1. Bidimensionality and Approximation Algorithms Mohammad T. Hajiaghayi UMD r r

2. Dealing with Hard Network Design Problems  Main (theoretical) approaches to solve NP-hard problems: ▪Special instances: Planar graphs (fiber networks in ground), etc. ▪Approximation algorithms (PTAS): Within a factor C of the optimal solution (PTAS if C= 1+ ε for arbitrary constant ε) ▪Fixed-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) )  We consider all above in Bidimentionality and aim for general algorithmic frameworks

3. Overview  For any network design problem in a large class (“bidimensional”) ▪Vertex cover, dominating set, connected dominating set, r-dominating set, feedback vertex set, TSP, k-cut, Steiner tree, Steiner forest, multiway cut,…  In broad classes of networks generalizing planar networks (most “minor-closed” graph families)  We Obtain (in a series of more than 25 papers): ▪Strong combinatorial properties ▪Fixed-parameter algorithms ◦Often subexponential: 2 O(√k) n O(1) where k=|OPT| ▪Approximation algorithms ◦Often PTASs (1+ ε approx): f(1/ε) n O(1)

4. Beyond Planar Graphs  A graph G has a minor H if H can be formed by removing and contracting edges of G delete contract H minor of G G * Otherwise, if G exclude H as a minor is called an H-minor-free graph For example, planar graphs are both K 3,3 -minor-free and K 5 -minor-free

5. Graph Minor Theory [Robertson & Seymour 1984–2004]  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”

6. Treewidth [GM2—Robertson & Seymour 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 overlapping subtrees ▪Width = maximum overlap − 1  Treewidth 1  tree; 2  series-parallel; … Graph Tree decomposition (width 3)

7. Treewidth Basics  Many fast algorithms for NP-hard problems on graphs of small treewidth ▪Typical running time: 2 O(treewidth) n O(1)  Computing treewidth is NP-hard  Computable in 2 2 O(treewidth) n time, including a tree decomposition [Bodlaender 1996]  O(1)-approximable in 2 O(treewidth) n O(1) time, including a tree decomposition [Amir 2001]  O(√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)

8. Treewidth Basics  Many fast algorithms for NP-hard problems on graphs of small treewidth ▪Typical running time: 2 O(treewidth) n O(1)  Computing treewidth is NP-hard  Computable in 2 2 O(treewidth) n time, including a tree decomposition [Bodlaender 1996]  O(1)-approximable in 2 O(treewidth) n O(1) time, including a tree decomposition [Amir 2001]  1.5-approximation for planar graphs and single- crossing-minor-free graphs [EDD,MTH,NN,PR,DMT]  O(|V(H)|^2)- approximable in n O(1) time in H-minor- free graphs [Feige, Hajiaghayi, Lee 2004]

9. Bidimensionality (version 1)  Parameter k is minor-bidimensional if ▪Closed under minors: k does not increase when deleting or contracting edges and ▪Large on grids: For the r  r grid, k = Ω(r 2 ) and more generally Ω(f(r)) r r vwvw delete vw contract

10. Example 1: Vertex Cover  k = minimum number of vertices required to cover every edge (on either endpoint)  Closed under minors:  still a cover (only fewer edges)  still a cover, possibly 1 smaller vwvw delete vw contract vwvwvwvw cover

11. Example 1: Vertex Cover  k = minimum number of vertices required to cover every edge (on either endpoint)  Large on grids: ▪Matching of size Ω(r 2 ) ▪Every edge must be covered by a different vertex vwvwvwvw cover r r

12. Bidimensionality (version 2)  Parameter k is contraction- bidimensional if ▪Closed under contractions: k does not increase when contracting edges and ▪Large on a grid-like graph: For naturally triangulated r  r grid graphs, k = Ω(r 2 ) vw vw contract

13. Example 2: Dominating Set  k = minimum number of vertices required to cover every vertex or its neighbor  Large on grids: ▪Ω(r 2 ) vertex-disjoint cycles ▪Every cycle must be covered by a different vertex vw cover u vw u vw u vw u … r r

14. Example 2: Dominating Set  k = minimum number of vertices required to cover every vertex or its neighbor  Closed under contraction but not minor:  Not necessarily a cover anymore  still a cover, possibly 1 smaller vwvw delete vw contract vw cover u vw u vw u vw u …

15. Contraction-Bidimensional Problems  Minimum maximal matching  Face cover (planar graphs)  Dominating set  Edge dominating set  R-dominating set  Connected … dominating set  Unweighted TSP tour  Chordal completion (fill-in) vw vw contract

16. Bidimensional  Relate Parameter & Treewidth  Theorem 1: If a parameter k is bidimensional, then it satisfies parameter-treewidth bound treewidth = O(√k) in any graph family excluding some minor [Demaine, Fomin, Hajiaghayi, Thilikos, JACM 2005; Demaine & Hajiaghayi, Combinatorica 2010]  Proof sketch: Large treewidth  very large grid [minor theory]  very large k [bidimensional] &

17. Bidimensional  Subexponential FPT  Theorem 2: If a parameter k is ▪bidimensional, and ▪fixed-parameter tractable on graphs of bounded treewidth: h(treewidth) n O(1) time then it has a subexponential fixed-parameter algorithm: h(√k) n O(1) time in any graph family excluding some minor ▪Typically 2 O(√k) n O(1) time (h(w) = 2 O(w) ) [Demaine, Fomin, Hajiaghayi, Thilikos 2004; Demaine & Hajiaghayi 2005]  Proof sketch: Run bounded-treewidth algorithm (tw = O(√k)) [If (approx.) treewidth is large, answer NO] &

18. Bidimensional  Subexponential FPT  Corollary 1: Vertex cover and feedback vertex set have subexponential fixed- parameter algorithms: 2 O(√k) n O(1) time in any graph family excluding some minor [Demaine, Fomin, Hajiaghayi, Thilikos 2004; Demaine & Hajiaghayi 2005] ▪Previously known for vertex cover (and some other problems) on 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] vw vw u

19. Bidimensional  PTAS  Theorem 3: If a parameter is ▪bidimensional, ▪fixed-parameter tractable on graphs of bounded treewidth: h(treewidth) n O(1) time, ▪O(1)-approximable in polynomial time, and ▪satisfies the “separation property” then it has an PTAS: (1+ε)-approximation in h(O(1/ε)) n O(1) time in any graph family excluding some minor [Demaine & Hajiaghayi, SODA’05] &

20. Bidimensional  PTAS  Corollary 3: Vertex cover and feedback vertex set have PTASs in any graph family excluding some minor [Demaine & Hajiaghayi 2005] ▪Previously known for vertex cover (and many, many other problems) on planar graphs ▪E.g., feedback vertex set result is new, even for planar graphs vw vw u

21. Consequence: Separator Theorem  Theorem: [Demaine, Fomin, Hajiaghayi, Thilikos 2004; Demaine & Hajiaghayi 2005] For every bidimensional parameter P, treewidth(G) ≤ √P(G)  Apply to P(G) = number of vertices in G  Corollary: For any fixed graph H, every H- minor-free graph has treewidth O(√8n) [Alon, Seymour, Thomas 1990; Grohe 2003]  Corollary: 1/3-2/3 separators, size O(√n) (A vertex set whose removal leaves no component of size greater than 2n/3)

22. Application to Independent Set (Lipton-Tarjan 1980)  Independent Set: a set of vertices with no edges in between  Note that OPT is at least n/4 since planar graphs are 4-colorable  For PTAS break each component of greater than εn (=log n) and ignore separator vertices  Solve each component individually and take their union as the final solution  Consider a laminar family: level 0 are leaves

23. Application to Ind. Set (cont’d) Note that l<= n/ ((3/2) i-1 ε n) since the size of a level I component is at least (3/2) i-1 ε n Let ε= log n/n, so εn= log n (and thus we can solve each component individually in 2 log n = n time) So the total number of ignored vertices is at most n/ (√ log n)< ε n/4<= ε OPT (In each component we are not worse than OPT) The maximum number of levels is at most log 3/2 n Say C is the union of all separator (ignored) vertices.

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

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

26. Polynomial-Time Approximation Schemes  Corollary: [Demaine & Hajiaghayi 2005] ▪PTAS in H-minor-free graphs for feedback vertex set, face cover, vertex cover, minimum maximal matching, and related vertex-removal problems ▪PTAS in apex-minor-free graphs for dominating set, edge dominating set, R- dominating set, connected … dominating set, clique-transversal set  No PTAS previously known for, e.g., feedback vertex set or connected dominating set, even in planar graphs

27. SIMPLIFYING DECOMPOSITIONS

28. Graph Decomposition Separator Decomposition Simplifying Decomposition Small separatorLarge interaction Small piecesSimple pieces (e.g. bounded treewidth) ………… … … … … [Lipton & Tarjan 1980; …]

29. Simplifying Graph Decomposition [Demaine, Hajiaghayi, Kawarabayashi, SODA 2010]  Theorem: Odd H-minor-free graphs can have their vertices or edges partitioned into two pieces such that each induced graph has bounded treewidth ▪Previously for planar graphs [Baker 1994], apex-minor-free [Eppstein 2000], H-minor-free [DeVos et al. 2004; Demaine, Hajiaghayi, Kawarabayashi, FOCS’05]

30. Example: Graph Coloring  Chromatic number: Use fewest colors to color the vertices of a graph such that no two equal colors connected by an edge ▪Classic NP-hard problem ▪Inapproximable within n 1−ε unless ZPP = NP Martin Gardner, April 1, 1975

31. Example: Graph Coloring [Demaine, Hajiaghayi, Kawarabayashi 2005/2010]  2-approximation for chromatic number in odd-H-minor-free graphs using decomposition into two bounded-treewidth pieces: General graphs: Inapprox. within n 1−ε unless ZPP = NP

32. Simplifying Graph Decompositions [DeVos et al. 2004; Demaine, Hajiaghayi, Kawarabayashi 2005]  Generalization to k pieces: H-minor-free graphs can have their vertices or edges partitioned into k pieces such that deleting any one piece results in bounded treewidth ▪Useful for PTASs for minor-closed properties (where k ~ 1/ε) ▪(Not true for odd-minor) ▪Application: e.g. PTAS for MaxCut

33. Many Problems Closed Under Contractions but not Deletions  Dominating set  Edge dominating set  R-dominating set  Connected … dominating set  Face cover (planar graphs)  Minimum maximal matching  Chordal completion (fill-in)  Traveling Salesman Problem  …

34. Contraction Decomposition [Demaine, Hajiaghayi, Kawarabayashi, STOC’11]  Theorem: H-minor-free graphs can have their edges partitioned into k pieces such that contracting any one piece results in bounded treewidth ▪Polynomial-time algorithm ▪Previously known for planar [Klein 2005, 2006], bounded-genus [Demaine, Hajiaghayi, Mohar 2007], apex- minor-free [Demaine, Hajiaghayi, Kawarabayashi 2009]

35. Applications  Lots of applications via a general theorem,e.g.  Corollary 1: PTAS for Traveling Salesman Problem in weighted H-minor-free graphs [Demaine, Hajiaghayi, Kawarabayashi 2011] solving an open problem of [Grohe 2001]  Coroallary 2: Fixed-Parameter Algorithm for k-cut and Bisection on planar graphs and H-minor-free graphs [Demaine, Hajiaghayi, Kawarabayashi 2011] solving an open problem of [Downey, Estivill-Castro, Fellows 2003]

36. Application to TSP  Corollary: PTAS for Traveling Salesman Problem in weighted H-minor-free graphs [Demaine, Hajiaghayi, Kawarabayashi 2011] ▪Existing bounded-treewidth algorithm [Dorn, Fomin, Thilikos 2006] ▪Existing spanner [Grigni, Sissokho 2002] ▪Decontraction: Euler tour (cost ≤ 2 weight) + perfect matching on odd-degree vxs (cost ≤ weight)

37. Graph TSP History  PTAS for unweighted planar [Grigni, Koutsoupias, Papadimitriou 1995]  PTAS for weighted planar [Arora, Grigni, Karger, Klein, Woloszyn 1998]  Linear PTAS for weighted planar [Klein 2005]  QPTAS (n (1/ε) O(log log n) time) for weighted bounded-genus / unweighted H-minor-free [Grigni 2000]  PTAS for weighted bounded genus [Demaine, Hajiaghayi, Mohar 2007]  PTAS for unweighted apex-minor-free [Demaine, Hajiaghayi, Kawarabayashi 2009]  PTAS for weighted H-minor-free [DHK 2011]

38. Application Beyond TSP  Corollary: PTAS for minimum-weight c-edge-connected submultigraph in H-minor-free graphs [Demaine, Hajiaghayi, Kawarabayashi 2011]  Previous results: ▪PTASs for 2-edge-connected in planar graphs [Klein 2005] (linear) [Berger, Czumaj, Grigni, Zhao 2005] [Czumaj, Grigni, Sissokho, Zhao 2004] ▪PTAS for c-edge-connected in bounded-genus graphs [Demaine, Hajiaghayi, Mohar 2007]

39. Fixed-Parameter Algorithmic Applications: k-cut  k-cut: Remove fewest edges to make at least k connected components  FPT in H-minor-free graphs: ▪Average degree c H = O(H lg H ) ▪  OPT ≤ c H k ▪  Contraction decomposition with c H k + 1 layers avoids OPT in some contraction ▪  Solve in 2 Õ(k) n + n O(1) time  Generalization to arbitrary graphs [Kawarabayashi & Thorup 2011] √‾‾‾

40. Proof Sketch  H-minor-free graph = “tree” of “almost-embeddable graphs” [Graph Minors]  Each almost-embeddable graph has contraction decomposition: ▪Bounded genus done ▪Apices easy: increase treewidth of anything by O(1) ▪Vortices similar [Demaine, Hajiaghayi, Mohar 2007]

41. Radial Coloring for Bounded Genus  Color edge at radial distance r as r mod k ▪Radial graph ≈ primal graph + dual graph  Any k consecutive layers have bounded treewidth, provided first k do

42. Neighborhoods of Shortest Paths have Bounded Treewidth

43. Contraction Decomposition [Demaine, Hajiaghayi, Kawarabayashi 2011]  Theorem: H-minor-free graphs can have their edges partitioned into k pieces such that contracting any one piece results in bounded treewidth ▪Polynomial-time algorithm  Seems a powerful tool for approximation & fixed-parameter algorithms  Let’s find more applications!

44. IMPROVING GRAPH MINORS

45. Graph Minors [Robertson&Seymour 1983–2004] in ≥ 20 papers … …

46. Nonconstructive Graph Minors  Theorem: Every H-minor- free graph can be written as a tree of graphs joined along f(H)-size cliques ▪Each term is a graph that can be almost embedded into a bounded-genus surface (f(H) “vortices” and “apices”) [GM16: Robertson & Seymour 2003]

47. Constructive Graph Minors  Theorem: Every H-minor- free graph can be written as a tree of graphs joined along f(H)-size cliques ▪Computable in n f(H) time [Demaine, Hajiaghayi, Kawarabayashi, FOCS 2005] ▪Weaker form in f(H) n O(1) time [Dawar, Grohe, Kreutzer 2007]

48. Grid Minors  Every H-minor-free graph of treewidth ≥ f(H) r has an r  r grid minor [Demaine & Hajiaghayi, SODA 2005, Combinatorica 2010] ▪Previous bounds exponential in r and H [GM5—Robertson & Seymour 1986; Robertson, Seymour, Thomas 1994; Reed 1997; Diestel, Jensen, Gorbunov, Thomassen 1999]  Open: What is f(H)? ▪Ω(√|V(H)| lg |V(H)|) ▪Conjecture: |V(H)| O(1) or even O(|V(H)|)

49. Beyond Bidimensionality  Nontrivial weights ▪Min-weight k disjoint paths?  Directed networks (with Rajesh) ▪Useful notion of treewidth?  Subset problems (with Rajesh and Marek) ▪Steiner tree, subset TSP, etc. have PTASs up to bounded-genus graphs [Borradaile, Mathieu, Klein 2007; Borradaile, Demaine, Tazari 2009] ▪Steiner forest has PTAS in planar graphs [Bateni, Hajiaghayi, Marx 2010] ▪Wanted: A more general framework k = 3

50. 50 Thanks for your attention…