Bidimensionality and Approximation Algorithms r r Mohammad T
Bidimensionality and Approximation Algorithms r r Mohammad T. Hajiaghayi UMD
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
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)
Beyond Planar Graphs · A graph G has a minor H if H can be formed by removing and contracting edges of G G delete * H minor of G contract • 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
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”
Treewidth [GM 2—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 Tree Graph decomposition overlapping subtrees ▪ Width = maximum overlap − 1 (width 3) · Treewidth 1 tree; 2 series-parallel; …
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 O(treewidth) 2 · Computable in 2 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)
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 O(treewidth) 2 · Computable in 2 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 singlecrossing-minor-free graphs [EDD, MTH, NN, PR, DMT] · O(|V(H)|^2)-approximable in n. O(1) time in H-minorfree graphs [Feige, Hajiaghayi, Lee 2004]
Bidimensionality (version 1) · Parameter k is minor-bidimensional if ▪ Closed under minors: k does not increase when deleting or contracting edges v and w v w delete vw contract ▪ Large on grids: For the r r grid, k = Ω(r 2) and more generally Ω(f(r)) r r
Example 1: Vertex Cover · k = minimum number of vertices required to cover every edge (on either endpoint) cover v w v w · Closed under minors: v w delete still a cover (only fewer edges) vw contract still a cover, possibly 1 smaller
Example 1: Vertex Cover · k = minimum number of vertices required to cover every edge (on either endpoint) cover v w v w · Large on grids: ▪ Matching of size Ω(r 2) ▪ Every edge must be covered by a different vertex r r
Bidimensionality (version 2) · Parameter k is contractionbidimensional 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) v w vw contract
Example 2: Dominating Set · k = minimum number of vertices required to cover every vertex or its neighbor v cover w u v w u … v w u · Large on grids: ▪ Ω(r 2) vertex-disjoint cycles ▪ Every cycle must be covered by a different vertex r r
Example 2: Dominating Set · k = minimum number of vertices required to cover every vertex or its neighbor v w cover v u w u v w u … v w u · Closed under contraction but not minor: v w delete Not necessarily a cover anymore vw contract still a cover, possibly 1 smaller
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) v w vw contract
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 very large k [minor theory] [bidimensional]
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]
Bidimensional Subexponential FPT v w u · Corollary 1: Vertex cover and feedback vertex set have subexponential fixedparameter 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]
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]
Bidimensional PTAS v w u · 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
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-minorfree graph has treewidth O(√ 8 n) [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 2 n/3)
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
Application to Ind. Set (cont’d) • The maximum number of levels is at most log 3/2 n • Say C is the union of all separator (ignored) vertices. • 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)
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
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|)
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, Rdominating 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
SIMPLIFYING DECOMPOSITIONS
Graph Decomposition Separator Decomposition Small separator Small pieces … … … … [Lipton & Tarjan 1980; …] Simplifying Decomposition Large interaction Simple pieces (e. g. bounded treewidth)
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 et al. 2004; Demaine, Hajiaghayi, Kawarabayashi, FOCS’ 05] [De. Vos
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
Example: Graph Coloring [Demaine, Hajiaghayi, Kawarabayashi 2005/2010] · 2 -approximation for chromatic number in odd-H-minor-free graphs General using decomposition into two graphs: Inapprox. bounded-treewidth pieces: within n 1−ε unless ZPP = NP
Simplifying Graph Decompositions [De. Vos 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 Max. Cut
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 …
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], apexminor-free [Demaine, Hajiaghayi, Kawarabayashi 2009]
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]
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)
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 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]
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]
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 2Õ(k) n + n. O(1) time · Generalization to arbitrary graphs [Kawarabayashi & Thorup 2011]
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]
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
Neighborhoods of Shortest Paths have Bounded Treewidth
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!
IMPROVING GRAPH MINORS
Graph Minors [Robertson&Seymour 1983– 2004] in ≥ 20 papers … …
Nonconstructive Graph Minors · Theorem: Every H-minorfree 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”) [GM 16: Robertson & Seymour 2003]
Constructive Graph Minors · Theorem: Every H-minorfree graph can be written as a tree of graphs joined along f(H)-size cliques ▪ Computable in nf(H) time [Demaine, Hajiaghayi, Kawarabayashi, FOCS 2005] ▪ Weaker form in f(H) n. O(1) time [Dawar, Grohe, Kreutzer 2007]
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 [GM 5—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)|)
Beyond Bidimensionality · Nontrivial weights ▪ Min-weight k disjoint paths? · Directed networks (with Rajesh) k=3 ▪ 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
Thanks for your attention… 50
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