1. Fast Algorithms for Hard Graph Problems: Bidimensionality, Minors, and (Local) Treewidth MohammadTaghi Hajiaghayi CS & AI Lab M.I.T. Joint work mainly with Erik Demaine

2. References zEDD, MTH, & DMT, “Exponential Speedup of Fixed-Parameter Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors”, Algorithmica 2005; ISAAC 2002. zEDD, FVF, MTH, & DMT, “Fixed-Parameter Algorithms for (k, r)-Center in Planar Graphs and Map Graphs”, ACM Trans. Algorithms 2005; ICALP 2003. zEDD, FVF, MTH, & DMT, “Bidimensional Parameters and Local Treewidth”, SIAM Journal on Discrete Mathematics 2005; LATIN 2004. zEDD, MTH, Naomi Nishimura, Prabhakar Ragde, & DMT, “Approximation algorithms for classes of graphs excluding single-crossing graphs as minors”, Journal of Computer and System Sciences 2004; APPROX 2002; COMB 2001. zEDD & MTH, “Bidimensionality: New Connections between FPT Algorithms and PTASs”, SODA 2005. zEDD & MTH, “Graphs Excluding a Fixed Minor have Grids as Large as Treewidth, with Combinatorial and Algorithmic Applications through Bidimensionality”, SODA 2005. zEDD & MTH, “Fast Algorithms for Hard Graph Problems: Bidimensionality, Minors, and Local Treewidth”, GD 2004. zEDD & MTH, “Diameter and Treewidth in Minor-Closed Graph Families, Revisited”, Algorithmica, 2004. zEDD, MTH, & DMT, “The Bidimensional Theory of Bounded-Genus Graphs”, MFCS 2004. zEDD & MTH, “Equivalence of Local Treewidth and Linear Local Treewidth and its Algorithmic Applications”, SODA 2004. zEDD, FVF, MTH, & DMT, “Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs”, SODA 2004. EDD = Erik D. Demaine MTH = MohammadTaghi Hajiaghayi FVF = Fedor V. Fomin DMT = Dimitrios M. Thilikos

3. References zEDD & MTH, “Bidimensionality: New Connections between FPT Algorithms and PTASs”, SODA 2005. zEDD & MTH, “Graphs Excluding a Fixed Minor have Grids as Large as Treewidth, with Combinatorial and Algorithmic Applications through Bidimensionality”, SODA 2005. zEDD, MTH & DMT, “Exponential Speed- up of FPT Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors”, Algorithmica 2005. zEDD, FVF, MTH, & DMT, “Fixed- Parameter Algorithms for (k, r)-Center in Planar Graphs and Map Graphs”, ACM Trans. on Algorithms 2005. zEDD, MTH, Naomi Nishimura, Prabhakar Ragde, & DMT, “Approx- imation algorithms for classes of graphs excluding single-crossing graphs as minors”, J. Comp & System Sci.2004. zEDD, MTH, Ken-ichi Kawarabayashi, “Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring’’, FOCS 2005. EDD = Erik D. Demaine, MTH = MohammadTaghi Hajiaghayi FVF = Fedor Fomin, DMT = Dimitrios Thilikos zEDD & MTH, “Fast Algorithms for Hard Graph Problems: Bidimen- sionality, Minors, and Local Treewidth”, GD 2004. zEDD, FVF, MTH, & DMT, “Bidimensional Parameters and Local Treewidth”, SIDMA 2005. zEDD & MTH, “Diameter and Tree- width in Minor-Closed Graph Fami- lies, Revisited”, Algorithmica 2004. zEDD, MTH, & DMT, “The Bidimen- sional Theory of Bounded-Genus Graphs”, SIDMA 2005. zEDD & MTH, “Equivalence of Local Treewidth and Linear Local Treewidth and its Algorithmic Applications”, SODA 2004. zEDD, FVF, MTH, & DMT, “Subexpo- nential parameterized algorithms on graphs of bounded genus and H- minor-free graphs”, JACM 2005. zUriel Feige, MTH & James R. Lee, “Improved approximation algorithms for vertex separators”, STOC 2005.

4. Motivation: Hard Graph Problems zMany NP-hard graph problems yFind minimum-cost Traveling Salesman tour in a graph yFind fewest radius-r facilities to cover graph yDraw a graph with fewest crossings yEtc. zNo polynomial-time algorithm unless P = NP zStill want to solve them

5. Dealing with Hard Graph Problems zMain theoretical approaches to solving NP-hard problems: yAverage case (phase transitions, smoothed analys.) ySpecial instances: Planar graphs, etc. yApproximation algorithms: Within some factor C of the optimal solution yFixed-parameter algorithms: Parameterize problem by parameter P (typically, the cost of the optimal solution) and aim for f(P) n O(1) (or even f(P) + n O(1) )

6. Our Results at a High Level zConsider vertex cover, dominating set, feedback vertex set, unweighted TSP, … zOr any graph optimization problem that’s yBidimensional: Large (Ω(r 2 )) on r  r grid, never increases when contracting an edge ySolvable on graphs of bounded treewidth zIn most minor-closed graph families: yTreewidth = O(√solution value) y2 O(√k) n O(1) subexponential fixed-parameter alg. y(1+ε)-approximation in 2 O(1/ε) n O(1) time r r

7. Our Results at a High Level zFor any graph problem in a large class (“bidimensional”) yVertex cover, dominating set, connected dominating set, r-dominating set, feedback vertex set, unweighted TSP, … zIn broad classes of graphs (most “minor-closed” graph families) zObtain: yStrong combinatorial properties ySubexponential fixed-parameter algorithms yPolynomial-time approximation schemes

8. Our Results at a High Level zFor any graph optimization problem that’s yBidimensional: Large (Ω(r 2 )) on r  r grid, never increases when contracting an edge ySolvable on graphs of bounded treewidth zIn most minor-closed graph families: yTreewidth = O(√solution value) y2 O(√k) n O(1) subexponential fixed-parameter alg. y(1+ε)-approximation in 2 O(1/ε) n O(1) time r r

9. Results at a very high level zFor any graph optimization problem that’s yBidimensional: Large (Ω(r 2 )) on r  r grid, never increases when contracting an edge ySolvable on graphs of bounded treewidth zThere is a fixed-parameter alg. 2 2 k 2.5 n O(1) zIn most minor-closed graph families: yTreewidth = O(√solution value) y2 O(√k) n O(1) subexponential fixed-parameter alg. yPTASs

10. delete Minors zA graph G has a minor H if H can be formed by removing and contracting edges of G zOtherwise, G is H-minor-free zFor example, planar graphs are both K 3,3 -minor-free and K 5 -minor-free contract H minor of G G *

11. Graph Minor Theory [Robertson & Seymour 1984–2004] zSeminal series of ≥ 20 papers zPowerful results on excluded minors: yEvery minor-closed graph property (preserved when taking minors) has a finite set of excluded minors [Wagner’s Conjecture] yEvery minor-closed graph property can be decided in polynomial time yFor fixed graph H, graphs minor-excluding H have a special structure: drawings on bounded-genus surfaces + “extra features”

12. Treewidth [GM2—Robertson & Seymour 1986] zTreewidth of a graph is the smallest possible width of a tree decomposition zTree decomposition spreads out each vertex as a connected subtree of a common tree, such that adjacent vertices have overlapping subtrees yWidth = maximum overlap − 1 zTreewidth 1  tree; 2  series-parallel; … Graph Tree decomposition (width 3)

13. Treewidth Basics zMany fast algorithms for NP-hard problems on graphs of small treewidth yTypical running time: 2 O(treewidth) n O(1) zComputing treewidth is NP-hard zComputable in 2 2 O(treewidth) n time, including a tree decomposition [Bodlaender 1996] zO(1)-approximable in 2 O(treewidth) n O(1) time, including a tree decomposition [Amir 2001] zO(√lg opt)-approximable in n O(1) time [Feige, Hajiaghayi, Lee 2004] (using a new framework for vertex separators based on embedding with minimum average distortion into line)

14. Treewidth Basics zMany fast algorithms for NP-hard problems on graphs of small treewidth yTypical running time: 2 O(treewidth) n O(1) zComputing treewidth is NP-hard zComputable in 2 2 O(treewidth) n time, including a tree decomposition [Bodlaender 1996] zO(1)-approximable in 2 O(treewidth) n O(1) time, including a tree decomposition [Amir 2001] z1.5-approximation for planar graphs and single-crossing- minor-free graphs zO(|V(H)|^2)- approximable in n O(1) time in H-minor- free graphs [Feige, Hajiaghayi, Lee 2004]

15. Treewidth Basics zMany fast algorithms for NP-hard problems on graphs of small treewidth yTypical running time: 2 O(treewidth) n O(1) zComputing treewidth is NP-hard zComputable in 2 2 O(treewidth) n time, including a tree decomposition [Bodlaender 1996] zO(1)-approximable in 2 O(treewidth) n O(1) time, including a tree decomposition [Amir 2001] zPlanar graphs and graphs of bounded genus have treewidth O(√‾) n

16. Grid Minors zr  r grid: yr 2 vertices, 2 r (r − 1) edges yTreewidth ~ r zr  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] ySo any planar graph of large treewidth has an r  r grid minor certifying large treewidth yWhat about nonplanar graphs? r r

17. r r Grid Minors zFor any fixed graph H, every H-minor-free graph of treewidth ≥ w(r) has an r  r grid minor [GM5—Robertson & Seymour 1986] yRe-proved & strengthened [Robertson, Seymour, Thomas 1994; Reed 1997; Diestel, Jensen, Gorbunov, Thomassen 1999] yBest bound of these: w(r) = 20 5 |V(H)| 3 r [Robertson, Seymour, Thomas 1994] yNew optimal bound: w(r) = Θ(r) [Demaine & Hajiaghayi 2005] xGrids certify large treewidth in H-minor-free graph

18. r r Bidimensionality at a High Level zParameter = function P assigning nonnegative integer P(G) to every graph G yParameter  graph optimization problem zParameter P is bidimensional if yNever increases when contracting an edge yP is “large” on r  r “grid-like graphs” typically, Ω(r 2 ) think e.g. partially triangulated grid with a few extra edges

19. Bidimensionality at a High Level zParameter = function P assigning nonnegative integer P(G) to every graph G yParameter  graph optimization problem zParameter P is bidimensional if yNever increases when contracting an edge yP is “large” on r  r “grid-like graphs” typically, Ω(r 2 ) think e.g. partially triangulated grid with a few extra edges r r

20. Graph Classes planar bounded-genus single-crossing- minor-free map graphs apex-minor-free H-minor-free general

21. Graph Classes zDrawings on surfaces: yPlanar graphs xDrawable in plane/sphere without crossings x= K 3,3 -minor-free & K 5 -minor-free graphs yBounded-genus graphs xOne family for each integer g ≥ 0 xDrawable in the genus-g orientable surface without crossings planar bounded-genus single-crossing- minor-free map graphs apex-minor-free H-minor-free general

22. Graph Classes zDrawings on surfaces: yMap graph xGiven embedded planar graph xGiven face 2-coloring as nations and lakes xVertex for each nation xEdge between nations that share a vertex yMore generally, fixed powers of graph family xMap graphs = half-square of planar bipartite planar bounded-genussingle-crossing- minor-free map graphs apex-minor-free H-minor-free general New York Times

23. Graph Classes zDrawings on surfaces: yMap graph xGiven embedded planar graph xGiven face 2-coloring as nations and lakes xVertex for each nation xEdge between nations that share a vertex yMore generally, fixed powers of graph family xMap graphs = half-square of planar bipartite planar bounded-genussingle-crossing- minor-free map graphs apex-minor-free H-minor-free general New York Times

24. Graph Classes zExcluding minors: yX-minor-free = class of graphs excluding a fixed graph from family X yH-minor-free = exclude any fixed graph H yApex graph = planar graph + one vertex + any incident edges ySingle-crossing graph = minor of a graph that can be drawn with at most one crossing xE.g., K 3,3 and K 5 planar bounded-genus single-crossing- minor-free map graphs apex-minor-free H-minor-free general

25. r r Bidimensionality zParameter P is g(r)-bidimensional (or just bidimensional) if either y“g(r)-minor-bidimensional” y“g(r)-contraction-bidimensional” zDepends on the graph class we consider zParameter P is g(r)-minor-bidimensional for an H-minor-free graph class if yP never increases when taking minors yP is at least g(r) on the r  r grid

26. r r Bidimensionality zParameter P is g(r)-contraction-bidimensional if yP never increases when contracting edges yP is at least g(r) on r  r “grid-like graphs” xPlanar graphs & single-crossing-minor-free graphs: partially triangulated r  r grid xBounded-genus graphs: partially triangulated r  r grid plus up to genus(G) additional edges xApex-minor-free graphs: r  r grid plus edges such that each vertex is adjacent to O(1) nonboundary vertices of the grid xUndefined for general graphs

27. Examples of Bidimensional Parameters zNumber of vertices zDiameter zSize of yFeedback vertex set yVertex cover yMaximal matching yFace cover yDominating set yEdge dominating set yR-dominating set yConnected … dominating set yUnweighted TSP tour (closed walk) g(r) = Θ(r 2 ) g(r) = Θ(lg r) g(r) = r 2

28. RESULTS zThe following slides include definitions of what we want, previous results, and our big improvements

29. Fixed-Parameter Tractability in General Graphs zAny minor-closed property can be decided in polynomial time [Robertson & Seymour GM] zSo any parameter that does not increase when taking minors can be decided ≤ k in polynomial time for any fixed k [Fellows & Langston 1988] zNot enough for fixed-parameter algorithm yDon’t know how to compute finite set of excluded minors given k yDon’t even know dependence of minor size on k

30. Fixed-Parameter Tractability in General Graphs zTheorem: [Demaine & Hajiaghayi 2004] Any Ω(1)-minor-bidimensional parameter P can be decided ≤ k in h(2 k 2.5 ) n O(1) time in a general graph if yP(G) ≥ sum over connected components of G yP(G) can be computed in h(treewidth(G)) n O(1)

31. Parameter-Treewidth Bounds zParameter-treewidth bound: treewidth(G) ≤ f(P(G)) for all G in family yTypically, f(k) = O(√k) yProved for various problems in planar graphs [Alber et al. 2002; Kanj & Perković 2002; Fomin & Thilikos 2003; Alber, Fernau, Niedermeier 2004; Chang, Kloks, Lee 2001; Kloks, Lee, Liu 2002; Gutin, Kloks, Lee 2001]

32. Parameter-Treewidth Bounds zParameter-treewidth bound: treewidth(G) ≤ f(P(G)) for all G in family yEvery g(r)-bidimensional parameter has a parameter-treewidth bound f(k) = g −1 (k) g −1 (k) [Demaine, Fomin, Hajiaghayi, Thilikos 2004] yImproved to f(k) = g −1 (k) for xPlanar graphs [Demaine, Fomin, Hajiaghayi, Thilikos 2003] xSingle-crossing-minor-free graphs [Demaine, Hajiaghayi, Thilikos 2002; Demaine, Hajiaghayi, Nishimura, Ragde, Thilikos 2004] xBounded-genus graphs [Demaine, Fomin, Hajiaghayi, Thilikos 2004; Demaine, Hajiaghayi, Thilikos 2004] √k

33. Parameter-Treewidth Bounds zTheorem: [Demaine, Fomin, Hajiaghayi, Thilikos 2004; Demaine & Hajiaghayi 2005] Every g(r)-bidimensional parameter has a parameter-treewidth bound with f(k) = O(g −1 (k)) yIn particular, Θ(r 2 )-bidimensional parameters have treewidth = O(√parameter)

34. Subexponential Fixed-Parameter Algorithms in H-minor-free Graphs zSubexponential fixed-parameter algorithm has running time 2 o(k) n O(1) yTypically, 2 O(√k) n O(1) yAttained for various problems in planar graphs [Alber et al. 2002; Kanj & Perković 2002; Fomin & Thilikos 2003; Alber, Fernau, Niedermeier 2004; Chang, Kloks, Lee 2001; Kloks, Lee, Liu 2002; Gutin, Kloks, Lee 2001] yExtended to bidimensional parameters in planar graphs, single-crossing-minor-free graphs, and bounded-genus graphs […]

35. Subexponential Fixed-Parameter Algorithms in H-minor-free Graphs zTheorem: [Demaine, Fomin, Hajiaghayi, Thilikos 2004; Demaine & Hajiaghayi 2005] Every g(r)-bidimensional parameter solvable in h(treewidth(G)) n O(1) time has a fixed-parameter algorithm with running time h(g -1 (k)) n O(1) yIn particular, Θ(r 2 )-bidimensional parameters lead to h(√k) n O(1) time, typically 2 O(√k) n O(1)

36. Subexponential Fixed-Parameter Algorithms in H-minor-free Graphs zFor contraction-bidimensional parameters, general result applies only to apex-minor-free graphs zDominating set has a fixed-parameter algorithm with running time 2 O(√k) n O(1) in yH-minor-free graphs [Demaine, Fomin, Hajiaghayi, Thilikos 2004] yMap graphs [Demaine, Fomin, Hajiaghayi, Thilikos 2003] yFixed powers of H-minor-free graphs

37. Polynomial-Time Approximation Schemes zSeparator approach [Lipton & Tarjan 1980] gives PTASs only when OPT (after kernelization) can be lower bounded in terms of n (typically, OPT = Ω(n)) yExamples: Various forms of TSP [Grigni, Koutsoupias, Papadimitriou 1995; Arora, Grigni, Karger, Klein, Woloszyn 1998; Grigni 2000; Grigni & Sissokho 2002] zParameter-treewidth bounds give separators in terms of OPT, not n

38. Polynomial-Time Approximation Schemes zTheorem: [Demaine & Hajiaghayi 2005] (1+ε)-approximation with running time h(O(1/ε)) n O(1) for any bidimensional optimization problem that is yComputable in h(treewidth(G)) n O(1) ySolution on disconnected graph = union of solutions of each connected component yGiven solution to G − C, can compute solution to G at an additional cost of ± O(|C|) ySolution S of G induced on connected component X of G − C has size |S  X| ± O(|C|)

39. Polynomial-Time Approximation Schemes zCorollary: [Demaine & Hajiaghayi 2005] yPTAS in H-minor-free graphs for feedback vertex set, face cover, vertex cover, minimum maximal matching, and related vertex-removal problems yPTAS in apex-minor-free graphs for dominating set, edge dominating set, R- dominating set, connected … dominating set, clique-transversal set zNo PTAS previously known for, e.g., feedback vertex set or connected dominating set, even in planar graphs

40. r r Grid Minors zFor any fixed graph H, every H-minor-free graph of treewidth w has an O(w)  O(w) grid minor [Demaine & Hajiaghayi 2004] zProof uses and extends structure of H- minor-free graphs from [GM16—Robertson & Seymour ’03]

41. Consequence: Separator Theorem zTheorem: [Demaine, Fomin, Hajiaghayi, Thilikos 2004; Demaine & Hajiaghayi 2005] For every g(r)-bidimensional parameter P, treewidth(G) ≤ g −1 (P(G)) zApply to P(G) = number of vertices in G zCorollary: For any fixed graph H, every H-minor-free graph has treewidth O(√n) [Alon, Seymour, Thomas 1990; Grohe 2003] zCorollary: 1/3-2/3 separators, size O(√n)

42. Consequence: Half-Integral Flow zTheorem: The ratio of max (fractional) multi- commodity flow and max half-integral multi- commodity flow is at most O(polylog n) in planar graphs [Chekuri, Khanna, Shepherd 2004] zAt high level they use hierarchical decomposition tree of Racke and linearity of size of grid minor in terms of treewidth. zTheorem: The ratio of max (fractional) multi- commodity flow and max half-integral multi- commodity flow is at most O(polylog n) in H-minor- free graphs for a fixed H (Using our result). zGive also the O(1) max-flow min-cut gap of [KPR’93]

43. Consequence: Local Treewidth zLocal treewidth ltw(v, r) = treewidth of radius-r neighborhood around v zBounded local treewidth: ltw(v,r) ≤ f(r) zMinor-closed graph family has bounded local treewidth iff it is apex-minor-free [Eppstein 2000; Demaine & Hajiaghayi 2004] yIn particular, if apex-minor-free, then bounded local treewidth with f(r) = 2 2 O(r) yParameter-treewidth bound with P = diameter gives f(r) = 2 O(r) (inverse of lg) yf(r) = O(r) [Demaine & Hajiaghayi 2004] G r=1 r=2 v

44. Approximation Algorithms based on Bounded Local Treewidth zBaker’s approach for approximation gives PTASs for planar graphs [Baker 1994] yMinimum vertex cover yMinimum dominating set  generally APX-hard yMaximum independent set yMaximum H-matching ySubgraph isomorphism for a fixed pattern zExtends to graphs of bounded local treewidth [Eppstein 2000] zImprove time 2 2 2 O(1/ε) n O(1)  2 O(1/ε) n O(1)

45. r r Grid Minors: High-level Proof zFor any fixed graph H, every H-minor-free graph of treewidth w has an O(w)  O(w) grid minor [Demaine & Hajiaghayi 2004] zProof uses and extends structure of H- minor-free graphs from [GM16—Robertson & Seymour ’03]

46. Structure of H-minor-free Graphs [GM16—Robertson & Seymour 2003] zEvery H-minor-free graph can be written as O(1)-clique sums of graphs zEach summand is a graph that can be O(1)-almost-embedded into a bounded-genus surface zO(1) constants depend only on |V(H)|

47. G2G2 G1G1 Clique Sums zConsider vertex-disjoint graphs G 1 and G 2 ySuppose both graphs have clique of size k ≥ 1 zk-clique sum combines G 1 & G 2 as follows: yLabel each clique with {1, …, k} yContract corresp. vertices between cliques yRemove any number of edges from new clique 1 2 34 5 K5K5 K5K5 G 1 2 34 5

48. Connection Between Clique Sums and Treewidth zLemma: [HNRT 2001] For any two graphs G 1 and G 2, the treewidth of their clique sum is at most the maximum of their resp. treewidths yAny tree decomp. of G 1 or G 2 has a vertex v through which all nodes of clique are spread yConnect v 1 and v 2 and combine tree decomps. 1 2 34 5 K5K5 v1v1 v2v2 G 1 ’ s tree decomp. G 2 ’ s tree decomp.

49. Almost-Embeddable Graphs zA graph is O(1)-almost-embeddable into a bounded-genus surface if it is yA bounded-genus graph y+ a bounded number of vortices: xVortex = Replace a face in the bounded-genus graph by a graph of bounded pathwidth xThe interiors of the replaced faces are disjoint y+ a bounded number of apices: xApex = extra vertex with any incident edges

50. A Series of Reductions zAt least one term has the same treewidth zRemove apices by deletion zContract each vortex down to a vertex z“Approximate” almost-embeddable graph so that it is a minor of original graph (and thus H-minor-free) yProblem: Edges added from clique sum zBounded-genus graph has large grid minor

51. A Series of Reductions zContract each vortex down to a vertex z“Approximate” almost-embeddable graph so that it is a minor of original graph  H-minor-free yProblem: Edges added from clique sum zIf bounded-genus part is “compact”, planarize by cutting small noose and making vertices apices zAlmost-planar graph has O(tw) × O(tw) grid minor zIf bounded-genus part is “spread out”, carve into almost-embeddable graphs, where all except one are almost-planar and have no vortices zRemaining almost-nonplanar-embeddable graph has all neighbors of apices in the vortices zNow apices restricted by H-minor-freeness

52. PTASs zGeneral results zBaker’s Approach, relation to Eppstein, generalization

53. Open Problems zGeneral graphs (beyond H-minor-free) yMinor-bidimensionality can be generalized: still grid-like graph = r  r grid yBest known grid bound: treewidth(G) ≥ 20 2r 5 implies G has an r  r grid minor [Robertson, Seymour, Thomas 1994] ySome graphs of treewidth Ω(r 2 lg r) have no grid larger than O(r)  O(r) [Robertson, Seymour, Thomas 1994] yConjecture: r Θ(1)

54. Open Problems zGeneral graphs (beyond H-minor-free) yContraction-bidimensionality cannot be generalized beyond apex-minor-free graphs and still obtain parameter-treewidth bounds, for e.g. dominating set [Demaine, Fomin, Hajiaghayi, Thilikos 2004] yIs there a theory of graph contractions for handling contraction-closed properties? yHave rough analogs of grid-minor theorems

55. Open Problems zDependence of constant factors on fixed excluded minor H yImportant: In exponents of running times yHeart of constants is in grid-minor bound: treewidth w implies O(w)  O(w) grid minor yΩ( √ |V(H)| lg |V(H)|) yConjecture: |V(H)| O(1) or even O(|V(H)|) xLatter bound would improve factor in H-minor-free separator theorem to conjectured bound O(|V(H)|) (from |V(H)| 3/2 [Alon, Seymour, Thomas 1990])

56. Open Problems zGeneralize PTASs beyond bidimensional parameters yAdd weights to vertices and/or edges and desire minimum-weight e.g. dominating set xBeing large on grid no longer well-defined; depends on weights of vertices in chosen grid ySubset-type problems e.g. Steiner tree or subset feedback vertex set xBeing large on grid no longer well-defined; depends on which vertices in grid are in the subset r r

57. References zEDD & MTH, “Bidimensionality: New Connections between FPT Algorithms and PTASs”, SODA 2005. zEDD & MTH, “Graphs Excluding a Fixed Minor have Grids as Large as Treewidth, with Combinatorial and Algorithmic Applications through Bidimensionality”, SODA 2005. zEDD, MTH & DMT, “Exponential Speed- up of FPT Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors”, Algorithmica 2005. zEDD, FVF, MTH, & DMT, “Fixed- Parameter Algorithms for (k, r)-Center in Planar Graphs and Map Graphs”, ACM Trans. on Algorithms 2005. zEDD, MTH, Naomi Nishimura, Prabhakar Ragde, & DMT, “Approx- imation algorithms for classes of graphs excluding single-crossing graphs as minors”, J. Comp & System Sci.2004. zEDD, MTH, Ken-ichi Kawarabayashi, “Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring’’, FOCS 2005. EDD = Erik D. Demaine, MTH = MohammadTaghi Hajiaghayi FVF = Fedor Fomin, DMT = Dimitrios Thilikos zEDD & MTH, “Fast Algorithms for Hard Graph Problems: Bidimen- sionality, Minors, and Local Treewidth”, GD 2004. zEDD, FVF, MTH, & DMT, “Bidimensional Parameters and Local Treewidth”, SIDMA 2005. zEDD & MTH, “Diameter and Tree- width in Minor-Closed Graph Fami- lies, Revisited”, Algorithmica 2004. zEDD, MTH, & DMT, “The Bidimen- sional Theory of Bounded-Genus Graphs”, SIDMA 2005. zEDD & MTH, “Equivalence of Local Treewidth and Linear Local Treewidth and its Algorithmic Applications”, SODA 2004. zEDD, FVF, MTH, & DMT, “Subexpo- nential parameterized algorithms on graphs of bounded genus and H- minor-free graphs”, JACM 2005. zUriel Feige, MTH & James R. Lee, “Improved approximation algorithms for vertex separators”, STOC 2005.

58. Thank you.