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The Square Root Phenomenon in Planar Graphs Survey and New Results The Square Root Phenomenon in Planar Graphs Survey and New Results Dániel Marx Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI) Budapest, Hungary Satisfiability Lower Bounds and Tight Results for Parameterized and Exponential-Time Algorithms Simons Institute, Berkeley, CA November 6, 2015 1 Main message NP-hard problems become easier on planar graphs and geometric objects, and usually exactly by a square root factor. Planar graphs Geometric objects 2 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. 3 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. 3 Overview We repeat some of the material from the boot camp, but will also see new results. Chapter 1: Subexponential algorithms using treewidth. Chapter 2: Grid minors and bidimensionality. Chapter 3: Beyond bidimensionality: Finding bounded-treewidth solutions. 4 Chapter 1: Subexponential algorithms using treewidth Treewidth is a measure of “how treelike the graph is.” We need only the following basic facts: Treewidth 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). 5 Treewidth — a measure of “tree-likeness” 6 A subtree communicates with the outside world only via the root of the subtree. 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 7 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. 7 + Corollary p 3-Coloring can be solved in time2 O( n) on planar graphs. textbook algorithm + combinatorial bound + subexponential algorithm Subexponential algorithm for 3-Coloring Theorem [textbook dynamic programming] 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). 8 Subexponential algorithm for 3-Coloring Theorem [textbook dynamic programming] 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 8 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? 9 Lower bounds based on ETH ETH + Sparsification Lemma 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(n + m) vertices m clauses O(n + m) edges Corollary Assuming ETH, there is no2 o(n) algorithm for 3-Coloring on an n-vertex graph G. 10 Lower bounds based on ETH ETH + Sparsification Lemma 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. 10 Transfering bounds There are polynomial-time reductions from, say, 3-Coloring to many other problems such that the reduction increases the number of vertices by at most a constant factor. Consequence: Assuming ETH, there is no2 o(n) time algorithm on n-vertex graphs for Independent Set Clique Dominating Set Vertex Cover Hamiltonian Path Feedback Vertex Set ::: 11 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” with4 external connectors: In every3-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 gadget. If two edges cross, replace them with a crossover gadget. 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” with4 external connectors: In every3-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 gadget. If two edges cross, replace them with a crossover gadget. 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” with4 external connectors: In every3-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 gadget. If two edges cross, replace them with a crossover gadget. 12 Lower bounds based on ETH The reduction from 3-Coloring to Planar 3-Coloring introduces O(1) new edges/vertices for each crossing. A graph with m edges can be drawn with O(m2) crossings. 3SAT formula φ Graph G Planar graph G 0 n variables ) O(m) vertices ) O(m2) vertices m clauses O(m) edges O(m2) edges Corollary p Assuming ETH, there is no2 o( n) algorithm for 3-Coloring on an n-vertex planar graph G. (Essentially observed by [Cai and Juedes 2001]) 13 Lower bounds for planar problems p Consequence: Assuming ETH, there is no2 o( n) time algorithm on n-vertex planar graphs for Independent Set Dominating Set Vertex Cover Hamiltonian Path Feedback Vertex Set ::: 14 Lower bounds for planar problems p Consequence: Assuming ETH, there is no2 o( k) · nO(1) time algo- rithm on planar graphs for k-Independent Set k-Dominating Set k-Vertex Cover k-Path k-Feedback Vertex Set ::: 14 Summary of Chapter 1 Streamlined way of obtaining tight upper and lower bounds for planar problems. Upper bound: Standard bounded-treewidthp algorithm + treewidth bound on planar graphs give2 O( n) time subexponential algorithms. Lower bound: Textbook NP-hardnessp proof with quadratic blow up + ETH rule out2 o( n) algorithms. Works for Hamiltonian Cycle, Vertex Cover, Independent Set, Feedback Vertex Set, Dominating Set, Steiner Tree, ::: 15 Chapter 2: Bidimensionality Bidimensionality theory [Demaine, Fomin, Hajiaghayi, Thilikos 2005] gives very elegant subexponential algorithms on planar graphs for parameterized problems such as k-Path Vertex Cover Feedback Vertex Set Independent Set Dominating Set p We already know that (assuming ETH), there are no2 o( k) · nO(1) time algorithms for these problems. 16 Minors Definition Graph H is a minor of G (H ≤ G) if H can be obtained from G by deleting edges, deleting vertices, and contracting edges. uv deleting uv contracting uv vu w Note: length of the longest path in H is at most the length of the longest path in G. 17 Planar Excluded Grid Theorem Theorem [Robertson, Seymour, Thomas 1994] Every planar graph with treewidth at least5 k has a k × k grid minor. Note: for general graphs, treewidth at least k100 or so guarantees a k × k grid minor [Chekuri and Chuzhoy 2013]! 18 We use this observation to find a path of length at least k on planar graphs: Bidimensionality for k-Path p Observation:p If thep treewidth of a planar graph G is at least5 k ) It has a k × k grid minor (Planar Excluded Grid Theorem) ) The grid has a path of length at least k. ) G has a path of length at least k. 19 Bidimensionality for k-Path p Observation:p If thep treewidth of a planar graph G is at least5 k ) It has a k × k grid minor (Planar Excluded Grid Theorem) ) The grid has a path of length at least k. ) G has a path of length at least k. We use this observation to find a path of length at least k on planar graphs: p Set w := 5 k.
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