Bidimensionality and Approximation Algorithms
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Bidimensionality and Approximation AlgorithmsMohammad T. HajiaghayiUMD

Dealing with Hard Network Design ProblemsMain (theoretical) approaches to solve NPhard 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 )Fixedparameter 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, rdominating set, feedback vertex set, TSP, kcut, Steiner tree, Steiner forest, multiway cut,In broad classes of networks generalizing planar networks (most minorclosed graph families)We Obtain (in a series of more than 25 papers):Strong combinatorial propertiesFixedparameter algorithmsOften subexponential: 2O(k) nO(1) where k=OPTApproximation 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 Hminorfree graph For example, planar graphs are both K3,3minorfree and K5minorfree

Graph Minor Theory[Robertson & Seymour 19842004]Seminal series of 20 papersPowerful results on excluded minors:Every minorclosed graph property (preserved when taking minors) has a finite set of excluded minors [Wagners Conjecture]Every minorclosed graph property can be decided in polynomial timeFor fixed graph H, graphs minorexcluding H have a special structure: drawings on boundedgenus 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 seriesparallel; GraphTree decomposition(width 3)

Treewidth BasicsMany fast algorithms for NPhard problems on graphs of small treewidthTypical running time: 2O(treewidth) nO(1)Computing treewidth is NPhardComputable 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 NPhard problems on graphs of small treewidthTypical running time: 2O(treewidth) nO(1)Computing treewidth is NPhardComputable 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]1.5approximation for planar graphs and singlecrossingminorfree graphs [EDD,MTH,NN,PR,DMT]O(V(H)^2)approximable in nO(1) time in Hminorfree graphs [Feige, Hajiaghayi, Lee 2004]

Bidimensionality (version 1)Parameter k is minorbidimensional 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 gridlike 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) vertexdisjoint 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

ContractionBidimensional ProblemsMinimum maximal matchingFace cover (planar graphs)Dominating setEdge dominating setRdominating setConnected dominating setUnweighted TSP tourChordal completion (fillin)

Bidimensional RelateParameter & TreewidthTheorem 1: If a parameter k is bidimensional, then it satisfies parametertreewidth 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, andfixedparameter tractable on graphs of bounded treewidth: h(treewidth) nO(1) time then it has a subexponential fixedparameter 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 boundedtreewidth algorithm (tw = O(k)) [If (approx.) treewidth is large, answer NO]
&

Bidimensional Subexponential FPTCorollary 1: Vertex cover and feedback vertex set have subexponential fixedparameter 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,fixedparameter 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 Hminorfree graph has treewidth O(8n) [Alon, Seymour, Thomas 1990; Grohe 2003]Corollary: 1/32/3 separators, size O(n)(A vertex set whose removal leaves no component of size greater than 2n/3)

Application to Independent Set (LiptonTarjan 1980)Independent Set: a set of vertices with no edges in betweenNote that OPT is at least n/4 since planar graphs are 4colorableFor 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

PolynomialTimeApproximation 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]Parametertreewidth bounds give separators in terms of OPT, not n

PolynomialTimeApproximation 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)

PolynomialTimeApproximation SchemesCorollary: [Demaine & Hajiaghayi 2005]PTAS in Hminorfree graphs for feedback vertex set, face cover, vertex cover, minimum maximal matching, and related vertexremoval problemsPTAS in apexminorfree graphs for dominating set, edge dominating set, Rdominating set, connected dominating set, cliquetransversal 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 Hminorfree 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], apexminorfree [Eppstein 2000], Hminorfree [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 NPhard problemInapproximable within n1 unless ZPP = NPMartin Gardner, April 1, 1975

Example: Graph Coloring[Demaine, Hajiaghayi, Kawarabayashi 2005/2010]2approximation for chromatic number in oddHminorfree graphs using decomposition into two boundedtreewidth pieces:General graphs: Inapprox. within n1 unless ZPP = NP

Simplifying Graph Decompositions[DeVos et al. 2004; Demaine, Hajiaghayi, Kawarabayashi 2005]Generalization to k pieces: Hminorfree graphs can have their vertices or edges partitioned into k pieces such that deleting any one piece results in bounded treewidthUseful for PTASs for minorclosed properties (where k ~ 1/)(Not true for oddminor)Application: e.g. PTAS for MaxCut

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

Contraction Decomposition[Demaine, Hajiaghayi, Kawarabayashi, STOC11]Theorem: Hminorfree graphs can have their edges partitioned into k pieces such that contracting any one piece results in bounded treewidth
Polynomialtime algorithmPreviously known for planar [Klein 2005, 2006], boundedgenus [Demaine, Hajiaghayi, Mohar 2007], apexminorfree [Demaine, Hajiaghayi, Kawarabayashi 2009]

ApplicationsLots of applications via a general theorem,e.g.
Corollary 1: PTAS for Traveling Salesman Problem in weighted Hminorfree graphs [Demaine, Hajiaghayi, Kawarabayashi 2011] solving an open problem of [Grohe 2001]
Coroallary 2: FixedParameter Algorithm for kcut and Bisection on planar graphs and Hminorfree graphs [Demaine, Hajiaghayi, Kawarabayashi 2011] solving an open problem of [Downey, EstivillCastro, Fellows 2003]

Application to TSPCorollary: PTAS for Traveling Salesman Problem in weighted Hminorfree graphs [Demaine, Hajiaghayi, Kawarabayashi 2011]Existing boundedtreewidth algorithm [Dorn, Fomin, Thilikos 2006]Existing spanner [Grigni, Sissokho 2002]Decontraction:Euler tour (cost 2 weight) + perfect matching on odddegree 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 boundedgenus / unweighted Hminorfree [Grigni 2000]PTAS for weighted bounded genus [Demaine, Hajiaghayi, Mohar 2007]PTAS for unweighted apexminorfree [Demaine, Hajiaghayi, Kawarabayashi 2009]PTAS for weighted Hminorfree [DHK 2011]

Application Beyond TSPCorollary: PTAS for minimumweight cedgeconnected submultigraph in Hminorfree graphs [Demaine, Hajiaghayi, Kawarabayashi 2011]
Previous results:PTASs for 2edgeconnected in planar graphs [Klein 2005] (linear) [Berger, Czumaj, Grigni, Zhao 2005] [Czumaj, Grigni, Sissokho, Zhao 2004]PTAS for cedgeconnected in boundedgenus graphs [Demaine, Hajiaghayi, Mohar 2007]

FixedParameter Algorithmic Applications: kcutkcut: Remove fewest edges to make at least k connected componentsFPT in Hminorfree 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 SketchHminorfree graph = tree of almostembeddable graphs [Graph Minors]Each almostembeddable 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: Hminorfree graphs can have their edges partitioned into k pieces such that contracting any one piece results in bounded treewidthPolynomialtime algorithm
Seems a powerful tool for approximation & fixedparameter algorithmsLets find more applications!

IMPROVINGGRAPH MINORS

Graph Minors[Robertson&Seymour 19832004] in 20 papers

NonconstructiveGraph MinorsTheorem: Every Hminor 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 boundedgenus surface (f(H) vortices and apices) [GM16: Robertson & Seymour 2003]

ConstructiveGraph MinorsTheorem: Every Hminor 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 Hminorfree 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 weightsMinweight 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 boundedgenus 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**