A dominating set in an undirected and simple graph is a set such that every vertex lies either in or has a neighbour in . The Minimum Dominating Set problem takes a graph as input and the objective is to find a minimum size dominating set of . The corresponding decision problem is NP-hard karp1972reducibility and this even holds in very restricted settings, e.g. on planar graphs of maximum degree garey2002computers .
The following greedy algorithm approximates Minimum Dominating Set in an -vertex graph up to a factor johnson1974approximation ; lovasz1975ratio . Starting with the empty dominating set , the algorithm iteratively adds vertices to according to the following greedy rule until all vertices are dominated: in each round, choose the vertex that dominates the largest number of vertices which still need to be dominated. The greedy algorithm on general graphs is almost optimal: it is NP-hard to approximate Minimum Dominating Set within factor for some constant raz1997sub , and by a recent result it is even NP-hard to approximate Minimum Dominating Set within factor for every dinur2014analytical .
On several restricted graph classes Minimum Dominating Set can be approximated much better. For instance, the problem admits a polynomial-time approximation scheme (PTAS) on planar graphs baker1994approximation and, more generally, on graph classes with subexponential expansion har2017approximation . It admits a constant factor approximation on classes of bounded arboricity bansal2017tight and an approximation (where denotes the size of a minimum dominating set) on classes of VC-dimension bronnimann1995almost ; even2005hitting . While the above algorithms on restricted graph classes yield good approximations, they are computationally much more complex than the greedy algorithm. Unfortunately, the greedy algorithm does not provide any better approximation on these restricted graph classes than on general graphs (see for example Section 4 of bronnimann1995almost for an instance of the set cover problem, which can easily be transformed into a planar instance of the dominating set problem, where the greedy algorithm achieves only an approximation). Jones et al. jones2013parameterized showed how to slightly change the classical greedy algorithm to obtain a constant factor approximation algorithm on sparse graphs, more precisely, the algorithm computes a approximation of Minimum Dominating Set on any graph of degeneracy at most .
We follow the approach of Jones et al. jones2013parameterized and study small modifications of the greedy algorithm which lead to improved approximations on restricted graph classes. We denote the complete bipartite graph with vertices on one side and vertices on the other side by .
We present a greedy algorithm which takes as input a graph and an optional parameter . If run with the integer parameter and excludes as a subgraph for some , then the algorithm computes an approximation of Minimum Dominating Set (where denotes the size of a minimum dominating set in ). If run without the integer parameter , the algorithm outputs the largest subgraph that it found during its computation, as well as an approximation of Minimum Dominating Set. By running the classical greedy algorithm in parallel, the approximation ratios can be improved to , and , respectively.
Based on a known hardness result for the set cover problem on families with intersection it is easy to show that it is unlikely that polynomial time constant factor approximations exist even on -free graphs.
Comparison to other algorithms
Every -free graph has VC-dimension at most , hence the algorithms of bronnimann1995almost ; even2005hitting achieve approximations on the graphs we consider. The algorithm presented in bronnimann1995almost is based on finding -nets with respect to a weight function and a polynomial number of reweighting steps. The algorithm presented in even2005hitting
requires solving a linear program. Hence, even though these algorithms achieve the same approximation bounds as our modified greedy algorithm, our algorithm is much easier to implement and has much better running times. On the other hand,-free-graphs are strictly more general than degenerate graphs. Hence, our algorithm is applicable to a more general class of graphs than the algorithm of Jones et al. jones2013parameterized .
2 The greedy algorithm on biclique-free graphs
We first consider the following greedy algorithm which takes as input an optional parameter and a graph . We start by presenting how the algorithm works if the parameter is given with the input.
We initialise and . The set denotes the initial dominating set and denotes the set of vertices that have to be dominated. The algorithm runs in rounds and in every round it makes a greedy choice on a few vertices to add to the dominating set, until no vertices remain to be dominated. Formally, in each round we construct a new set which is obtained from by adding at most vertices . The set is obtained from by removing and their neighbours. We output the set as a dominating set, when .
Let us describe a round of the modified greedy algorithm. Assume that after round we have constructed a partial dominating set and vertices remain to be dominated. We choose vertices , , as follows. We choose as an arbitrary vertex that dominates the largest number of vertices which still need to be dominated, i.e., a vertex which maximises . Here, denotes the neighbourhood of a vertex , including the vertex . Let . We continue to choose vertices inductively as follows. If the vertices and sets have been defined, we choose the next vertex as an arbitrary vertex not in that dominates the largest number of vertices of , i.e., a vertex which maximises and let . We terminate this round and add to if either we have , or for each . We mark the vertices and their neighbours as dominated, i.e., we remove from the set all vertices of to obtain the set and start the next round.
The crucial difference between the above modified greedy algorithm and the classical greedy algorithm is that the former is guaranteed to choose in every round at least one vertex from every minimum dominating set for , given that is still large. This is made precise in the following lemma.
Let be a graph which excludes as a subgraph. Let be a set of vertices to be dominated and let be a dominating set of of size in . If , then the algorithm applied to will find vertices with .
By assumption, is dominated by the set of size . Hence there must exist a vertex which dominates at least a fraction of , that is, at least vertices of . Let , hence .
Assume . We repeat the same argument as above for . Also is dominated by of size , hence there must exist a vertex which dominates at least a fraction of , that is, at least vertices of . Let , hence . We repeat the argument for and , each for (set ) of size at least , ending with a set of size at least .
Hence, assuming that , we have and there must exist a vertex with . Fix any subset of of size exactly . Then the vertices and the vertices form a subgraph , contradicting that such a subgraph does not exist in . Hence, one of must be contained in .
Hence, as long as it remains to dominate a large set , the modified greedy algorithm makes an almost optimal choice. Once we are left with a small set , it performs only slightly worse than the classical greedy algorithm.
If is a graph which excludes as a subgraph, then the modified greedy algorithm called with parameter computes an approximation of a minimum dominating set of , where is the size of a minimum dominating set of .
Fix any minimum dominating set of size of . By lemma 1, as long as it remains to dominate a set of size at least , the modified greedy algorithm chooses in every round at least one vertex of . Hence, when it remains to dominate a set of size smaller than , the algorithm has chosen at most vertices.
Once we have reached this situation, let . We argue just as in the proof of lemma 1 that there exists a vertex which dominates at least a fraction of , that is, a subset of of size at least . The algorithm chooses such a vertex together with at most other vertices which in the worst case dominate nothing else. Hence after the first round we are left to dominate at most vertices. In the second round, we find again a vertex which dominates at least a fraction of the remaining vertices, hence after the second round we are left to dominate at most vertices. We repeat this argumentation and conclude that after executing rounds of the algorithm it remains to dominate at most elements. Let us determine for what value of we have , in which case we have dominated all vertices.
We have , where the last inequality follows from the bound , which holds for all . Thus, for we have . We conclude that the algorithm terminates after at most steps, in particular, it computes a dominating set of size at most . Now, as , we have . Hence, in total the set has size at most .
With slightly more computational effort we can compute an approximation on -free graphs (and an approximation on -free graphs, respectively) as follows. For each of the sets constructed in the course of the algorithm, run the standard greedy algorithm to extend it to a dominating set, and return the smallest of the sets obtained in this way. Letting be the first index such that dominates all but at most vertices of the graph, the above argument shows that . The standard greedy algorithm then adds at most further vertices to the dominating set, resulting in a dominating set of size .
We now modify the algorithm slightly to work without the parameter . In each round let the algorithm choose elements , defining sets in the above notation, until we do not find a vertex defining a set with any more. Let for the largest that was encountered in any round. Hence, the modified algorithm chooses at most elements in every round. Observe that in this construction, when we are at step and the corresponding set has size at least , , , then we have found a subgraph . Hence, is the least number such that the algorithm did not find as a subgraph and we can argue as above that the algorithm performs as if was excluded from . We output as a witness for this performance guarantee.
Finally, note that the algorithm can be used to approximate the minimum size of a set which dominates a given subset of vertices of the graph, by initializing instead of .
3 Hardness beyond degenerate graphs
By the result of Bansal and Umboh bansal2017tight one can compute a approximation of a minimum dominating set on any -degenerate graph. The approximation factor was improved to by Dvořák dvovrak2017distance . To the best of our knowledge, degenerate graphs are currently the most general graphs on which polynomial time constant factor approximation algorithms for the dominating set problem are known. It is easy to see that the existence of such algorithms on bi-clique free graphs, even on -free graphs, is unlikely. This result is a simple consequence of the following result of Kumar et al. kumar2000hardness . Given a family of subsets of a set , a set cover is a subset such that . The Minimum Set Cover problem is to find a minimum size set cover. The intersection of a set family is the maximum size of the intersection of two sets from .
Theorem 3 (Kumar et al. kumar2000hardness ).
The Minimum Set Cover problem on set families of intersection cannot be approximated to within a factor of for some constant in polynomial time unless for some constant it holds that .
Now it is easy to derive the following theorem.
The Minimum Dominating Set problem on -free graphs cannot be approximated to within a factor of for some constant in polynomial time unless for some constant it holds that .
We present an approximation preserving reduction from Minimum Set Cover on instances of intersection to Minimum Dominating Set on -free graphs. Let be an instance of Minimum Set Cover with intersection . Let . We compute in polynomial time an instance of Minimum Dominating Set on a graph as follows. We let , where are new vertices that do not appear in . We add all edges if , as well as all edges for and the edge .
Now if is a feasible solution for the Minimum Set Cover instance, then (as a subset of ) together with the vertex is a dominating set for of size at most . Conversely, let be a dominating set for . We construct another dominating set such that and . We simply replace each by a neighbour . Furthermore, if , we replace by . Observe that or must belong to , as must be dominated. Hence, in any case, . Now is a set cover of size . Hence, the reduction preserves approximations.
Let us show that excludes as a subgraph. Assume towards a contradiction that . Then . Since is bipartite we find elements and (as vertices of ) with , , which form a subgraph of this graph. By construction of we have , contradicting that is a Minimum Set Cover instance with intersection .
Finally observe that the reduction is obviously polynomial time computable.
I thank Saket Saurabh for pointing me to the work of Jones et al. jones2013parameterized . I thank the anonymous reviewers for their valuable comments, and in particular for pointing out that the modified greedy algorithm can be improved to an approximation by running the classical greedy algorithm in parallel.
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