Grids containment and treewidth, PTASs

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1 Lecture 8 (30..0) Grids containment and treewidth, PTASs Author: Jakub Oćwieja Definition. A graph G is a t t grid if it consists of t vertices forming a square t t such that every two consecutive vertices in a row or in a colum are connected. As a result of previous excercises we know that a treewidth of a t t grid is t. A treewidth of a graph can only decrease as a result of taking minors so we can conclude that if a graph contain a t t grid as a minor it s treewidth is at least t. Following series of theorems gives us a symetric dependence. Theorem. (Robertson, Seymour [6]) ] c R [tw(g) > c t5 G contains a t t grid as a minor Theorem 3. (Robertson, Seymour [7], Grigoriev [4]) Given a planar graph G, V (G) = n and a number t it is posibble to find a t t grid as a minor of G or produce a decomposition of G of width 5t 6 in O(n log n) time. Theorem 4. (Demaine, Hajiaghayi []) graph H c(h) R graph G t if H minor G then G contains a t t grid as a minor or tw(g) c(h) t Definition 5. A graph G is an apex graph if there exists a vertex v such that G \ v is planar. Theorem 6. (Fomin, Golovach, Thilikos [3]) apex graph H c(h) R graph G t if G cannot be contracted to H then G can be contracted to one of each of size t t, or tw(g) c(h) t It is worthy to mention that there exists algorithm which for a given planar graph G construct its planar embedding in linear time [5]. From now we will assume a planar graph comes with its embedding.

2 or Proof of Theorem 3. To avoid technical parts of the proof we will only prove 5t boundary. We will give an algorithm that construct a tree decomposition recursively. If on some stage of recursion it fails we will produce a representation of a t t grid minor of a graph. In each step of recursion a created vertex T of tree decomposition will satisfy following condition: each vertex associated to T lies on the outer face of the remaining subgraph G. We will call this property circular cuts. Moreover, T 4t. To create a decomposition it is enough to consider following cases:. G is not biconnected Let v be a vertex which splits G into two disconnected subgraphs G and G, G G = {v}. There are two cases: G T and G T If so, v must lie on the outer face of G. v G G In this case we can create a decomposition vertex T = T {v} and then split it into two vertices: T G and T G, each consisting of no more than 4t vertices of G. G T = We can simply continue decomposing G using our algorithm. The same way we can decompose G using {v} as the beginning vertex of decomposition. As v is included in some vertex of the first decomposition we can easily merge these decompositions to get satisfactory result.. G is biconnected and T < 4t T contains all vertices of the outer face of G. In this case we can forget about (remove) some edge of the outer face of G as it connects two vertices of T.

3 v G G Otherwise. Let v be a vertex of the outer face such that v T. We can then create a new decomposition vertex T = T {v}. 3. G is biconnected and T = 4t We may split T into subsets N, E, S, W using clockwise order such that N = E = S = W = t. There exists vertex cut X between sets N and S such that < t. N W X G G E S Assume that X splits graph G into G and G such that G G = X. We can extend T of X ( T X 5t ) and create two tree decomposition subvertices: G (T X) and G (T X). Moreover G (T X) N E X W 4 t and the same inequality holds for the second created vertex. There exists vertex cut X between sets E and W such that < t. Analogously to the previous case. There exists t disjoint paths ( ) t i= connecting pairwise vertices of N and S and t disjoint paths (Q i ) t i= connecting pairwise vertices of W and E. In this case we will finally find a t t subgrid with respect to minors.

4 Consider sets of paths minimalizing M = Q i. We will prove this paths induces a t t subgrid. We have to prove that s intersects only with Q i s and each pair intersects exactly once. From the construction of paths the first property holds and it should be obvious that each pair (, Q j ) intersects at least one time. If any of such pairs intersects more than once, we will show a transformation to another set of paths with lesser value of M. Consider some and its intersections with paths (Q j ) t j=. If intersects path Q j twice in a row we can perform following transformation: Q j+ Q j+ Q j Q j Q j Q j Assume now that is the first (counting from the left) path that intersects some Q j more than once. Then there exists Q j such that intersects it, but doesn t cross it. Depending on the case we can perform one of the following transformations (there always exists j that fit one of these situations): Q j Q j Q j Q j As each of given transformations decrease value of M contradicting minimality of the selection, thus none of these cases occures. To sumarize, the choice of paths ( ) and (Q i ) induces a t t subgrid Q.E.D. Using these theorems we can solve many problems using the same idea combining grids containment and treewidth. Theorem 7. Vertex cover problem for a planar graph G can be solved in O ( O( k) ) time.

5 Q j + Q j + Q j Q j Proof. Using Theorem 3 for t = k + we get that either G contains a t t grid as a minor or has O( k) treewidth. In the first case, a t t grid contains a subset of t disjoint edges and thus minimal vertex cover s size cannot be less than t k+ = k +. As taking minors only shrink a minimal vertex cover we can simply answer no. In the second case, with bounded treewidth we can use a dynamic programming to achieve the goal in the assumed time. Theorem 8. K-path problem for a planar graph G can be solved in O ( O( k) ) time. Proof. Using Theorem 3 for t = k we find a t t grid minor representation of G or construct tree decomposition of width O( k). In the first case the subgrid contains t t k vertices and there obviously exists a path of length k. In the second case we can use bounded treewidth and a dynamic programming to construct O ( O( k log k) ) algorithm. To achieve O ( O(k) ) we have to be more precise and use a construction of tree decomposition - circular cuts. During dynamic programming the form of circular cats limit the space of states for each vertex of decomposition. Specifically, in each state it is enough to remember pairs of vertices which are already connected by a path. In this case connections between vertices creates a parenthesis over a cyclic set of vertices. There always exists a vertex that lies on the outer face of the embedding of a graph of connections (image above). Starting from it and moving clockwise it is enough to tell for each vertex if it is already connected with some other vertex and remember the direction (left, right) of this connection. This observation limits the size of the space of states to O(t 3 t ) = O( O( k) ) and thus dynamic programming can perform in O ( O( k) ) time Q.E.D.

6 At the end of this lecture we will focus on Baker s technique, which is a method for designing polynomial-time approximation schemes for problems on planar graphs. Theorem 9. (Baker []) For ɛ > 0 there exists a polynomial time algorithm which gives ( ɛ)-approximation of independent set for planar graphs. Proof. For a given ɛ let l = ɛ. Consider a partitioning of vertices of a planar graph G into layers given by bfs search starting from a vertex of the outer face of G. Let L i (for 0 i < l) be sum of layers which depth equals i modulo l. V = L i thus there exists j such that L j OP T ɛ OP T, where OP T is maximal stable set for G. Otherwise l l OP T = OP T V = OP T L i > ɛ OP T = l ɛ OP T > OP T i=0 Given j focus on H = G[V \ L j ]. H consists of l of l layer groups of a planar G and thus its treewidth is O(l). Thus, we can run a polynomial algorithm of finding maximal independent set for H running in polynomial time. The result RES holds RES OP T (V \ L j ) = OP T OP T L j ( ɛ) OP T i=0 Theorem 0. (Baker []) For ɛ > 0 there exists a polynomial time algotihm which gives ( + ɛ)-approximation of vertex cover for planar graphs. Proof. Given a vertex cover X of a planar graph G and ɛ > 0 we will construct a Y V (G) such that Y ɛ tw(g \ Y ) = O( ɛ ) Using this construction we will find ( + ɛ)-approximation by. Find a -approximation X of a minimal vertex cover of G. Use the construction to obtain Y, Y ɛ 3. Using dynamic programming find an optimal solution Z in G \ Y 4. Return Z Y Assume we are given a vertex cover X of a planar graph G and ɛ > 0. From Theorem 6 for any appex graph H and t = 7 + we can construct in polynomial time a t t almost subgrid of G with respect to contractions (as G is planar only one type of presented grids is possible) or a tree decomposition of width t.

7 However, a size of minimal vertex cover can only shrink during contractions and a grid s of size t t minimal vertex cover contains at least (t ) vertices which is in contradiction with a given vertex cover. Thus G has treewidth O( ). Given a tree decomposition of G we can find its vertex T such that its removal leaves components which vertices contain less than vertices of. For such T we can add T to Y, remove T form G and run the procedure recursively for each connected component of G. For each component branching procedure induces its vertex cover of size less than and we can recalculate tree ( ) decomposition of width O. We stop our recursion when each of connected components contains a part of X of size factor than ɛ. ɛ ɛ ɛ ɛ ( Obviously, the algorithm returns such Y, that tw(g \ Y ) O of Y can be bouded by the sum of geometric sequence ɛ ) = O ( ɛ ). Moreover, size Y i i i = i X i = O ( X : ɛ X ) = O (ɛ ) Balancing a value of ɛ we can remove a factor of O() from this boundary. It is easy to check that procedure given at the beginning of the proof returns (+ɛ)-approximation of the optimal solution of vertex cover problem. References [] Brenda S. Baker. Approximation algorithms for np-complete problems on planar graphs. J. ACM, 4():53 80, 994. [] Erik D. Demaine and MohammadTaghi Hajiaghayi. Linearity of grid minors in treewidth with applications through bidimensionality. Combinatorica, 8():9 36, 008. [3] Fedor V. Fomin, Petr A. Golovach, and Dimitrios M. Thilikos. Contraction obstructions for treewidth. J. Comb. Theory, Ser. B, 0(5):30 34, 0.

8 [4] Alexander Grigoriev. Tree-width and large grid minors in planar graphs. Discrete Mathematics & Theoretical Computer Science, 3():3 0, 0. [5] John Hopcroft and Robert Tarjan. Efficient planarity testing. J. ACM, (4): , October 974. [6] Neil Robertson and Paul D. Seymour. Graph minors. x. obstructions to tree-decomposition. J. Comb. Theory, Ser. B, 5():53 90, 99. [7] Neil Robertson, Paul D. Seymour, and Robin Thomas. Quickly excluding a planar graph. J. Comb. Theory, Ser. B, 6():33 348, 994.