Parameterized tractability of some (efficient) Y -domination variants for planar graphs and t-degenerate graphs
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(2) In [8] an algorithm that runs in O(n2 ) for the subclass of cocomparability graphs (which are properly contained in AT-free graphs) can be found. For complexity results on the minus domination problem, (i.e., Y = {−1, 0, +1}) we refer to [11]. For example, in this paper it is shown that the minus domination problem is NP-complete for chordal graphs and chordal bipartite graphs and a linear time algorithm is given for the minimum minus dominating function on trees. In [17] it is shown that the question whether there exists an efficient minus domination problem is NPcomplete, even when restricted to chordal graphs, chordal bipartite graphs, planar bipartite graphs, and planar graphs with maximum degree 4. The method they use to show NP-completeness is very simple and was used earlier in the paper [11] to show that the minus domination problem is NP-complete even when restricted to bipartite graphs and chordal graphs by a transformation from the dominating set problem. (The idea used in [17] is to attach a P3 to each vertex of the graph and transform the problem to the efficient domination problem. The same method was used also in [11]) In their paper [17] the authors also show that, for trees this problem coincides with the ”ordinary” efficient domination problem. It is known that the efficient signed domination (i.e., Y = {−1, +1}) and the efficient minus domination problem can be solved in linear time for special cases of interval graphs (see [17]). In this paper linear time algorithms are given for these special classes of interval graphs (called ”chain interval graphs” defined as those interval graphs for which every pair of minimal separators have an empty intersection). As far as we know, the complexity of the efficient minus domination, the minus domination and the efficient signed domination problem are still open for AT-free graphs, even for interval graphs. In Section 4 of this paper we show a general method that shows that the FP version of the efficient domination problem can also be solved for t-degenerate graphs for fixed values of t. This class of graphs is very general and contains for example bounded treewidth graphs, chordal graphs with bounded clique number, graphs with bounded bandwidth, cutwidth etc., general planar graphs, and many other classes. We now turn to some remarks on the methods we use to solve the Y -domination problem for planar graphs. Recently it was shown in [1], that if the domination number of a planar graph is√at most k, then the treewidth can be bounded by δ k for some reasonably. small constant δ (for a detailed proof see [1]). This result was used to improve a result by Downey and Fellows [10], who showed that the domination number for planar graphs is in FPT and can be solved in O(11k n) time. The relation between planar domination and treewidth as expressed above was used to obtain a new uniformly polynomial time algorithm for the √ planar dominating set problem which runs in time O(c k n) for some reasonable constant c [1] (for present best bounds on c we refer to [1]). The same method (as will be described in [7]) gives a somewhat more direct method (at least to describe it) with a similar (small) worst case constant d, but runs √ k 2 in time O(d n ). This method uses the result of [19] which gives an algorithm for the branchwidth of a planar graph in O(n2 ) time. At the moment it is not clear which of these two algorithms could be of more practical use (if any of the two). Clearly, both algorithms are not hard to implement and both have as an output an approximate tree decomposition which is then used to solve the problem. The advantage of the latter one is that it gives a treewidth value which is sure to be close to the optimal one whereas the first algorithm gives only an approximate tree decomposition of which the approximation factor is not guaranteed. In this paper we show that this result can be used also to obtain ”fast” FP algorithms for the Y -dominating problem on planar graphs by simply applying the result stated above.. 2. Preliminaries. For detailed information on fixed parameter complexity we refer to the recent work of Downey and Fellows [10]. Readers who are unfamiliar with treewidth can have a look in [15]. For more information on special classes of graphs and problems not defined here, we refer to [5]. In this section we define some of the problems that have been mostly studied as special cases of the Y domination problem that were mentioned in the introduction. But we start with the most important results for the problem in it’s full generality. For a graph G = (V, E) and a vertex x ∈ V , we write N (x) for the neighborhood of x and N [x] for its closed neighborhood , i.e., N [x] = N (x) ∪ {x}. By the way, in the rest of this paper, for convenience we turn to the modern abbreviated style of writing this as N [x] = N (x) + x. We use the same notation for sets, and also we use A − x (if A is some set) for example instead of the more cumbersome notation A \ {x}..
(3) If f is an assignment of integers to the vertices of a graph GP = (V, E) and A ⊆ V then we use the notation f (A) = x∈A f (x).. • The efficient signed domination problem asks for an {+1, −1}-assignment to all vertices such that f (N [x]) = 1 for every vertex x.. Definition 1 Let Y be a finite set of integers. The Y -domination problem asks for an assignment f : V → Y set Y of integers such that for each vertex x, f (N [x]) ≥ 1. The efficient Y -domination problem asks for such an assignment with f (N [x]) = 1 for all vertices x ∈ V .. Notice that not every graph has an efficient assignment. A square may serve as a trivial example of this. The parameterized version of these type of problems we consider in this paper asks for an assignment with value at most k, where k is the parameter of the FP problem.. These problems were first described investigated in the paper by [2]. We mention here the most important results of this paper. Two Y -dominating functions are equivalent if they have the same closed neighborhood sum at every vertex. D. W. Bange in his paper proved that G has an efficient Y -dominating function if and only if all equivalent Y -dominating functions have the same weight. Furthermore, in this paper it is shown that if the closed neighborhood matrix of G is invertible then G has an efficient Y -dominating function for some set Y . This is quite obvious, but unfortunately, the solution is in most cases not integer. Special cases that would be interesting to look at also from a point of view in coding theory, are certain strongly regular graphs (see, e.g., [9]). An analysis of the spectrum of the closed neighborhood matrices should give some insight which of these graphs are worth some detailed analysis. (For the moment we only mention that strongly regular graphs for which the ordinary adjacency matrix has smallest eigenvalue at least -2 were characterized, and they all seem to have efficient dominating sets.) We don’t go into cases where the set Y also has values which are real or fractional numbers. For some results of this type (i.e., real numbers) the reader is referred to [14]. The problem of finding Y -assignments is of theoretical interest at least and for various sets Y of great importance for many practical and important applications, as was pointed out for example in [2, 17, 20] and in many other papers. The most researched cases of Y -assignments are listed in the following definition. • An efficient assignment is a {0, 1}-assignment f of the vertices such that for every vertex x ∈ V , f (|N [x]) = 1.. Definition 2 If Y is a set of integers then we define the value of the Y -assignment f as || {x | f (x) > 0} P ||. The weight of the Y -assignment is defined as x∈V f (x). Clearly, for many cases there is a close relation between the weight and value of assignments. As a trivial example we mention that these values coincide for efficient and efficient signed domination. Consider an Y -assignment f of a graph G = (V, E) and let D = {x | f (x) > 0}. Clearly, by definition, D is a dominating set, since every vertex must have at least one vertex with a positive number assigned to it in it’s closed neighborhood. Unfortunately dominating set is W [2]-hard, (see, e.g., [10]) and hence it is not expected that this problem has a fixed parameter solution. However, when restricted to planar graphs, the problem is in FPT. A proof of this appears for example in [10], and the result was recently improved by [1], using graph separator techniques (for more information and results on graph separators, see the forthcoming book [7]). The most important tool we use is the following theorem, which appeared in [1]. The technique used to prove it will appear in a more general setting in [7]. Theorem 3 If a planar graph has a dominating set √ of size at most k, then the treewidth is bounded by δ k, with some small constant δ. A constructive algorithm finding an approximate tree decomposition with width √ δ k can be given in linear time. At the moment the constant δ is already quite small (smaller than 50), but work on it is under construction. The proof of this theorem is based on methods that will appear in [7], and a preliminary full proof can be found in [1].. • The minus domination problem asks for an {0, −1, +1}-assignment of the vertices of the graphs such that f (N [x]) ≥ 1 for every vertex x.. 3. • The efficient minus domination problem asks for an {0, +1, −1}-assignment f to the vertices such that f (N [x]) = 1 for every vertex x.. In this section we show a uniformly polynomial time algorithm for the efficient Y -assignment on planar graphs.. (Efficient) Y -domination on planar graphs.
(4) Let k be the parameter, i.e., we try to find an efficient Y -assignment f with value at most k in a planar graph, (i.e., the number of vertices which are assigned a positive value is at most k). We mentioned already that if f is an Y -efficient assignment with value at most k, then clearly it also must have a dominating set with size at most k. Hence we can apply Theorem 3√ to find a tree decomposition with width at most ` = δ k for some reasonable constant δ. The following, somewhat surprising result shows that we don’t have to worry about the actual weight of an efficient or efficient minus or efficient signed dominating set. The theorem below is due to [2] (and in fact a more general result was proved in this paper; we state it in this way, since it suffices for our purposes here). Theorem 4 A graph G = (V, E) has an efficient Y dominating assignment if and only if all equivalent Y dominating functions have the same weight. Hence if our algorithm finds an efficient Y -dominating assignment then it’s weight is fixed. Our algorithm first computes an √ approximate tree decomposition of width at most c k (see, e.g., [1, 7]). For practical aplications we recommend to optimize the treewidth from this approximation as much as possible with ad-hoc methods, since this could improve the time bound of our algorithm considerably. At the moment we don’t have a general method to improve the treewidth given by Theorem 3 but we feel that the exact treewidth could be considerably smaller than the approximate one given above. (Maybe for particular cases of the Y -domination problem even some definite improvements on the treewidth are possible.) Let (T, S) be a √ tree decomposition with bounded width ` (where ` = c k) as given output of the approximation algorithm mentioned in Theorem 3. (For the current best value of the constant c, we refer to [1].) Here S is a set of subsets Si , such that there is a one-to-one correspondents with the vertices in the tree and the subsets of S. It is easy to see, (see, e.g., [15]) that we may assume that a rooted tree decomposition can be made binary and that the so-called ”bags” Si can be made of four different types. • An introduce node i; The node i in the tree has only one child and the corresponding bag Si has a ”new” vertex x which does not appear in the bag corresponding with the child j of i. The rest of the bag Si − x is exactly the same as the bag corresponding with the bag Sj , i.e., Si = Sj + x.. • A forget node i; The node i in the tree has only one child j and its bag Si is exactly the same as the bag of its child except for one node that is missing; i.e., Si = Sj − x. • A join node i has two children p and q. The three corresponding bags are exactly the same. • A leaf node i; which is simply a leaf of the rooted tree T . (If the root of T is also a pendant vertex, it is not considered as a leaf.) We use this tree decomposition to find an efficient dominating set. Definition 5 A Y -assignment fi to the vertices of a bag Si is called feasible for a node i in the tree, if there exists an extension f of fi to all vertices from all bags corresponding to the nodes in the subtree of i such that for every vertex x of a bag of a node in the subtree, which is not in Si , |f (N [x])| = 1. We use dynamic programming and work our way up in the tree and keep track of all possible feasible assignments with their respective values. Let us start with the leaves. For a leaf node i we simply compute a table of all possible assignments of the vertices in Si with elements from Y and keep track of it’s value. Now consider a join node i and let p and q be its children. An assignemt at i is feasible if and only if it is feasible for both it’s children p and q. The value of a feasible assignment at i is simply the weight of the assignments of the assignments at p and q, not double counting the values at i of course. Consider an introduce node i. Let x be the vertex being introduced. Consider all possible assignments of Si which are feasible when restricted to Si and which are feasible when restricted to the child j of i. If the assignment remains feasible within Si then this is added to the table of feasible assignments of i. If x is assigned a positive value, then the value of this feasible assignment is updated. Finally consider a forget node i and let x be the vertex which disappears from the child. Because we want the final set to be efficient, we now must insist that f (N [x]) = 1 for a feasible assignment f , since this value cannot be changed when we go further up in the tree. If this does not hold, the feasible assignment is not put in the table of feasible assignments at node i (otherwise it is again feasible for node i). Our algorithm comes to an end when we reach the root r of the subtree. At the root, we only have to compute the feasible assignments, as described above, and make a final check if the value is at most k..
(5) Theorem 6 There exists an algorithm which runs in time O(|Y |` n) to check if a planar graph G has an effi√ cient Y -assignment of value at most k. Here ` = c k for some small constant k. Remark 7 In this section we have given the exact description of the algorithm for the efficient Y domination problem, but it should be immediately clear that only slight modifications are necessary to describe the algorithm for ”ordinary” y-domination.. 4. Efficient domination of t-degenerate graphs. In this section we present a uniformly polynomial-time algorithm for solving the fixed parameter version of the efficient domination problem on t-degenerate graphs (t fixed). Definition 8 A graph G is t-degenerate if there is a linear ordering L of its vertices v1 , v2 , · · · , vn such that each vi , 1 ≤ i ≤ n, is adjacent to at most t vertices that succeed vi in L. Equivalently, G is t-degenerate if every induced subgraph has a vertex of degree at most t. The class of t-degenerate graphs contains many well-known classes of graphs; for example, graphs embeddable on some fixed surface, graphs of bounded treewidth, and graphs that are H-minor free. For some recent results on tdegenerate graphs, see [4]. The Efficient Domination k-Set problem asks whether an input graph G contains an efficient domination set with k vertices. Recall that for any vertex v of G, an efficient dominating set S contains exactly one vertex in the closed neighborhood NG [v]. Furthermore, if a vertex v is in S then v is the only vertex of S inside the closed neighborhood NG [NG [v]] of NG [v]. We now give our algorithm. First, we compute an ordering L = v1 , v2 , . . . , vn of the vertices of G that satisfies the definition of a t-degenerate graph, which can be done easily in linear time. Then we build a search tree to search for an efficient dominating set S of size k according to this ordering.. in S no other vertices of NG [NG [v]] are in S. Therefore we label the nodes of the search tree as follows: The children of the root are labeled by (vi , Gi ), where vi ∈ NG [v1 ] and Gi = G − NG [NG [v1 ]]. For a node labeled by (vi , Gi ), we pick the first vertex u of Gi in the linear ordering L, and label its children by (vj , Gj ), where vj ∈ NGi [u] and Gj = G − NGi [NGi [u]]. For each root-to-leaf path in the search tree, the set of vertices in the labels is a candidate for an efficient dominating set of size k. Therefore we need only build the search tree up to depth k. Once we build the search tree, we check for each such candidate if it is indeed an efficient dominating set of G. Since each node of the search tree has at most t + 1 children, each node takes O(tn) time, and checking whether a candidate is an efficient dominating set takes O(tn) time, the total running time of the algorithm is O((t + 1)k+1 n). Theorem 9 The Efficient Domination k-Set problem can be solved in O((t + 1)k+1 n) time for tdegenerate graphs.. 5. Concluding remarks. Notice that a vast amount of problems remain open. We only mention some of the most obvious ones. As far as we know no complexity results are known for the efficient minus and efficient signed domination problem for (classes of) AT-free graphs. It seems not likely that the method of [6, 18] can be used also to solve these problems in polynomial time, unless we require additionally that the separator sets have bounded size. As mentioned, all equivalent Y -dominating functions have the same weight if and only if the graph has an efficient Y -dominating function (see [2]). It would be interesting to know if a similar statement can be made for the value of efficient Y -dominating functions. A final interesting open problem is the complexity of the signed domination problem on planar graphs.. References. The root of the search tree is labeled by (∗, G). Each of the remaining nodes in the search tree is labeled by a pair (v, G), where v is the vertex selected to be in S and G is the graph in which we search for the remaining vertices of S.. [1] Alber, J., H. Bodlaender, R. Niedermeier, and H. Fernau, Faster algorithms for domination of planar graphs, Proc. 17th SWAT’00 , SpringerVerlag, LNCS 1851, pp. 97–110.. As we mentioned, for a vertex vi , exactly one of the vertices in NG [vi ] belongs to S, and once a vertex vi is. [2] Bange, D. W., A. Barkauskas, L. H. Post, and P. J. Slater, Generalized domination and efficient.
(6) domination in graphs, Discrete Mathematics 159, (1996), pp. 1–11. [3] Bodlaender, H., A tourist guide through treewidth, Acta Cybernetica 11, (1993), pp. 1–23. [4] Borodin, O. V., A. V. Kostochka, B. Toft, Variable degeneracy: extensions of Brooks’ and Gallai’s theorems, Discrete Mathematics 214, (2000), pp. 101–112. [5] Brandstaedt, A., V. B. Le, and J. P. Spinrad, Graph classes–A Survey, SIAM Monographs on Discrete Mathematics and Applications, Philadelphia, 1999. [6] Broersma, H. J., T. Kloks, D. Kratsch, and H. M¨ uller, Independent sets in AT-free graphs, To appear in: SIAM J. of Discr. Math.. [7] Broersma, Hajo and Ton Kloks, Separators in graphs. To appear. [8] Chang, M.-S., Weighted domination of cocomparability graphs, Discrete Applied Mathematics 80, (1997), pp. 135–148. [9] Corneil, Derek G. and Rudi mathon (eds.), Geometry and combinatorics: selected works of J. J. Seidel , London Ac. Press, 1991. [10] Downey, R. G., and M. R. Fellows, Parameterized complexity, Monographs in Computer Science, Springer-Verlag 1999. [11] Dunbar Jean, Wayne Goddard, Stephen Hedetniemi, Alice McRae, and Michael A. Henning, The algorithmic complexity of minus domination in graphs, Discrete applied mathematics 68, (1996), pp. 73–84. [12] Dunbar, Jean, Stephen Hedetniemi, Michael A. Henning, and Allice McRae, Minus domination in graphs, Discrete mathematics 99, pp. 35–47. [13] Fellows, M. R. and M. N. Hoover, perfect dominations, Australian Journal of Combinatorics 3, (1991), pp. 141–150. [14] Goddard, W. and Michael A. Henning, Real and integer domination in graphs, Discrete mathematics 199, (1999), pp. 61–75. [15] Kloks, T., Treewidth—Computations and Approximations, Springer–Verlag, LNCS 842, 1994.. [16] Lu, C. L., On the study of weighted efficient domination, PhD-thesis, Department of Computer Science, National Tsing Hua University, Taiwan, 1998. [17] Lu, C. L., S. -L. Peng and C. Y. Tang, Efficient minus and signed domination in graphs, To appear at ISAAC’00. [18] M¨ uller, Haiko, Asteroidal sets in graphs, Habilitation, Friedrich–Sciller–Universit¨at Jena, 1999. [19] Seymour, P. D. and R. Thomas, Call routing and the ratcatcher, combinatorica 14, (1994), pp. 217– 241. [20] Yen, C. C. and R. C. T. Lee, The weighted perfect domination problem and its variants, Discrete Applied Mathematics 66, (1996), pp. 147–160..
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