in time2 O(w) · n [Bodlaender et al. 2013]. O(plog w)-approximation in polynomial time [Feige, Hajiaghayi, Lee 2008]. As we are mostly interested in algorithms with running time 2O(w) · nO(1), we may assume that we have a decomposition. 7 3-Coloring and tree decompositions Theorem Given a tree decomposition of width w, 3-Coloring can be solved in time O(3w · w O(1) · n). Bx : vertices appearing in node x. Vx : vertices appearing in the subtree rooted at x. c; d; f For every node x and coloring c : Bx ! f1; 2; 3g, we compute the Boolean value b; c; fd ; f ; g E[x; c], which is true if and only if c can be extended to a proper 3-coloring of Vx . a; b; cb ; e; fg ; h Claim: bcf=T bcf=F We can determine E[x; c] if all the values are bcf=T bcf=F known for the children of x. :::::: 8 Subexponential algorithm for 3-Coloring Theorem 3-Coloring can be solved in time2 O(w) · nO(1) on graphs of treewidth w. Theorem [Robertson and Seymour] p A planar graph on n vertices has treewidth O( n). Corollary p 3-Coloring can be solved in time2 O( n) on planar graphs. textbook algorithm + combinatorial bound + subexponential algorithm 9 P 6= NP is not a sufficiently strong hypothesis: it is compatible with 1=1000 3SAT being solvable in time2 O(n ) or even in time nO(log n). We need a stronger hypothesis! Lower bounds Corollary p 3-Coloring can be solved in time2 O( n) on planar graphs. Two natural questions: Can we achieve this running time on general graphs? p Can we achieve even better running time (e.g.,2 O( 3 n)) on planar graphs? 10 Lower bounds Corollary p 3-Coloring can be solved in time2 O( n) on planar graphs. Two natural questions: Can we achieve this running time on general graphs? p Can we achieve even better running time (e.g.,2 O( 3 n)) on planar graphs? P 6= NP is not a sufficiently strong hypothesis: it is compatible with 1=1000 3SAT being solvable in time2 O(n ) or even in time nO(log n). We need a stronger hypothesis! 10 Sparsification Lemma [Impagliazzo, Paturi, Zane 2001] There is a2 o(n)-time algorithm for n-variable 3SAT. m There is a2 o(m)-time algorithm for m-clause 3SAT. Exponential Time Hypothesis (ETH) Hypothesis introduced by Impagliazzo, Paturi, and Zane: Exponential Time Hypothesis (ETH) There is no2 o(n)-time algorithm for n-variable 3SAT. Note: current best algorithm is1 :30704n [Hertli 2011]. Note: an n-variable 3SAT formula can have Ω(n3) clauses. 11 Exponential Time Hypothesis (ETH) Hypothesis introduced by Impagliazzo, Paturi, and Zane: Exponential Time Hypothesis (ETH) There is no2 o(n)-time algorithm for n-variable 3SAT. Note: current best algorithm is1 :30704n [Hertli 2011]. Note: an n-variable 3SAT formula can have Ω(n3) clauses. Sparsification Lemma [Impagliazzo, Paturi, Zane 2001] There is a2 o(n)-time algorithm for n-variable 3SAT. m There is a2 o(m)-time algorithm for m-clause 3SAT. 11 Lower bounds based on ETH Exponential Time Hypothesis (ETH) There is no2 o(m)-time algorithm for m-clause 3SAT. The textbook reduction from 3SAT to 3-Coloring: 3SAT formula φ Graph G n variables ) O(m) vertices m clauses O(m) edges Corollary Assuming ETH, there is no2 o(n) algorithm for 3-Coloring on an n-vertex graph G. 12 Lower bounds based on ETH What about 3-Coloring on planar graphs? The textbook reduction from 3-Coloring to Planar 3-Coloring uses a “crossover gadget” with 4 external connectors: In every 3-coloring of the gadget, opposite external connectors have the same color. Every coloring of the external connectors where the opposite vertices have the same color can be extended to the whole gadgets. If two edges cross, replace them with a crossover gadget. 13 Lower bounds based on ETH What about 3-Coloring on planar graphs? The textbook reduction from 3-Coloring to Planar 3-Coloring uses a “crossover gadget” with 4 external connectors: In every 3-coloring of the gadget, opposite external connectors have the same color. Every coloring of the external connectors where the opposite vertices have the same color can be extended to the whole gadgets. If two edges cross, replace them with a crossover gadget. 13 Lower bounds based on ETH What about 3-Coloring on planar graphs? The textbook reduction from 3-Coloring to Planar 3-Coloring uses a “crossover gadget” with 4 external connectors: In every 3-coloring of the gadget, opposite external connectors have the same color.
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