A lower bound for the tree-width of planar graphs with vital linkages
AA lower bound for the tree-width of planargraphs with vital linkages
Isolde Adler, Philipp Klaus KrauseNovember 2, 2018
Abstract
The disjoint paths problem asks, given an graph G and k + 1 pairsof terminals ( s , t ) , . . . , ( s k , t k ), whether there are k + 1 pairwise disjointpaths P , . . . , P k , such that P i connects s i to t i . Robertson and Seymourhave proven that the problem can be solved in polynomial time if k isfixed. Nevertheless, the constants involved are huge, and the algorithmis far from implementable. The algorithm uses a bound on the tree-width of graphs with vital linkages, and deletion of irrelevant vertices.We give single exponential lower bounds both for the tree-width of planargraphs with vital linkages, and for the size of the grid necessary for findingirrelevant vertices. The disjoint paths problem is the following problem.
Input : Graph G , terminals ( s , t ) , . . . , ( s k , t k ) ∈ V ( G ) k +1) Question : Are there k + 1 pairwise vertex disjoint paths P , . . . , P k in G such that P i has endpoints s i and t i ?It is a classic problem in algorithmic graph theory and it has many applications,e. g. in routing problems, PCB design and VLSI layout [1, 9]. It is NP-hard [7],and it remains NP-hard on planar graphs [10]. Robertson and Seymour provedthat for fixed k it is decidable in polynomial time [14]. More precisely, theyshowed that it can be decided in FPT time f ( k ) · | V ( G ) | , where f is a com-putable function. For planar graphs, Reed et al. [11] gave an algorithm thatsolves the problem in time g ( k ) · | V ( G ) | for some computable function g . Bothfunctions f and g are not really made explicit. They are huge towers of ex-ponentiations, and the algorithms are far from being implementable, even forsmall k . Hence an important task is to improve the algorithms towards a betterdependency on the parameter k .We address this problem, exhibiting a lower bound on the parameter de-pendency arising from an essential technique used in both algorithms [14, 11].1 a r X i v : . [ c s . D S ] N ov G s t i i P i st jj P j Figure 1: DPP in planar Graph G containing grid G This technique is finding irrelevant vertices [12], i.e. vertices in the input graph G , such that their deletion does not affect the answer to the problem. This isclosely related to the theorem on vital linkages [15], and it is a main source ofthe impractical parameter dependency. The irrelevant vertices can be guaran-teed if G contains a sufficiently large subdivided grid as a subgraph (dependingon k ). Hence an interesting open question is to determine tight bounds on thesize of the subdivided grid, necessary for G to contain an irrelevant vertex. Inthis paper we give a lower bound by showing that a (2 k + 1) × (2 k + 1) grid maynot suffice – even in planar graphs. Despite recent progress [8], the quest forgood upper bounds is still open.Indeed, grids occur naturally in many graphs that are relevant in practicalapplications of the disjoint paths problem, such as PCB and VLSI design. Formany classes of NP-hard graph problems, attempts have been made to reducethe computational complexity by restricting the class of input graphs by anupper bound on a width parameter, such as tree-width and rank-width [5].However, grids remain notoriously hard to deal with [13, 4, 6]. Graphs are finite, undirected and simple. We denote the vertex set of a graph G by V ( G ) and the edge set by E ( G ). Every edge is a two-element subset of V ( G ).A graph H is a subgraph of a graph G , denoted by H ⊆ G , if V ( H ) ⊆ V ( G ) and2 ( H ) ⊆ E ( G ). A path in a graph G is a sequence P = v , . . . , v n of pairwisedistinct vertices of G , such that { v i , v i +1 } ∈ E ( G ) for all 1 ≤ i ≤ n −
1. We usestandard graph terminology as in [3].
Definition 1 (Grid) . Let m, n ≥ . The ( m × n ) grid is the graph G m,n givenby V ( G m,n ) := (cid:8) , . . . , m (cid:9) × (cid:8) , . . . , n (cid:9) , and E ( G m,n ) := (cid:8) { ( i , i ) , ( j , j ) } (cid:12)(cid:12) ( i = j and | j − i | = 1) or ( | j − i | = 1 and i = j ) (cid:9) A subdivided grid is a graph obtained from a grid by replacing some edgesof the grid by pairwise internally vertex disjoint paths of length at least one. Embeddings of graphs in the plane, planar graphs and faces are defined in theusual way.
Definition 2 (Inner vertex) . Let G be a grid embedded in the plane. An innervertex of G is a vertex that does not lie on the outer face of G . Definition 3 (Crossing) . We say that a path crosses the grid G if it containsan inner vertex of G and its endpoints are not inner vertices of G . For k ∈ N wesay that a path P = p , p , . . . , p n crosses G k times, if it can be split into k paths P = p , p , . . . , p i , P = p i , p i +1 , . . . p i , . . . , P k − = p i k − , p i k − +1 , . . . , p n with each P i , i = 0 , . . . k − crossing G . From now on we will consider the case of a planar graph G containing a grid G with all s i and t i lying on the edge of or outside the grid G (Figure 1).Intuitively, we construct our example from a grid G of sufficient size. We addendpoints s and t on the boundary of the grid, mark the areas opposite tothe grid as not part of the graph and connect s to t without crossing the grid.Now we continue to mark vertices by s i and t i in such a way that P i has tocross G as often as possible (in order to avoid crossing P j , j < i ). Once s i and t i have been added we remove the area opposite to the grid from s i from thegraph. Figure 2(a) shows the situation after doing this for i up to 2. In thisconstruction P does not cross the grid at all, while P crosses it once and P i +1 crosses it twice as often as P i for i >
0: Let k i be the number of times P i crossesthe grid. k = 0 , k = 1 , k i +1 = 2 k i , k i = 2 i − , i >
0. After the last P i has beenadded, the areas opposite to the grid from both s i and t i are removed from thegraph as seen in Figure 2(c).Formally, to construct problem and graph with k + 1 terminals, we use a(2 k + 1) × (2 k + 1) grid. Let the vertices on the left border of the grid be n , . . . , n k . Terminals are assigned as follows: t is the topmost vertices on theleft border on the grid, t the middle vertices on the right border. For all otherterminals: s i := n k − i , t i := n · k − i . Then add edges going around the t i to the3raph: For i > , t i = n j add n j − n j +1 , n j − n j +1 , . . . , n j − k − i − n j +2 k − i − , andon the right border of G do the analogue for t . See Figure 2(d) for a graphconstructed this way. Theorem 1.
There is only one solution to the constructed DPP, all verticesof the graph lie on paths of the solution and the grid is crossed k − times bysuch paths.Proof. To connect s k to t k we need to cross G at least once (Figure 3(a)).However, a solution in which P K crosses G only once would block P k − . Toconnect s k − to t k − P k has to be routed around t k − , which requires leavingand reentering G (Figure 3(b)). Thus inductively constructing a solution each P i , i > G and doubles the number of crossings in each P j , j > i . The solution uses all edges on the left side of G and uses all but oneof the edges on the right side. Thus the only way to connect s to t is withoutcrossing the grid. q. e. d.In particular, G has no irrelevant vertex in the sense of [12]. Corollary 1.
There is a planar graph G with k + 1 pairs of terminals such that • G contains a (2 k + 1) × (2 k + 1) grid as a subgraph, • the disjoint paths problem on this input has a unique solution, • the solution uses all vertices of G ; in particular, no vertex of G is irrele-vant. Conjecture.
There is a function f ∈ O ( k ) such that for every planar input G together with k + 1 pairs of terminals: if G contains a subdivided f ( k ) × f ( k ) grid as a subgraph, then G contains an irrelevant vertex. We refer the reader to [2] for the definitions of tree-width and path-width . Definition 4 (Vital linkage) . Let L be a subgraph of G such that every compo-nent of L is a path and all vertex of G are vertices of L . The pattern of L isthe set of vertices of degree in L . L is a vital linkage in G if there is no othersuch L that has the same pattern. Theorem 2 (Robertson and Seymour [15]) . There are functions f and g suchthat if G has a vital linkage with k components then G has tree-width at most f ( k ) and path-width at most g ( k ) . Recall that the n × n grid has path-width n and tree-width n . Our exampleyields a lower bound for f and g : Corollary 2.
Let f and g be as in Theorem 2. Then k − + 1 ≤ f ( k ) and k − + 1 ≤ g ( k ) . ss t s t (a) P , . . . , P t s st s t s t (b) P , . . . , P t s s t st s t s t (c) P , . . . , P t ss st s t t s t (d) Actual graph with grid Figure 2: Construction of graph and solution5 roof.
Looking at the graph G and DPP constructed above the solution to theDPP is, due to its uniqueness, a vital linkage for the graph G . G contains a(2 k + 1) × (2 k + 1) grid as a minor. The tree-width of such a grid is 2 k + 1 , itspath-width 2 k + 1 [4]. Thus we get lower bounds 2 k − + 1 ≤ f ( k ) , g ( k ) for thefunctions f and g . q. e. d.6 s s t st s t s t (a) P , P t s s t st s t s t (b) P , P , . . . , P t s s t st s t s t (c) P , P , . . . , P t s s s t st s t t (d) P , P , . . . , P Figure 3: Optimality of solution7 eferences [1] Alok Aggarwal, Jon M. Kleinberg, and David P. Williamson. Node-disjointpaths on the mesh and a new trade-off in vlsi layout.
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