Bidimensionality and Approximation AlgorithmsMohammad T. HajiaghayiUMD

Dealing with Hard Network Design ProblemsMain (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+ for arbitrary constant )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)) We consider all above in Bidimentionality and aim for general algorithmic frameworks

OverviewFor 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 propertiesFixed-parameter algorithmsOften subexponential: 2O(k) nO(1) where k=|OPT|Approximation algorithmsOften PTASs (1+ approx): f(1/) nO(1)

Beyond Planar GraphsA graph G has a minor H if H can be formed by removing and contracting edges of G

* Otherwise, if G exclude H as a minor is called an H-minor-free graph For example, planar graphs are both K3,3-minor-free and K5-minor-free

Graph Minor Theory[Robertson & Seymour 19842004]Seminal series of 20 papersPowerful results on excluded minors:Every minor-closed graph property (preserved when taking minors) has a finite set of excluded minors [Wagners Conjecture]Every minor-closed graph property can be decided in polynomial timeFor fixed graph H, graphs minor-excluding H have a special structure: drawings on bounded-genus surfaces + extra features

Treewidth[GM2Robertson & Seymour 1986]Treewidth of a graph is the smallest possible width of a tree decompositionTree decomposition spreads out each vertex as a connected subtree of a common tree, such that adjacent vertices have overlapping subtreesWidth = maximum overlap 1Treewidth 1 tree; 2 series-parallel; GraphTree decomposition(width 3)

Treewidth BasicsMany fast algorithms for NP-hard problems on graphs of small treewidthTypical running time: 2O(treewidth) nO(1)Computing treewidth is NP-hardComputable in 22O(treewidth) 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)

Treewidth BasicsMany fast algorithms for NP-hard problems on graphs of small treewidthTypical running time: 2O(treewidth) nO(1)Computing treewidth is NP-hardComputable in 22O(treewidth) 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 [EDD,MTH,NN,PR,DMT]O(|V(H)|^2)-approximable in nO(1) time in H-minor-free graphs [Feige, Hajiaghayi, Lee 2004]

Bidimensionality (version 1)Parameter k is minor-bidimensional ifClosed under minors: k does not increase when deleting or contracting edges

and

Large on grids: For the r r grid, k = (r2) and more generally (f(r))

Example 1: Vertex Coverk = 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 smallervwvwvwvwcover

Example 1: Vertex Coverk = minimum number of vertices required to cover every edge (on either endpoint)

Large on grids:Matching of size (r2)Every edge must be covered by a different vertexvwvwvwvwcover

Bidimensionality (version 2)

Parameter k is contraction- bidimensional ifClosed under contractions: k does not increase when contracting edges

and

Large on a grid-like graph: For naturally triangulated r r grid graphs, k = (r2)

Example 2: Dominating Setk = minimum number of vertices required to cover every vertex or its neighbor

Large on grids:(r2) vertex-disjoint cyclesEvery cycle must be covered by a different vertexvwcoveruvwuvwuvwu

Example 2: Dominating Setk = 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 smallervwcoveruvwuvwuvwu

Contraction-Bidimensional ProblemsMinimum maximal matchingFace cover (planar graphs)Dominating setEdge dominating setR-dominating setConnected dominating setUnweighted TSP tourChordal completion (fill-in)

Bidimensional RelateParameter & TreewidthTheorem 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]&

Bidimensional Subexponential FPTTheorem 2: If a parameter k isbidimensional, andfixed-parameter tractable on graphs of bounded treewidth: h(treewidth) nO(1) time then it has a subexponential fixed-parameter algorithm: h(k) nO(1) time in any graph family excluding some minorTypically 2O(k) nO(1) time (h(w) = 2O(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 FPTCorollary 1: Vertex cover and feedback vertex set have subexponential fixed-parameter algorithms: 2O(k) nO(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 & Perkovi 2002; Fomin & Thilikos 2003; Alber, Fernau, Niedermeier 2004; Chang, Kloks, Lee 2001; Kloks, Lee, Liu 2002; Gutin, Kloks, Lee 2001]

vwvwu

Bidimensional PTASTheorem 3: If a parameter isbidimensional,fixed-parameter tractable on graphs of bounded treewidth: h(treewidth) nO(1) time,O(1)-approximable in polynomial time, andsatisfies the separation property then it has an PTAS: (1+)-approximation in h(O(1/)) nO(1) time in any graph family excluding some minor [Demaine & Hajiaghayi, SODA05]&

Bidimensional PTASCorollary 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 graphsE.g., feedback vertex set result is new, even for planar graphsvwvwu

Consequence: Separator TheoremTheorem: [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 GCorollary: 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)

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

Application to Ind. Set (contd) Note that l

Polynomial-TimeApproximation SchemesSeparator approach [Lipton & Tarjan 1980] gives PTASs only when OPT (after kernelization) can be lower bounded in terms of n (typically, OPT = (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-TimeApproximation SchemesTheorem: [Demaine & Hajiaghayi 2005] (1+)-approximation with running time h(O(1/)) nO(1) for any bidimensional optimization problem that isComputable in h(treewidth(G)) nO(1)Solution on disconnected graph = union of solutions of each connected componentGiven 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-TimeApproximation SchemesCorollary: [Demaine & Hajiaghayi 2005]PTAS in H-minor-free graphs for feedback vertex set, face cover, vertex cover, minimum maximal matching, and related vertex-removal problemsPTAS in apex-minor-free graphs for dominating set, edge dominating set, R-dominating set, connected dominating set, clique-transversal setNo PTAS previously known for, e.g., feedback vertex set or connected dominating set, even in planar graphs

SIMPLIFYINGDECOMPOSITIONS

Graph Decomposition[Lipton & Tarjan 1980; ]

Separator DecompositionSimplifying DecompositionSmall separatorLarge interactionSmall piecesSimple 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 [DeVos et al. 2004; Demaine, Hajiaghayi, Kawarabayashi, FOCS05]

Example: Graph ColoringChromatic number: Use fewest colors to color the vertices of a graph such that no two equal colors connected by an edgeClassic NP-hard problemInapproximable within n1 unless ZPP = NPMartin Gardner, April 1, 1975

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 n1 unless ZPP = NP

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 treewidthUseful for PTASs for minor-closed properties (where k ~ 1/)(Not true for odd-minor)Application: e.g. PTAS for MaxCut

Many Problems Closed Under Contractions but not DeletionsDominating setEdge dominating setR-dominating setConnected dominating setFace cover (planar graphs)Minimum maximal matchingChordal completion (fill-in)Traveling Salesman Problem

Contraction Decomposition[Demaine, Hajiaghayi, Kawarabayashi, STOC11]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 algorithmPreviously known for planar [Klein 2005, 2006], bounded-genus [Demaine, Hajiaghayi, Mohar 2007], apex-minor-free [Demaine, Hajiaghayi, Kawarabayashi 2009]

ApplicationsLots 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 TSPCorollary: 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 HistoryPTAS 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/) 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]

Application Beyond TSPCorollary: 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-cutk-cut: Remove fewest edges to make at least k connected componentsFPT in H-minor-free graphs:Average degree cH = O(H lg H ) OPT cH k Contraction decomposition with cH k + 1 layers avoids OPT in some contraction Solve in 2(k) n + nO(1) timeGeneralization to arbitrary graphs [Kawarabayashi & Thorup 2011]

Proof SketchH-minor-free graph = tree of almost-embeddable graphs [Graph Minors]Each almost-embeddable graph has contraction decomposition:Bounded genus doneApices easy: increase treewidth of anything by O(1)Vortices similar[Demaine, Hajiaghayi, Mohar 2007]

Radial Coloring for Bounded GenusColor edge at radial distance r as r mod kRadial graph primal graph + dual graphAny 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 treewidthPolynomial-time algorithm

Seems a powerful tool for approximation & fixed-parameter algorithmsLets find more applications!

IMPROVINGGRAPH MINORS

Graph Minors[Robertson&Seymour 19832004] in 20 papers

NonconstructiveGraph MinorsTheorem: Every H-minor- free graph can be written as a tree of graphs joined along f(H)-size cliquesEach term is a graph that can be almost embedded into a bounded-genus surface (f(H) vortices and apices) [GM16: Robertson & Seymour 2003]

ConstructiveGraph MinorsTheorem: Every H-minor- free graph can be written as a tree of graphs joined along f(H)-size cliquesComputable in nf(H) time [Demaine, Hajiaghayi, Kawarabayashi, FOCS 2005]Weaker form in f(H) nO(1) time [Dawar, Grohe, Kreutzer 2007]

Grid MinorsEvery 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 [GM5Robertson & Seymour 1986; Robertson, Seymour, Thomas 1994; Reed 1997; Diestel, Jensen, Gorbunov, Thomassen 1999]Open: What is f(H)?(|V(H)| lg |V(H)|)Conjecture: |V(H)|O(1) or even O(|V(H)|)

Beyond BidimensionalityNontrivial weightsMin-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 frameworkk = 3

*Thanks for your attention

***xxx check 1984 datethis is min of GM2 and GM3**