FastFast AlgorithmsAlgorithms foforr HardHard GraphGraph Problems:Problems: BidimensionalityBidimensionality,, Minors,Minors, andand (Local)(Local) TreewidthTreewidth MohammadTaghiMohammadTaghi HajiaghayiHajiaghayi CSCS && AIAI LabLab M.I.T.M.I.T. JointJoint workwork mainlymainly withwith ErikErik DemaineDemaine EDD = Erik D. Demaine, MTH = MohammadTaghi Hajiaghayi ReferencesReferences FVF = Fedor Fomin, DMT = Dimitrios Thilikos ❚ EDD & MTH, “Bidimensionality: New ❚ EDD & MTH, “Fast Algorithms for Connections between FPT Algorithms Hard Graph Problems: Bidimen- and PTASs”, SODA 2005. sionality, Minors, and Local ❚ EDD & MTH, “Graphs Excluding a Fixed Treewidth”, GD 2004. Minor have Grids as Large as ❚ EDD, FVF, MTH, & DMT, Treewidth, with Combinatorial and “Bidimensional Parameters and Local Algorithmic Applications through Treewidth”, SIDMA 2005. Bidimensionality”, SODA 2005. ❚ ❚ EDD & MTH, “Diameter and Tree- EDD, MTH & DMT, “Exponential Speed- width in Minor-Closed Graph Fami- up of FPT Algorithms for Classes of lies, Revisited”, Algorithmica 2004. Graphs Excluding Single-Crossing ❚ Graphs as Minors”, Algorithmica 2005. EDD, MTH, & DMT, “The Bidimen- ❚ sional Theory of Bounded-Genus EDD, FVF, MTH, & DMT, “Fixed- Graphs”, SIDMA 2005. Parameter Algorithms for (k, r)-Center ❚ in Planar Graphs and Map Graphs”, ACM EDD & MTH, “Equivalence of Local Trans. on Algorithms 2005. Treewidth and Linear Local ❚ Treewidth and its Algorithmic EDD, MTH, Naomi Nishimura, Applications”, SODA 2004. Prabhakar Ragde, & DMT, “Approx- ❚ imation algorithms for classes of EDD, FVF, MTH, & DMT, “Subexpo- graphs excluding single-crossing graphs nential parameterized algorithms on as minors”, J. Comp & System Sci.2004. graphs of bounded genus and H- ❚ minor-free graphs”, JACM 2005. EDD, MTH, Ken-ichi Kawarabayashi, ❚ “Algorithmic Graph Minor Theory: Uriel Feige, MTH & James R. Lee, Decomposition, Approximation, and “Improved approximation algorithms Coloring’’, FOCS 2005. for vertex separators”, STOC 2005. Motivation:Motivation: HardHard GraphGraph ProblemsProblems ❚ Many NP-hard graph problems ❙ Find minimum-cost Traveling Salesman tour in a graph ❙ Find fewest radius-r facilities to cover graph ❙ Draw a graph with fewest crossings ❙ Etc. ❚ No polynomial-time algorithm unless P = NP ❚ Still want to solve them DealingDealing withwith HardHard GraphGraph ProblemsProblems ❚ Main theoretical approaches to solving NP-hard problems: ❙ Average case (phase transitions, smoothed analys.) ❙ Special instances: Planar graphs, etc. ❙ Approximation algorithms: Within some factor C of the optimal solution ❙ Fixed-parameter algorithms: Parameterize problem by parameter P (typically, the cost of the optimal solution) and aim for f(P) nO(1) (or even f(P) + nO(1)) OurOur ResultsResults atat aa HighHigh LevelLevel ❚ Consider vertex cover, dominating set, feedback vertex set, unweighted TSP, … ❚ Or any graph optimization problem that’s ❙ 2 × Bidimensional: Large (Ω(r )) on r r grid, r never increases when contracting an edge r ❙ Solvable on graphs of bounded treewidth ❚ In most minor-closed graph families: ❙ Treewidth = O(√solution value) ❙ 2O(√k) nO(1) subexponential fixed-parameter alg. ❙ (1+ε)-approximation in 2O(1/ε) nO(1) time OurOur ResultsResults atat aa HighHigh LevelLevel ❚ For any graph problem in a large class (“bidimensional”) ❙ Vertex cover, dominating set, connected dominating set, r-dominating set, feedback vertex set, unweighted TSP, … ❚ In broad classes of graphs (most “minor-closed”graph families) ❚ Obtain: ❙ Strong combinatorial properties ❙ Subexponential fixed-parameter algorithms ❙ Polynomial-time approximation schemes OurOur ResultsResults atat aa HighHigh LevelLevel ❚ For any graph optimization problem that’s ❙ 2 × Bidimensional: Large (Ω(r )) on r r grid, r never increases when contracting an edge r ❙ Solvable on graphs of bounded treewidth ❚ In most minor-closed graph families: ❙ Treewidth = O(√solution value) ❙ 2O(√k) nO(1) subexponential fixed-parameter alg. ❙ (1+ε)-approximation in 2O(1/ε) nO(1) time ResultsResults atat aa veryvery highhigh levellevel ❚ For any graph optimization problem that’s ❙ Bidimensional: Large (Ω(r2)) on r × r grid, never increases when contracting an edge ❙ Solvable on graphs of bounded treewidth 2.5 ❚ There is a fixed-parameter alg. 22k nO(1) ❚ In most minor-closed graph families: ❙ Treewidth = O(√solution value) ❙ 2O(√k) nO(1) subexponential fixed-parameter alg. ❙ PTASs MinorsMinors ❚ A graph G has a minor H if H can be formed by removing and contracting edges of G * delete G H minor of G contract ❚ Otherwise, G is H-minor-free ❚ For example, planar graphs are both K3,3-minor-free and K5-minor-free GraphGraph MinorMinor TheoryTheory [Robertson[Robertson && SeymourSeymour 19841984––2004]2004] ❚ Seminal series of ≥ 20 papers ❚ Powerful results on excluded minors: ❙ Every minor-closed graph property (preserved when taking minors) has a finite set of excluded minors [Wagner’s Conjecture] ❙ Every minor-closed graph property can be decided in polynomial time ❙ For fixed graph H, graphs minor-excluding H have a special structure: drawings on bounded-genus surfaces + “extra features” TreewidthTreewidth [GM2[GM2——RobertsonRobertson && SeymourSeymour 1986]1986] ❚ Treewidth of a graph is the smallest possible width of a tree decomposition ❚ Tree decomposition spreads out each vertex as a connected subtree of a common tree, such that adjacent vertices have Tree Graph overlapping subtrees decomposition ❙ Width = maximum overlap − 1 (width 3) ❚ Treewidth 1 ≡ tree; 2 ≡ series-parallel; … TreewidthTreewidth BasicsBasics ❚ Many fast algorithms for NP-hard problems on graphs of small treewidth ❙ Typical running time: 2O(treewidth) nO(1) ❚ Computing treewidth is NP-hard O(treewidth) ❚ Computable in 22 n time, including a tree decomposition [Bodlaender 1996] ❚ O(1)-approximable in 2O(treewidth) nO(1) time, including a tree decomposition [Amir 2001] ❚ O(√lg opt)-approximable in nO(1) time [Feige, Hajiaghayi, Lee 2004] (using a new framework for vertex separators based on embedding with minimum average distortion into line) TreewidthTreewidth BasicsBasics ❚ Many fast algorithms for NP-hard problems on graphs of small treewidth ❙ Typical running time: 2O(treewidth) nO(1) ❚ Computing treewidth is NP-hard O(treewidth) ❚ Computable in 22 n time, including a tree decomposition [Bodlaender 1996] ❚ O(1)-approximable in 2O(treewidth) nO(1) time, including a tree decomposition [Amir 2001] ❚ 1.5-approximation for planar graphs and single-crossing- minor-free graphs ❚ O(|V(H)|^2)-approximable in nO(1) time in H-minor- free graphs [Feige, Hajiaghayi, Lee 2004] GridGrid MinorsMinors r ❚ r × r grid: ❙ r2 vertices, 2 r (r − 1) edges r ❙ Treewidth ~ r ❚ r × r grid is the canonical planar graph of treewidth Θ(r): every planar graph of treewidth w has an Ω(w) × Ω(w) grid minor [Robertson, Seymour, Thomas 1994] ❙ So any planar graph of large treewidth has an r × r grid minor certifying large treewidth ❙ What about nonplanar graphs? GridGrid MinorsMinors r r ❚ For any fixed graph H, every H-minor-free graph of treewidth ≥ w(r) has an r × r grid minor [GM5—Robertson & Seymour 1986] ❙ Re-proved & strengthened [Robertson, Seymour, Thomas 1994; Reed 1997; Diestel, Jensen, Gorbunov, Thomassen 1999] ❙ Best bound of these: w(r) = 20 5|V(H)|3 r [Robertson, Seymour, Thomas 1994] ❙ New optimal bound: w(r) = Θ(r) [Demaine & Hajiaghayi 2005] ❘ Grids certify large treewidth in H-minor-free graph BidimensionalityBidimensionality atat aa HighHigh LevelLevel ❚ Parameter = function P assigning nonnegative integer P(G) to every graph G ❙ Parameter ↔ graph optimization problem ❚ Parameter P is bidimensional if ❙ Never increases when contracting an edge ❙ P is “large” on r × r “grid-like graphs” typically, think e.g. partially r Ω(r2) triangulated grid with a few extra edges r GraphGraph ClassesClasses planar bounded-genus single-crossing- minor-free apex-minor-free map graphs H-minor-free general planar GraphGraph ClassesClasses bounded-genus single-crossing- minor-free apex-minor-free map graphs ❚ Drawings on H-minor-free surfaces: ❙ Planar graphs general ❘ Drawable in plane/sphere without crossings ❘ = K3,3-minor-free & K5-minor-free graphs ❙ Bounded-genus graphs ❘ One family for each integer g ≥ 0 ❘ Drawable in the genus-g orientable surface without crossings planar GraphGraph ClassesClasses bounded-genus single-crossing- minor-free apex-minor-free map graphs ❚ Drawings on H-minor-free surfaces: ❙ general Map graph New York Times ❘ Given embedded planar graph ❘ Given face 2-coloring as nations and lakes ❘ Vertex for each nation ❘ Edge between nations that share a vertex ❙ More generally, fixed powers of graph family ❘ Map graphs = half-square of planar bipartite planar GraphGraph ClassesClasses bounded-genus single-crossing- minor-free apex-minor-free map graphs ❚ Drawings on H-minor-free surfaces: ❙ general Map graph New York Times ❘ Given embedded planar graph ❘ Given face 2-coloring as nations and lakes ❘ Vertex for each nation ❘ Edge between nations that share a vertex ❙ More generally, fixed powers of graph family ❘ Map graphs = half-square of planar bipartite planar GraphGraph ClassesClasses bounded-genus single-crossing- minor-free ❚ Excluding minors: apex-minor-free map graphs ❙ X-minor-free = class of graphs H-minor-free excluding a fixed general graph from family X ❙ H-minor-free = exclude any fixed graph H ❙ Apex graph = planar graph + one vertex + any incident edges ❙ Single-crossing graph = minor of a graph that can be drawn with at most one crossing