Bidimensionality and EPTAS
Fedor V. Fomin∗ Daniel Lokshtanov†‡ Venkatesh Raman§ Saket Saurabh§
Abstract 1 Introduction Bidimensionality theory appears to be a powerful framework While most interesting graph problems remain NP com- for the development of meta-algorithmic techniques. It was plete even when restricted to planar graphs, the restric- introduced by Demaine et al. [J. ACM 2005 ] as a tool tion of a problem to planar graphs is usually consider- to obtain sub-exponential time parameterized algorithms ably more tractable algorithmically than the problem on for problems on H-minor free graphs. Demaine and Haji- general graphs. Over the last four decades, it has been aghayi [SODA 2005 ] extended the theory to obtain polyno- proved that many graph problems on planar graphs mial time approximation schemes (PTASs) for bidimensional admit subexponential time algorithms [18, 25, 33], problems, and subsequently improved these results to EP- subexponential time parameterized algorithms [1, 26], TASs. Fomin et. al [SODA 2010 ] established a third meta- (Efficient) Polynomial Time Approximation Schemes algorithmic direction for bidimensionality theory by relating ((E)PTAS) [4, 28, 11, 20, 27, 29] and linear ker- it to the existence of linear kernels for parameterized prob- nels [2, 5, 8]. Amazingly, the emerging theory of Bidi- lems. In this paper we revisit bidimensionality theory from mensionality developed by Demaine et al. [15, 16, 13] is the perspective of approximation algorithms and redesign able to simultaneously explain the tractability of these the framework for obtaining EPTASs to be more powerful, problems within the paradigms of parameterized algo- easier to apply and easier to understand. rithms [13], approximation [14] and kernelization [24]. One of the important conditions required in the framework The theory is built on cornerstone theorems from Graph developed by Demaine and Hajiaghayi [SODA 2005 ] is that Minors Theory of Robertson and Seymour, and allows to obtain an EPTAS for a graph optimization problem Π, we not only to explain the tractability of many problems, have to know a constant-factor approximation algorithm for but also to generalize the results from planar graphs and Π. Our approach eliminates this strong requirement, which graphs of bounded genus to graphs excluding a fixed makes it amenable to more problems. At the heart of our minor. Roughly speaking, a problem is bidimensional framework is a decomposition lemma which states that for if the solution value for the problem on a k × k-grid is “most” bidimensional problems, there is a polynomial time Ω(k2), and the contraction or removal of an edge does algorithm which given an H-minor-free graph G as input not increase solution value. Many natural problems and an > 0 outputs a vertex set X of size · OPT such are bidimensional, including Dominating Set, Feed- that the treewidth of G \ X is O(1/). Here, OPT is the ob- back Vertex Set, Edge Dominating Set, Ver- jective function value of the problem in question This allows tex Cover, Connected Dominating Set, Cycle us to obtain EPTASs on (apex)-minor-free graphs for all Packing, Connected Vertex Cover, and Graph problems covered by the previous framework, as well as for Metric TSP. a wide range of packing problems, partial covering problems A PTAS is an algorithm which takes an instance and problems that are neither closed under taking minors, I of an optimization problem and a parameter > 0 nor contractions. To the best of our knowledge for many and, runs in time nO(f(1/)), produces a solution that of these problems including Cycle Packing, Vertex-H- is within a factor of being optimal. A PTAS with Packing, Maximum Leaf Spanning Tree, and Partial running time f(1/) · nO(1), is called efficient PTAS r-Dominating Set no EPTASs on planar graphs were pre- (EPTAS). Prior to bidimensionality [14], there were viously known. two main approaches to design (E)PTASs on planar graphs. The first one was based on the classical Lipton- Tarjan planar separator theorem [32]. The second, more widely used approach was given by Baker [4]. In the ∗Department of Informatics, University of Bergen, Norway. Lipton-Tarjan based approach we split the input n- [email protected] vertex graph into two pieces of approximately equal size †Department of Computer Science and Engineering, University √ of California, San Diego, USA. [email protected] using a separator of size O( n). Then we recursively §The Institute of Mathematical Sciences, Chennai, India. approximate the problem on the two smaller instances {vraman|saket}@imsc.res.in and glue the approximate solutions at the separator. the input graph leaves a graph on which the problem This approach was only applicable to problems where can be solved optimally in polynomial time. The set X the size of the optimal solutions was at least a constant could be as large as O(n) which in some cases makes fraction of n. it difficult to bound the amount of interaction between The main idea in Baker’s approach is to decompose the set X and the optimum solution. the planar graph into overlapping subgraphs of bounded In this paper we propose a framework which com- outerplanarity and then solve the problem optimally bines the best of both worlds—the generalized Lipton- in each of these subgraphs using dynamic program- Tarjan and generalized Baker’s approaches. In particu- ming. Later Eppstein [20] generalized this approach lar, we show that for most bidimensional problems there to work for larger class of graphs, namely apex minor is a polynomial time algorithm that given a graph G free graphs. Khanna and Motwani [29] used Baker’s ap- and an > 0 outputs a vertex set X of size · OPT proach in an attempt to syntactically characterize the such that the treewidth of G \ X is O(1/). Because complexity class of problems admitting PTASs, estab- the size of X is bounded, the interaction between X lishing a family of problems on planar graphs to which and the optimum solution is bounded trivially. Since it applies. The same kind of approach is also used by X is only removed once, the difficulty faced by a re- Dawar et al. [11] to obtain EPTASs for every minimiza- cursive approach vanishes. In our framework to obtain tion problem definable in first-order logic on every class EPTASs, we demand that the problem in question is of graphs excluding a fixed minor. Baker’s approach “reducible”, which is nothing else than that the set X seemed to be limited to “local” graph problems–where can be removed from the graph, disturbing the optimum one is interested in finding a vertex/edge set satisfying solution by at most O( · OPT ). Finally, our algorithm a property that can be checked by looking at constant to compute X does not require an approximation algo- size neighborhood around each vertex. rithm for the problem in question, and relies only on Demaine and Hajiaghayi [14] used bidimensionality a sublinear treewidth bound. For most problems, such theory to strengthen and generalize both approaches. a bound can be obtained via bidimensionality, whereas In particular they strengthened the Lipton-Tarjan ap- for some problems that are not bidimensional, one can proach significantly by showing that for a magnitude√ obtain the sublinear treewidth bound directly. This is of problems one can find a separator of size O( OPT ) where our framework differs significantly from the one that splits the optimum solution evenly into two pieces. proposed by Demaine and Hajiaghayi [14]. One of the Here OPT is the size of the optimum solution. This important condition required to obtain an EPTAS for allowed them to give EPTASs for several problems on a graph optimization problem Π using the framework planar graphs and more generally on apex-minor-free developed in [14] is to have a constant-factor approxi- graphs or H-minor free graphs. Two important prob- mation algorithm for Π. Thus our framework removes lems to which their approach applies are Feedback this stringent condition of demanding approximation al- vertex Set and Connected Dominating Set. Ear- gorithm, which makes it amenable to more problems. lier only a PTAS and no EPTAS for Feedback Ver- Our new framework allows to obtain EPTASs on tex Set on planar graphs was known [30]. In addition, (apex)-minor-free graphs for all problems covered by they also generalize Baker’s approach by allowing more the previous framework, as well as for several packing interaction between the overlapping subgraphs. problems, partial covering problems and problems that Comparing the generalized versions of the two ap- are neither closed under taking minors nor contractions. proaches, it seems that each has its strengths and weak- For an example consider the Maximum Degree Pre- nesses. In the generalized Lipton-Tarjan approach of serving Spanning Tree problem where given a graph Demaine and Hajiaghayi [14] one splits the graph into G the objective is to find a spanning tree such that the two pieces recursively. To ensure that the repeated ap- number of vertices which have the same degree in the plication does not “increase” the approximation factor, tree as in the input graph is maximized. Maximum De- in each recursive step one needs to carefully reconstruct gree Preserving Spanning Tree is neither closed the solution from the smaller ones. Additionally, to under taking minors nor contractions, but one can still ensure that the separator splits the optimum solution apply our framework to obtain an EPTAS for this prob- evenly, the framework of Demaine and Hajiaghayi [14] lem. For another example, consider Cycle Packing, requires a constant factor approximation for the prob- where given a graph G the objective is to find the max- lem in question. On the other hand, their generaliza- imum number of vertex disjoint cycles in G. For this tion of Baker’s approach essentially identifies a set of problem, it is not clear how to directly apply the previ- vertices or edges that interacts in a limited way with ous framework to obtain an EPTAS. On the other hand, the optimum solution, such that the removal of X from using our framework to obtain an EPTAS for this prob- lem is easy. More generally, we give an EPTAS for the sequence of edge-contractions is said to be a contraction Vertex-H-Packing problem, defined as follows. Let of G. We denote it by H ≤c G. A graph H is a minor H be a finite set of connected graphs such that at least of a graph G if H is the contraction of some subgraph one graph in H is planar. Input to Vertex-H-Packing of G and we denote it by H ≤m G. We say that a graph is a graph G and the objective is to find a maximum size G is H-minor-free if G does not contain H as a minor. collection of vertex disjoint subgraphs G1,...,Gk of G We also say that a graph class GH is H-minor-free (or, such that each of them contains some graph from H as a excludes H as a minor) when all its members are H- minor. To the best of our knowledge no EPTASs for Cy- minor-free. An apex graph is a graph obtained from cle Packing, Vertex-H-Packing, Maximum Leaf a planar graph G by adding a vertex and making it Spanning Tree, or Partial r-Dominating Set were adjacent to some of the vertices of G. A graph class GH previously known, even on planar graphs. Our frame- is apex-minor-free if GH excludes a fixed apex graph H work to obtain EPTASs seems to be the most general as a minor. Let us remark that every planar, and more one could hope for in the context of bidimensionality generally, graph of bounded genus, is an H-minor-free and approximation. graph for some fixed apex graph H. Our results are less prone to the impracticality is- Grids and their triangulations. Let r be a positive sues that follow most results in algorithmic Graph Mi- integer, r ≥ 2. The (r×r)-grid is the Cartesian product nors. In particular, it is easy to implement our algo- of two paths of lengths r − 1. A vertex of a grid is a rithms in a manner completely independent from results corner if it has degree 2. Thus each (r × r)-grid has 4 in Graph Minors and only use bounds from graph mi- corners. A vertex of a (r × r)-grid is called internal if it nor in the analysis. Furthermore, there is evidence [35] has degree 4, otherwise it is called external. Let Γ be that the large hidden constants only arise in the analy- r the graph obtained from the (r × r)-grid by triangulat- sis, and that with correct fine tuning algorithms could ing internal faces of the (r×r)-grid such that all internal be made to yield good approximations in practice—at vertices become of degree 6, all non-corner external ver- the expense of the rigorous approximation guarantee. tices are of degree 4, and then one corner of degree two is joined by edges with all vertices of the external face. 2 Definitions and Notations The graph Γ6 is shown in In this section we give various definitions which we Fig. 1. make use of in the paper. Let G be a graph with vertex set V (G) and edge set E(G). A graph G0 is a Counting Monadic Second subgraph of G if V (G0) ⊆ V (G) and E(G0) ⊆ E(G). Order Logic. Count- The subgraph G0 is called an induced subgraph of G ing monadic second-order if E(G0) = {uv ∈ E(G) | u, v ∈ V (G0)}, in this logic (CMSO) is monadic case, G0 is also called the subgraph induced by V 0 and second-order logic (MSO) denoted by G[V 0]. For a vertex set S, by G \ S we additionally equipped with an Figure 1: The graph Γ6. denote G[V (G) \ S]. A graph class G is hereditary if atomic formula cardn,p(U) for for any graph G ∈ G all induced subgraphs of G are testing whether the cardinality of a set U is congruent in G. By N(u) we denote (open) neighborhood of u, to n modulo p, where n and p are integers independent that is, the set of all vertices adjacent to u. Similarly, of the input graph such that 0 ≤ n < p and p ≥ 2. We by N[u] we denote N(u) ∪ {u}. For a subset D ⊆ V , refer to [3, 9, 10] for a detailed introduction to CMSO. For reader’s convenience, we provide the definition of we define N[D] = ∪v∈DN[v] and N(D) = N[D] \ D. MSO in Appendix. and The distance dG(u, v) between two vertices u and v of Min-CMSO Max-CMSO G is the length of the shortest path in G from u to v. problems are graph optimization problems where the r objective is to find a maximum or minimum sized vertex Define BG(v) to be the set of vertices within distance at most r from v, including v itself. For a vertex set S, or edge set satisfying a CMSO-expressible property. In r S r particular, in a Min/Max-CMSO graph problem Π define BG(v) = v∈S BG(v). We denote by tw(G) the treewidth of a graph G. (The definition of treewidth can we are given a graph G as input. The objective is to be found in Appendix.) find a minimum/maximum cardinality vertex/edge set S such that the CMSO-expressible predicate PΠ(G, S) is satisfied. Minors. Given an edge e = xy of a graph G, the graph G/e is obtained from G by contracting the edge Bidimensionality and Separability. Our results e. That is, the endpoints x and y are replaced by a new concern graph optimization problems where the objec- vertex vxy which is adjacent to the old neighbors of x tive is to find a vertex or edge set that satisfies a feasibil- and y (except from x and y). A graph H obtained by a ity constraint and maximizes or minimizes a problem- specific objective function. For a problem Π and ver- In Definitions 2 and 3 we slightly misused notation. tex (edge) set S let φΠ(G, S) be the feasibility con- Specifically, in the case that OPT is an edge set we straint returning true if S is feasible and false oth- should not be considering OPT \ R and OPT \ L but erwise. Let κΠ(G, S) be the objective function. In most OPT \ E(G[R]) and OPT \ E(G[L]) respectively. cases, κΠ(G, S) will return |S|. We will only consider Reducibility, η-Transversability and Graph problems where every instance has at least one feasible Classes with Truly Sublinear Treewidth. We now solution. Let U be the set of all graphs. For a graph define three of the central notions of this article. optimization problem Π let π : U → N be a function re- turning the objective function value of the optimal solu- Definition 4. A graph optimization problem Π with tion of Π on G. We say that a problem Π is minor-closed objective function κΠ is called reducible if there exists a if π(H) ≤ π(G) whenever H is a minor of G. Similarly, Min/Max-CMSO problem Π0 and a function f : N → we say Π is contraction-closed if π(H) ≤ π(G) whenever N such that H is a contraction of G. We now define bidimensional 1. there is a polynomial time algorithm that given G problems. and X ⊆ V (G) outputs G0 such that π0(G0) = 0 Definition 1. ([13, 22]) A graph optimization prob- π(G) ± O|X|) and tw(G ) ≤ f(tw(G \ X)), lem Π is minor-bidimensional if Π is minor-closed and 2. there is a polynomial time algorithm that given G there is a δ > 0 such that π(R) ≥ δr2 for the (r × r)- and X ⊆ V (G), G0 and a vertex (edge) set S0 0 0 grid R. In other words, the value of the solution on R such that PΠ0 (G ,S ) holds, outputs S such that 0 should be at least δr2. A graph optimization problem Π φΠ(G, S) = true and κΠ(G, S) = |S | ± O|X|). is called contraction-bidimensional if Π is contraction- Definition 5. A graph optimization problem Π is closed and there is δ > 0 such that π(Γ ) ≥ δr2. In r called η-Transversable if there is a polynomial time al- either case, Π is called bidimensional. gorithm that given a graph G outputs a set X of size Demaine and Hajiaghayi [14] define the separation O(π(G)) such that tw(G \ X) ≤ η. property for problems, and show how separability to- Definition 6. A class G has truly sublinear treewidth gether with bidimensionality is useful to obtain EPTASs if there exist constants η, β and λ, such that λ < 1 on H-minor-free graphs. In our setting a slightly weaker and for any graph G ∈ G and X ⊆ V (G) such that notion of separability is sufficient. In particular the fol- tw(G \ X) ≤ η, we have tw(G) ≤ η + β|X|λ. We call lowing definition is a reformulation of the requirement η, β and λ the parameters of G. 3 of the definition of separability in [14] and similar to the definition used in [24] to obtain kernels for bidimen- 3 Partitioning Graphs of Truly Sublinear sional problems. Treewidth Definition 2. A minor-bidimensional problem Π has We need the following well known lemma, see e.g. [6], the separation property if given any graph G and a on separators in graphs of bounded treewidth. partition of V (G) into L ] S ] R such that N(L) ⊆ S Lemma 3.1. Let G be a graph of treewidth at most t and N(R) ⊆ S, and given an optimal solution OPT + and w : V (G) → R ∪ {0} be a weight function. Then to G, π(G[L]) ≤ κΠ(G[L],OPT ∩ L) + O(|S|) and there is a partition of V (G) into L ] S ] R such that π(G[R]) ≤ κΠ(G[R],OPT ∩ R) + O(|S|). • |S| ≤ t + 1, N(L) ⊆ S and N(R) ⊆ S, For contraction-bidimensional parameters we have a • every connected component G[C] of G \ S has slightly different definition of the separation property. w(C) ≤ w(V )/2, For a graph G and a partition of V (G) into L ] S ] R • w(V (G))−w(S) ≤ w(L) ≤ 2(w(V (G))−w(S)) and such that N(L) ⊆ S and N(R) ⊆ S, we define G (G 3 3 L R w(V (G))−w(S) ≤ w(R) ≤ 2(w(V (G)−w(S)) . ) to be the graph obtained from G by contracting every 3 3 connected component of G[R](G[L]) into the vertex of The next lemma is crucial in our analysis. S with the lowest index. Lemma 3.2. Let G be a hereditary graph class of truly Definition 3. A contraction-bidimensional problem sublinear treewidth with parameters η, λ and β. For any has the separation property if given any graph G and > 1 there is a γ such that for any G ∈ G and X ⊆ a partition of V (G) into L ] S ] R such that N(L) ⊆ S V (G) with tw(G\X) ≤ η there is X0 ⊆ V (G) satisfying and N(R) ⊆ S, and given an optimal solution OPT to |X0| ≤ |X| and for every connected component C of 0 G, π(GL) ≤ κΠ(GL,OPT \ R) + O(|S|) and π(GR) ≤ G \ X , we have |C ∩ X| ≤ γ and |N(C)| ≤ γ. Moreover 0 κΠ(GR,OPT \ L) + O(|S|). X can be computed from G and X in polynomial time. λ λ Proof. For any γ ≥ 1, define Tγ : N → N such that Tγ (k) ≤ max k − δ(αk) − δ((1 − α)k) is the smallest integer such that if G ∈ G and there is a 1/3≤α≤2/3 X ⊆ V (G) with tw(G \ X) ≤ η and |X| ≤ k, then there + (2 + 1)(βkλ + η + 1) 0 is a X ⊆ V (G) of size at most Tγ (k) such that for every ≤ max k − δkλ(αλ + (1 − α)λ) connected component C of G \ X0 we have |C ∩ X| ≤ γ 1/3≤α≤2/3 λ and |N(C)| ≤ γ. Informally, Tγ (k) is the minimum size + (2 + 1)(βk + η + 1) of a vertex set X0 such that every connected component ≤ k − δkλ − δ(ρ − 1)kλ C of G\X0 has at most γ neighbours in X0 and contains at most γ vertices of X, if we know that deleting the k- + (2 + 1)(βkλ + η + 1) sized vertex set X from G yields a graph of treewidth η. ≤ k − δkλ. We will make choices for the constants δ, γ and ρ based on η, λ, β and . Our aim is to show that T (k) ≤ k γ The last inequality holds whenever δ(ρ − 1)kλ ≥ for every k. (2 + 1)(βkλ + η + 1), which is ensured by the choice of Observe that for any numbers a > 0, b > 0, we δ and the fact that kλ ≥ 1. Thus T (k) ≤ k for all k. have aλ + bλ > (a + b)λ since λ < 1. Thus we have γ Hence there exists a set X0 of size at most k such that ρ = min αλ + (1 − α)λ > 1. We choose 1/3≤α≤2/3 for every component C of G \ X0 we have C ∩ X ≤ γ (2+1)(β+η+1)) 1 3δ 1−λ δ = ρ−1 and γ = ( ) . If tw(G \ X) ≤ η and |N(C)| ≤ γ. 0 0 and |X| ≤ γ then we set X = ∅, so Tγ (k) = 0 ≤ k What remains is to show that X can be com- for k ≤ γ. We now show that if k ≥ γ/3 then puted from G and X in polynomial time. The in- λ Tγ (k) = 0 ≤ k − δk by induction on k. For the base ductive proof can be converted into a recursive algo- case if γ/3 ≤ k ≤ γ then the choice of γ implies that rithm. The only computationally hard step of the proof λ γ λ k − δk ≥ 3 − δγ ≥ 0 = Tγ (k). is the construction of a tree-decompositon of G in each We now consider Tγ (k) for k > γ. Let G ∈ G inductive step. Instead of computing the treewidth and X ⊆ V (G) with tw(G \ X) ≤ η and |X| ≤ k. exactly we use the d∗plog tw(G)-approximation algo- The treewidth of G is at most η + βkλ. Construct a rithm by Feige et al. [21], where d∗ is a fixed con- weight function w : V (G) → N such that w(v) = 1, stant. Thus when we partition V (G) into L, S, and when v ∈ X and w(v) = 0 otherwise. By Lemma 3.1, R using Lemma 3.1, the upper bound on the size of S there is a partition of V (G) into L, S and R such that will be d∗(η + βkλ)plog(η + βkλ) instead of η + βkλ. |S| ≤ η+βkλ +1, N(L) ⊆ S, N(R) ⊆ S, |L∩X| ≤ 2k/3 However, for any λ < λ0 < 1 there is a β0 such that 0 and |R∩X| ≤ 2k/3. Deleting S from the graph G yields d∗(η + βkλ)plog(η + βkλ) < η + β0kλ . Now we can two graphs G[L] and G[R] with no edges between them. apply the above analysis with β0 instead of β and λ0 Thus we put S into X0 and then proceed recursively in instead of λ to bound the size of the set X0 output by G[L∪S] and G[R∪S] starting from the sets S ∪(X ∩L) the algorithm. This concludes the proof of the lemma. and S ∪ (X ∩ R) in G[L ∪ S] and G[R ∪ S] respectively. This yields the following recurrence for Tγ . The following corollary is a direct consequence of Lemma 3.2. Nevertheless, we find it worthwhile to λ Tγ (k) ≤ max T (αk + η + βk + 1) mention it separately. 1/3≤α≤2/3 λ + T ((1 − α)k + η + βk + 1) Corollary 3.1. Let G be a hereditary graph class of + η + βkλ + 1 truly sublinear treewidth with parameters η, λ and β. λ 1 1−λ For any > 1 there is a τ with τ = O( ) ) such that for any G ∈ G and X ⊆ V (G) with tw(G \ X) ≤ η Observe that since k ≥ γ we have αk ≥ γ/3 and there is a X0 ⊆ V (G) satisfying |X0| ≤ |X| such that (1 − αk) ≥ γ/3. The induction hypothesis then yields tw(G \ X0) ≤ τ. the following inequality. Proof. We apply Lemma 3.2 on G and X to obtain the set X0 of size |X|. Observe that in the proof λ 1 Tγ (k) ≤ max T (αk + η + βk + 1) 1 1−λ 1/3≤α≤2/3 of Lemma 3.2, γ = O( ) ). The treewidth of G \ X0 equals the maximum treewidth of a connected + T ((1 − α)k + η + βkλ + 1) component C of G \ X0. However |C ∩ X| ≤ γ and so λ λ + η + βk + 1 tw(G[C]) = Oγ ), concluding the proof. 4 Approximation Schemes Proposition 4.1. ([12, 17, 22]) Let G be a connected Approximation Schemes for η-Transversable graph excluding a fixed graph H as a minor. Then there 2 problems. exists some constant c such that if tw(G) ≥ c·r , then G contains the r ×r-grid as a minor. Moreover, if H is an Theorem 4.1. Let Π be an η-transversable, reducible apex graph, then G does not contain Γr as a contraction. graph optimization problem. Then Π has an EPTAS on Corollary 4.1. Let G be a class of graphs excluding every graph class G with truly sublinear treewidth. H a fixed graph H as a minor. Then GH has truly sublinear treewidth with λ = 1 . Proof. Let G be the input to Π and > 0 be fixed. 2
Since Π is η-transversable there is a polynomial time Proof. Let ρ be a constant such that any graph G ∈ GH algorithm that outputs a set X such that |X| ≤ ρ1π(G) of treewidth at least t contains a ρt × ρt grid as a 0 and tw(G) ≤ η, for a fixed constant ρ1. Let be a minor. Let G ∈ GH have a vertex subset X such constant to be selected later. By Lemma 3.2, there exist that tw(G \ X) ≤ η for a fixed constant η. We prove γ, λ0 < 1 and β0 depending on 0, λ, η and β such that that tw(G) ≤ η+1 dp|X| + 1e. Suppose not. Then, 0 ρ given G and X a set X with the following properties by Proposition 4.1, G contains a (η + 1)dp|X| + 1e × can be found in polynomial time. First |X0| ≤ 0|X|, (η + 1)dp|X| + 1e grid as a minor. Hence G contains and secondly for every component C of G \ X0 we have 0 at least |X| + 1 disjoint η + 1 × η + 1 grids as a minor. that C ∩ X ≤ γ. Thus tw(G \ X0) = τ ≤ β0γλ + η. The set X is disjoint from at least one of these grids, Since Π is reducible there exists a Min/Max-CMSO and hence G \ X contains a η + 1 × η + 1 grid as a minor problem Π0, a constant ρ and a function f : → 2 N N and has treewidth at least η + 1, yielding the desired such that: contradiction. 1. there is a polynomial time algorithm that given G For every fixed integer η we define the η- and X0 ⊆ V (G) outputs G0 such that |π0(G0) − 0 0 Transversal problem as follows. Input is a graph π(G)| ≤ ρ2|X | and tw(G ) ≤ f(τ); G, and the objective is to find a minimum cardinality 2. there is a polynomial time algorithm that given G vertex set S ⊆ V (G) such that tw(G \ S) ≤ η. We 0 0 0 and X ⊆ V (G), G and a vertex (edge) set S now give a polynomial time constant factor approxima- 0 0 such that PΠ0 (G ,S ) holds outputs S such that tion for the η-Transversal problem on H-minor-free 0 0 φΠ(G, S) holds and |κΠ(G, S) − |S || ≤ ρ2|X |. graphs. 0 0 We constuct G from G and X using the first polyno- Lemma 4.1. For every integer η and fixed graph H 0 mial time algorithm. Since tw(G ) ≤ f(τ) we can use there is a constant c and a polynomial time c- an extended version of Courcelle’s theorem [9, 10] given approximation algorithm for the η-Transversal prob- 0 0 by Borie et al. [7] to find an optimal solution S to Π lem on H-minor-free graphs. in g(0)|V (G0)| time. By the properties of the first poly- nomial time algorithm, ||S0| − π(G)| ≤ ρ|X0| where ρ = Proof. Let X be a smallest vertex set in G such that max(ρ1, ρ2). We now use the second polynomial time al- tw(G \ X) ≤ η. By Lemma 3.2 with = 1/2 there gorithm to construct a solution S to Π from G, X0, G0 exists a γ depending only on H and η and a set X0 with and S0. The properties of the second algorithm ensure |X0| ≤ |X|/2 such that for any component C of G \ X0, 0 0 φΠ(G, S) holds and that |κΠ(G, S) − |S || ≤ ρ|X |, and |C ∩ X| ≤ γ and |N(C)| ≤ γ. Since X is the smallest 0 2 0 hence |κΠ(G, S) − π(G)| ≤ 2ρ|X | ≤ 2ρ π(G). Choos- set such that tw(G \ X) ≤ η, there is a component 0 C of G \ X0 with treewidth strictly more than η. Let ing = 2 yields |κΠ(G, S) − π(G)| ≤ π(G), proving 2ρ 0 the theorem. Z = N(C) and observe that Z ⊆ X is a set of size at most γ such that C is a conneced component of G \ Z. Approximation Schemes for Bidimensional prob- The algorithm proceeds as follows. It tries all possibilities for Z and looks for a connected component lems. Now we use Theorem 4.1 to give EPTASs √ for bidimensional, separable and reducible problems on C of G \ Z such that η < tw(G[C]) = O γ). It graphs excluding a fixed H as a minor. In order to do solves the η-Transversal problem optimally on G[C] this, we show that H-minor-free graphs have truly sub- by noting that η-Transversal can be formulated as linear treewidth and that bidimensional and separable a Min-CMSO problem and applying the algorithm by problems are η-transversible. To show that H-minor Borie et al [7]. Let XC be the solution obtained for free graphs have truly sublinear treewidth we use the G[C]. The algorithm adds XC and N(C) to the solution following result. and repeats this step on G \ (C ∪ N(C)) as long as tw(G) ≥ η. Clearly, the set returned by the algorithm is a Z be a minimum size r-dominating set of G and ZL be feasible solution. We now argue that the algorithm a minimum size r-dominating set of GL. We have that is a (γ + 1)-approximation algorithm. Let C1, C2, |ZL| ≤ |Z\R|+|S| because (Z\R)∪S is an r-dominating ...Ct be the components found by the algorithm in this set of GL and hence |ZL| > |Z \ R| + |S| contradicts manner. Since X must contain at least one vertex in the choice of ZL. Hence |ZL| ≤ |Z \ R| + O|S|) and each Ci it follows that t ≤ |X|. Now, for each i, N(Ci) r-Dominating Set is separable. S contains at most γ vertices outside of jmatching in linear time and output the such that Br (S) = V (G). For r = 1, if we demand G endpoints of the matching as X. Any vertex cover must that G[S] is connected we obtain the Connected contain at least one endpoint from each edge in the Dominating Set problem. In Connected Vertex matching, and thus |X| ≤ 2π(G). Also, tw(G \ X) = 0. Cover we are given a graph G and the objective is to find a minimum size subset S ⊆ V (G) such that G[S] is connected and every edge in E(G) has at least Lemma 5.1. r-Dominating Set, Connected Dom- one endpoint in S. It well-known that r-Dominating inating Set and Connected Vertex Cover Set, Connected Dominating Set and Connected are contraction-bidimensional, separable and reducible. Vertex Cover are contraction-bidimensional [13]. Thus they are η-transversable on apex-minor-free Let V (G) = L ] S ] R with N(L) ⊆ S and N(R) ⊆ graphs. Furthermore, Connected Vertex Cover is S. Let GL and GR be defined as in Definition 3. Let 0-transversible on general graphs. Max Leaf Spanning Tree. In the Max Leaf Vertex-H-Packing Spanning Tree problem we are given a connected Input: A graph G. graph G and asked to find a spanning tree T of G Objective: Find a maximum size collection of vertex maximizing the number of leaves of T . disjoint subgraphs G1,...,Gk of G such that each of them contains some graph from H as a minor. Lemma 5.2. Max Leaf Spanning Tree is 2- transversible and reducible. It it easy to see that both Vertex-H-Covering and Vertex-H-Packing are minor-closed problems. Now, let h be the size of the smallest planar graph H Proof. We could have shown that the problem is minor- in H. By a result of Robertson et al. [34], H is a minor bidimensional and separable, however, just as for Con- of the (t × t)-grid, where t = 14|V (H)| − 24. Consider a nected Vertex Cover, it is easier to show that the (r×r)-grid F . F contains r2/t2 disjoint H-minors. Any problem is 2-transversible directly. In particular, Kleit- covering of F must pick at least one vertex from each man and West [31] have shown that a connected graph of these minors, therefore both Vertex-H-Covering which contains no spanning tree with at least k leaves and Vertex-H-Packing are minor-bidimensional. has at most 4k + 2 vertices of degree at least 3. Thus We now prove that Vertex-H-Covering is sepa- given a graph we can just return all vertices of degree rable. Given a graph G and a partition of V (G) into at least 3, and the remaining graph will have treewidth L, S and R such that N(L) ⊆ S and N(R) ⊆ S, let at most 2. Hence, Max Leaf Spanning Tree is 2- Z be a set of minimum size such that G \ Z contains transversible. no graph of H as a minor. Consider the smallest set We prove that Max Leaf Spanning Tree is ZL such that G[L] \ ZL contains no graph of H as a 0 reducible. Given a graph G and set X, let G = G \ X minor. If |Z ∩ L| < |ZL|, this contradicts the choice 0 and let R = N(X). The annotated problem Π is to of ZL since G[L] \ (Z ∩ L) does not contain a graph 0 0 find a maximum size set S ⊆ V (G ) such that every of H as a minor. The proof for ZR is identical, thus vertex in S0 \ R has a neighbour outside of S0 and every Vertex-H-Covering is separable. connected component of G \ S contains at least one The proof of separability of Vertex-H-Packing vertex of R \ S. For a spanning tree T of G let S be goes along the same lines as the proof for Vertex-H- the set of leaves of T . Then S \ X is a feasible solution Covering, but with a few notable differences. In par- to the annotated problem. On the other hand, given a ticular, we formalize Vertex-H-Packing as a graph feasible solution S0 to Π0, observe every component of optimization problem where we seek an edge set Z ⊆ 0 G \ S contains a vertex of X. We construct a spanning E(G). The objective function, κCOV , counts the num- forest F of G with at most |X| components by picking ber of connected components of the subgraph formed by a spanning tree for every component of G \ S and for Z that contain at least one copy of a graph in H as a every vertex v in S \ R we connect v to a neighbour minor, and all edge subsets are considered feasible. outside of S. Notice that all vertices of S are leaves Given a graph G and a partition of V (G) into L, of the spanning forest F . From F we can construct a S and R such that N(L) ⊆ S and N(R) ⊆ S, let Z be spanning tree T by adding at most |X| − 1 edges. Thus an edge set maximizing κCOV (G, Z) and ZL be an edge T has at least |S| − 2(|X| − 1) leaves and we conclude set maximizing κCOV (G[L],ZL). By the choice of ZL we that Max Leaf Spanning Tree is reducible. have κCOV (G[L],ZL) ≥ κCOV (G[L],Z ∪E(G[L])). The proof for ZR is identical, hence Vertex-H-Packing is 5.2 Covering and Packing Problems Minor separable. Covering and Packing. We give below a few generic Finally, it is easy to see that both Vertex- problems each of which subsumes many problems in it- H-Covering and Vertex-H-Packing are reducible. self and fit in our framework. Let H be a finite set of Given G and X we let G0 = G \ X. For Vertex-H- connected graphs such that at least one graph in H is Covering X can be added to the an optimal solution in planar. G0 at the cost of |X|. For Vertex-H-Packing at most |X| of the minors of graphs in H contained a vertex in X and got removed when X was deleted. Expressing both Vertex-H-Covering problems as Min/Max-CMSO problems is routine. Input: A graph G Objective: Find a minimum size set S ⊆ V (G) such Lemma 5.3. Vertex-H-Covering and Vertex-H- that G \ S does not contain any of the graphs from Packing are minor-bidimensional, separable and re- H as a minor. ducible. Vertex-H-Covering contains various problems to a graph in S that contains v. This algorithm to check as a special case, for example: (a) Vertex Cover a subgraph isomorphic to a given graph containing a by letting H contain a single graph on a single edge, particular vertex appears in an article of Eppstein [19]. (b) Feedback Vertex Set by setting H to contain a Consider an instance G of Vertex-S-Covering single graph, the complete graph on 3 vertices K3; (c) reduced according to the Redundant Vertex Rule, and Diamond Hitting Set by letting H contain a single let S be an optimal solution to G. Since X hits all graph, the comlete graph on 4 vertices K4 minus a sin- copies of graphs in S occuring in G and every vertex gle edge. Other choices for H lead to vertex deletion in G appears in some copy of a graph in S it follows into outerplanar graphs, series-parallel graphs, graphs that X is a r-dominating set of G, where r is the of constant treewidth (η-Transversal) or pathwidth. maximum size of a graph in S. Finally, consider an On the other hand, Vertex-H-Packing contains prob- instance G of Vertex-S-Packing reduced according lems like Cycle Packing as a special case. to the Redundant Vertex Rule, and consider an optimal solution G ,...,G such that for every i, G contains Subgraph Covering and Packing. Now we consider 1 OPT i a graph. problems about covering and packing subgraphs. These problems can be handled in much the same way as 5.3 Partial Domination and Covering In the covering and packing minors. Let S be a finite set of Partial r-Dominating Set problem we are given a connected graphs. graph G together with an integer t ≤ |V (G)|. The objective is to find a minimum size set S such that Vertex-S-Covering |Br (S)| ≥ t. In we are given Input: A graph G G Partial Vertex Cover a graph G together with an integer t ≤ |E(G)| and the Objective: Find a minimum size set S ⊆ V (G) such objective is to find a minimum size vertex set S such that G \ S does not contain any of the graphs from that |{uv ∈ E : u ∈ S ∨ v ∈ S}| ≥ t. We will call S as a subgraph. {uv ∈ E : u ∈ S ∨ v ∈ S} the set of edges covered by S. Vertex-S-Packing PTAS for Partial Vertex Cover on planar graphs Input: A graph G. was given in [27]. We are not aware of any PTAS for Objective: Find a maximum size collection of vertex Partial r-Dominating Set. disjoint subgraphs We will not show that Partial r-Dominating G1,...,Gk of G such that each of them Set and Partial Vertex Cover are bidimensional, contains some graph from S as a subgraph. instead we will directly construct a EPTASs for these problems on apex-minor-free graphs using the tools developed so far. We will use OPT for the size of Lemma 5.4. Vertex-S-Covering or Vertex-S- an optimal solution to our instances. Let H be a Packing pre-processed with the Redundant Vertex Rule fixed apex graph, our input graphs will exclude H are η-transversible and reducible. as a minor. We employ an algorithm of Fomin et al. [23]. They give an algorithm for solving Partial r- Proof. We will not show that S or √ Vertex- -Covering O(r OPT ) O1) S are bidimensional. Instead, we Dominating Set in time 2 √ n and Partial Vertex- -Packing O( OPT ) O1) will give a reduction rule, and show that instances Vertex Cover in time 2 n . A key part of reduced according to this rule have an r-dominating their algorithm for Partial r-Dominating Set is a set of size O(OPT ), where r is the maximum size polynomial time algorithm ([23], Lemma 5) that given of a graph in S. Since r- is η- a graph G together with integers t and k returns an Dominating Set 0 transversible there is an algorithm that in polynomial induced subgraph G of G such that G has a k-sized r 0 time outputs a set X ⊆ V (G) of size O(OPT ) such vertex set S such that |BG(S)| ≥ t if and only if G 0 r 0 that tw(G \ X) ≤ η. Hence the pre-processed version has a k-sized vertex set S such that |BG0 (S )| ≥ t. 0 of Vertex-S-Covering and Vertex-S-Packing is η- Furthermore, G has a 3r-dominating set of size k. Our transversible. EPTAS loops over all possible values of k and for each 0 Consider the following rule, the Redundant Vertex such value produces Gk from G, t and k using Lemma 5 of [23]. If G0 has less than t vertices, then G0 cannot Rule. Given as input G to Vertex-S-Covering or k k S remove all vertices that are not have any set which covers at least t vertices, and so, Vertex- -Packing 0 part of any subgraph isomorphic to any graph in S. We neither can G. If Gk has at least t vertices, we proceed can perform the Redundant Vertex Rule in O(|V | · |S|) with the following subroutine. 0 time by looking at a small ball around every vertex v and Just as in the proof of Theorem 4.1, let be a 0 check whether the ball contains a subgraph isomorphic constant to be chosen later. By construction Gk has 3r-dominating set of size k. Since 3r-Dominating Set angle Packing, Vertex-S-Covering, Vertex-S- is η-transversible there is a polynomial time algorithm Packing Partial r-Dominating Set and Partial that outputs a set X of size at most ρk such that Vertex Cover admit an EPTAS on apex-minor-free 0 tw(Gk \ X) ≤ η. By Lemma 3.2 there is a polynomial graphs. time algorithm that computes a set X0 of size at most 0 0 0 It should be noted that for a fixed , the treewidth τ ρk such that tw(Gk \ X ) ≤ δ for a constant δ depending only on η and H. We put all vertices in X0 in Corollary 3.1 is O(1/). For H-minor-free graphs 0 0 one can compute the set X0 from G and X using the in our solution. Specifically, we remove X from Gk and r 0 procedure described in the last paragraph of the proof of put all other vertices of B 0 (X ) into a set R. Using Gk standard dynamic programming (or by formulating the Lemma 3.2 but using the constant factor approximation problem in an extended version of MSO [3]) on graphs for treewidth on H-minor-free graphs [21] instead of of bounded treewidth, we can find a minimum size set the approximation algorithm for general graphs. For 0 0 0 0 r 0 many problems discussed in this paper, the MSO-based S ⊆ V (Gk)\X such that |X |+|R∪BG0 \X0 (S )| ≥ t in k algorithms on graphs of bounded treewidth could be O1) 0 0 time f(δ)n . The subroutine returns the set S ∪ X replaced by standard dynamic programming algorithms as a solution. with running time 2Otw(G))n or 2Otw(G) log(tw(G)))n. On 0 Since G is an induced subgraph of G, any solution H-minor-free graphs this leads to EPTASs with running 0 0 S = S ∪ X returned by the subroutine covers at times on the form 2O1/)n + nO1) or 2O1/ log(1/))n + least t vertices in G. We return the smallest S as nO1), which is comparable to the fastest previously our (1 + )-approximate solution. In the iteration of known results. 0 the outer loop where k = OPT we have that Gk has a set Z of size OPT that covers t vertices in G0. 0 r 0 References Observe that Z \ X covers at least t − |B 0 (X )| of Gk 0 r 0 0 0 V (G )\B 0 (X ) in the graph G \X . Thus the solution k Gk k returned by the dynamic programming algorithm has [1] J. Alber, H. L. Bodlaender, H. Fernau, 0 size at most |Z \ X | ≤ |Z| = OPT and the solution T. Kloks, and R. Niedermeier, Fixed parameter al- returned by the subroutine in this iteration is at most gorithms for dominating set and related problems on OPT +|X0| ≤ OPT (1+0ρ). Chosing 0 = /ρ concludes planar graphs, Algorithmica, 33 (2002), pp. 461–493. the analysis of our EPTAS for Partial r-Dominating [2] J. Alber, M. R. Fellows, and R. Niedermeier, Set. An EPTAS for Partial Vertex Cover can be Polynomial-time data reduction for dominating set, constructed in a similar manner. Journal of the ACM, 51 (2004), pp. 363–384. [3] S. Arnborg, J. Lagergren, and D. Seese, Easy Lemma 5.5. There is an EPTAS for Partial r- problems for tree-decomposable graphs, J. Algorithms, Dominating Set and Partial Vertex Cover on 12 (1991), pp. 308–340. apex-minor-free graphs. [4] B. S. Baker, Approximation algorithms for NP- complete problems on planar graphs, J. ACM, 41 Finally by applying Theorems 4.1 and 4.2 together (1994), pp. 153–180. with Lemmata 5.1, 5.2, 5.3, 5.4 and 5.5 we get the [5] H. Bodlaender, F. V. Fomin, D. Lokshtanov, following corollary. E. Penninkx, S. Saurabh, and D. M. Thilikos, (Meta) Kernelization, in FOCS 2009, IEEE, 2009, Corollary 5.1. Feedback Vertex Set, Ver- pp. 629–638. tex Cover, Connected Vertex Cover, Cy- [6] H. L. Bodlaender, A partial k-arboretum of graphs cle Packing, Diamond Hitting Set, Minimum- with bounded treewidth, Theor. Comp. Sc., 209 (1998), Vertex Feedback Edge Set, Vertex-H-Packing, pp. 1–45. Vertex-H-Covering, Maximum Induced Forest, [7] R. B. Borie, R. G. Parker, and C. A. Tovey, Au- Maximum Induced Bipartite Subgraph and Max- tomatic generation of linear-time algorithms from pred- imum Induced Planar Subgraph admit an EPTAS icate calculus descriptions of problems on recursively on H-minor-free graphs. Edge Dominating Set, constructed graph families, Algorithmica, 7 (1992), pp. 555–581. Dominating Set, r-Dominating Set, q-Threshold [8] J. Chen, H. Fernau, I. A. Kanj, and G. Xia, , , Dominating Set Connected Dominating Set Parametric duality and kernelization: Lower bounds Directed Domination, r-Scattered Set, Mini- and upper bounds on kernel size, SIAM J. Comput., mum Maximal Matching, Independent Set, Max- 37 (2007), pp. 1077–1106. imum Full-Degree Spanning Tree, Max In- [9] B. Courcelle, The monadic second-order logic of duced at most d-Degree Subgraph, Max In- graphs I: Recognizable sets of finite graphs, Information ternal Spanning Tree, Induced Matching, Tri- and Computation, 85 (1990), pp. 12–75. [10] , The expression of graph properties and graph [28] M. Grohe, Local tree-width, excluded minors, and transformations in monadic second-order logic, Hand- approximation algorithms, Combinatorica, 23 (2003), book of Graph Grammars, (1997), pp. 313–400. pp. 613–632. [11] A. Dawar, M. Grohe, S. Kreutzer, and [29] S. Khanna and R. Motwani, Towards a syntactic N. Schweikardt, Approximation schemes for first- characterization of ptas, in STOC 1996, ACM, 1996, order definable optimisation problems, LICS 2006, pp. 329–337. (2006), pp. 411–420. [30] J. Kleinberg and A. Kumar, Wavelength conversion [12] E. D. Demaine, F. V. Fomin, M. Hajiaghayi, in optical networks, Journal of Algorithms, 38 (2001), and D. M. Thilikos, Bidimensional parameters and pp. 25–50. local treewidth, SIAM J. Discrete Math., 18 (2004/05), [31] D. J. Kleitman and D. B. West, Spanning trees pp. 501–511. with many leaves, SIAM J. Discrete Math., 4 (1991), [13] E. D. Demaine, F. V. Fomin, M. Hajiaghayi, pp. 99–106. and D. M. Thilikos, Subexponential parameterized [32] R. J. Lipton and R. E. Tarjan, A separator theorem algorithms on bounded-genus graphs and H-minor-free for planar graphs, SIAM J. Appl. Math., 36 (1979), graphs, J. ACM, 52 (2005), pp. 866–893. pp. 177–189. [14] E. D. Demaine and M. Hajiaghayi, Bidimension- [33] , Applications of a planar separator theorem, ality: New connections between FPT algorithms and SIAM J. Comput., 9 (1980), pp. 615–627. PTASs, in SODA 2005, ACM-SIAM, 2005, pp. 590– [34] N. Robertson, P. D. Seymour, and R. Thomas, 601. Quickly excluding a planar graph, J. Comb. Theory [15] , Bidimensionality, in Encyclopedia of Algo- Series B, 62 (1994), pp. 323–348. rithms, M.-Y. Kao, ed., Springer, 2008. [35] S. Tazari and M. Muller-Hannemann¨ , Dealing [16] , The bidimensionality theory and its algorithmic with large hidden constants: Engineering a planar applications, Comput. J., 51 (2008), pp. 292–302. steiner tree ptas, in ALENEX, SIAM, 2009, pp. 120– [17] E. D. Demaine and M. Hajiaghayi, Linearity of grid 131. minors in treewidth with applications through bidimen- sionality, Combinatorica, 28 (2008), pp. 19–36. A Treewidth. [18] F. Dorn, F. V. Fomin, and D. M. Thilikos, Subex- A tree decomposition of a graph G is a pair (X ,T ), where ponential parameterized algorithms, Computer Science Review, 2 (2008), pp. 29–39. T is a tree and X = {Xi | i ∈ V (T )} is a collection [19] D. Eppstein, Subgraph isomorphism in planar graphs of subsets of V such that the following conditions are and related problems, J. Graph Algorithms and Appli- satisfied. cations, 3 (1999), pp. 1–27. S [20] , Diameter and treewidth in minor-closed graph 1. i∈V (T ) Xi = V (G). families, Algorithmica, 27 (2000), pp. 275–291. 2. For each edge xy ∈ (G), {x, y} ⊆ Xi for some [21] U. Feige, M. Hajiaghayi, and J. R. Lee, Improved i ∈ V (T ). approximation algorithms for minimum-weight vertex 3. For each x ∈ V (G) the set {i | x ∈ X } induces a separators, in Proceedings of the 37th annual ACM i connected subtree of T . Symposium on Theory of computing (STOC 2005), New York, 2005, ACM Press, pp. 563–572. The width of the tree decomposition is max |X |− [22] , i∈V (T ) i F. V. Fomin, P. A. Golovach, and D. M. Thilikos 1. The treewidth of a graph G, tw(G), is the minimum Contraction bidimensionality: the accurate picture, in ESA 2009, vol. 5757 of LNCS, Sptinger, 2009, pp. 706– width over all tree decompositions of G. 717. [23] F. V. Fomin, D. Lokshtanov, V. Raman, and B MSO S. Saurabh, Subexponential algorithms for partial The syntax of MSO of graphs includes the logical cover problems, in FSTTCS, 2009, pp. 193–201. connectives ∨, ∧, ¬, ⇔, ⇒, variables for vertices, edges, [24] F. V. Fomin, D. Lokshtanov, S. Saurabh, and set of vertices and set of edges, the quantifiers ∀, ∃ that D. M. Thilikos, Bidimensionality and kernels, in can be applied to these variables, and the following five SODA 2010, ACM-SIAM, 2010, pp. 503–510. binary relations: [25] F. V. Fomin and D. M. Thilikos, New upper bounds on the decomposability of planar graphs, Journal of 1. u ∈ U where u is a vertex variable and U is a vertex Graph Theory, 51 (2006), pp. 53–81. set variable. [26] F. V. Fomin and D. M. Thilikos, New upper bounds on the decomposability of planar graphs, J. Graph 2. d ∈ D where d is an edge variable and D is an edge Theory, 51 (2006), pp. 53–81. set variable. [27] R. Gandhi, S. Khuller, and A. Srinivasan, Ap- 3. inc(d, u), where d is an edge variable, u is a vertex proximation algorithms for partial covering problems, variable, and the interpretation is that the edge d J. Algorithms, 53 (2004), pp. 55 – 84. is incident on the vertex u. 4. adj(u, v), where u and v are vertex variables u, and the interpretation is that u and v are adjacent. 5. Equality of variables, =, representing vertices, edges, set of vertices and set of edges.