The problem of computing an approximation to a maximum independent set (MIS) of an arbitrary graph is notoriously hard. It is known [hastad1999] that, for every fixed , MIS cannot be efficiently approximated (unless ) within a multiplicative factor of for an arbitrary . Naturally, there have been algorithmic studies of this problem on special classes of graphs. One such graph class is denoted by -VPG.
Vertex intersection graphs of Paths on Grid thm:aprx-mis-b1-vpg-gen (or, in short, VPG graphs) were first introduced by Asinowski et. al. [Asin]. For a member of this class of graphs, its vertices represent paths joining grid-points on a rectangular planar grid and two such vertices are adjacent if and only if the corresponding paths intersect in a grid-point. In particular, -VPG graphs denotes the class of intersection graphs of paths on a grid where each path has one of the following shapes: and , commonly referred to as an . By an arm of an , we mean either a horizontal or a vertical line segment associated with . An is said to be equilateral if its vertical and horizontal arms are of equal length . In this paper, we focus on equilateral -VPG graphs formed by equilateral ’s. For a set of equilateral ’s, we denote by and the maximum and minimum values of over .
VPG graphs are a special type of string graphs, which are intersection graphs of simple curves in the plane [Asin]. The best known approximation algorithm for MIS on string graphs is only known to have a guarantee of , for some [FoxP]. There are few subclasses of string graphs like outerstring graphs, planar graphs, -EPG, etc., classes which admit efficient computation of either a MIS or a constant factor approximation of a MIS [bakerJACM1994, epst, keil17]. Recently, in [vpgipl17], we presented a -time, -approximation algorithm for MIS over -VPG graphs on vertices.
Our Results : In this paper, we present new approximation algorithms for the class of equilateral -VPG graphs. The decision version of this problem is NP-complete even if restricted to instances where the arms of of all ’s are of equal length (see Theorem in [lahiri]). Throughout the paper, we assume that the input is in the form of a set of equilateral ’s. Precisely, we obtain the following results.
There exists an -time and -approximation algorithm for MIS restricted to equilateral -VPG graphs, where .
In particular, for all equilateral with , the algorithm of Theorem 1 yields an -approximation of MIS. Also, when this is result combined with the approximation algorithm of [vpgipl17], we obtain a slightly slower -time and -approximation algorithm for MIS over equilateral -VPG graphs.
When all members have a uniformly common arm length, the following corollary can be inferred by slightly modifying the proof arguments of Theorem 1. We provide a brief sketch of of this modification in Section 5.
There exists an -time and -approximation algorithm for computing a MIS over equilateral -VPG graphs formed by sets of ’s of uniformly common arm lengths.
It is easy to see from the description of the algorithm and its analysis that the algorithm can be suitably generalized to obtain an approximate MIS algorithm over arbitrary -VPG graphs. Precisely, we obtain the following theorem. A brief sketch of its proof is provided in Section 3.
There exists an -time and approximation algorithm for MIS restricted to -VPG graphs, where , . is the maximum length of the horizontal arm of any member of . , and are similarly defined.
When combined with the approximation algorithm of [vpgipl17], this yields an -time and approximation algorithm for MIS over arbitrary -VPG graphs. In particular, for -VPG graphs having , we obtain an efficient, -approximation algorithm for MIS. To the best of our knowledge, no such -factor approximation of MIS is known for any class of -VPG graphs. We also infer the following corollary by slightly modifying the proof arguments of Theorem 2. A sketch of this modification is provided in Section 5
There exists an -time and -approximation algorithm for computing a MIS over arbitrary -VPG graphs formed by sets of ’s having a uniformly common arm length for each of the horizontal and vertical arms of all members . The vertical and horizontal arm lengths may however differ for any in the input .
We introduce some conventions and notations in Section 2. In Section 3, we present the MIS approximation algorithm and its analysis for equilateral -VPG graphs. This constitutes the proof of Theorem 1. In Section 4, we present a sketch of the generalization of the approach of Section 3 to arbitrary -VPG graphs and its analysis. This constitutes the proof of Theorem 2. In Section 5, we provide a sketch of the proof arguments of Corollaries 1 and 2. In Section 6, we conclude with some remarks.
We work with geometric shapes and . For ease of further discussion, we refer to them as follows. refers to the shape , refers to , refers to and to . Henceforth, we use to denote a geometric object with one of the four shapes and .
The corner of an is defined to be the point where the two arms meet and is denoted by , the tip of the horizontal arm is denoted by and that of the vertical arm is denoted by . For an object , we use to denote respectively the - and - coordinates of , the - coordinate of and the - coordinate of . This 4-tuple completely and uniquely describes . The set of points constituting is denoted by and is given by (when is of shape )
We say that two distinct objects and intersect if . Given a set of ’s, the intersection graph formed by is defined to be where consists of all those unordered pairs such that and intersect. A set of ’s such that no two of them form an intersecting pair is said to be an independent set.
For each , we refer to an intersection graph formed by objects each of shape as a -graph. For ease of description, we refer to a -graph as a -graph. By symmetry (based on rotations), one can adapt any efficeint and exact/approximate MIS algorithm for -graphs to a similar algorithm (with same time and approximation guarantee) for -graphs, for every . This enables us to focus only on -graphs (at the cost of increasing the approximation guarantee by a multiplicative factor of 4) as is established by the following Claim 1. The claim is essentially a formal statement (in the context of equilateral -VPG graphs) of an approach that has been employed in [epst] for -EPG graphs. Let denote the size of a MIS of graph and let denote the size of an independent set of returned by an algorithm .
If there is an efficient algorithm which approximates MIS over equilateral -graphs within a multiplicative factor of for some increasing function , then there is an efficient algorithm which approximates MIS over equilateral -VPG graphs within a multiplicative factor of . Throughout, denotes the input to the corresponding algorithm.
Hence, from now on, we focus only on the subclass of equilateral -graphs.
All logarithms used below are with respect to base 2. We denote a set by . A permutation over is any bijection . An inversion of is any unordered pair satisfying . The graph associated with is defined as where is set of all inversions of . A permutation graph on vertices is any graph which is isomorphic to for some permutation .
3 MIS-approximation over equilateral VPG-graphs
Maximum Independent Set over equilateral -graphs Input: A set of -shaped equilateral ’s Output: a set such that is independent and is maximized.
Before we present an approximation algorithm for this problem, we present the following claim (proved and employed in [vpgipl17]).
Without loss of generality, we can assume that :
and for any pair of distinct , where is the input set of -shaped ’s.
We will also be making use of the following lemma (Lemma 1 of [vpgipl17]) in the design of new approximation algorithms.
([vpgipl17]) Suppose is a set of ’s, each being of type . Suppose there exist a horizontal line and a vertical line such that each intersects both and . Then, the intersection graph of members of is a permutation graph.
We will also be making use of the following fact from [kim1990].
[kim1990] Given an arbitrary permutation graph (in the form of two permutations defining ), a MIS of can be computed in time where .
We will also be making use of the following claim.
Without loss of generality, assume that input satisfies .
(sketch:) Rescale the the coordinates of -axis and -axis by stretching both of them by a multiplicative factor of . ∎
The algorithm begins by dividing the input set into disjoint sets where , . This split is to exploit the fact that , for any . We further partition each into nine subsets as follows.
For the set, we do the following. We place a sufficiently large but finite grid structure on the plane covering all members of . The grid is chosen in such a way so that grid-length in each of the and directions is . What we get is a rectangular array of square boxes of side length each. We number the rows of boxes from bottom and the columns of boxes from left, with numbers .
We denote a box by if it is in the intersection of row and column. We say lies inside a box if its corner lies either in the interior or on the boundary of the box, except that it should not lie either on its top horizontal boundary or on its right vertical boundary. If lies inside a box we denote it by .
For every , define
For a pictorial representation of the partition of into 9 subsets, see the figure below.
Thus, we partition in the input into induced subgraphs where . Each is a subgraph induced by for some and . In Lemma 2, we establish that a MIS can be computed in efficiently for the intersection graph induced by and hence, by symmetry, a MIS can be computed for each of the induced subgraphs. More precisely, for each of the induced subgraphs, a MIS can be computed in time, where represents the number of vertices in the respective induced subgraph, leading to an overall running time. We choose the largest of these independent sets and return it as the output. Since , we deduce that the algorithm just outlined above returns an independent set of size at least for an arbitrary equilateral -graph . Now, combining this observation with Claim 1, we deduce Theorem 1.
It now remains only to prove that a MIS can be efficiently computed for , as stated in Lemma 2 below.
A MIS can be computed in time for . Here, .
Recall that . From the definitions of boxes given above, the following can be seen immediately : if and , then for any and any , and are independent. Hence, computing a MIS for reduces to computing, for each such that , a MIS for the subgraph of induced by those . Since, for any such , each intersects the vertical line forming the left-border of the -th column of boxes as well as the horizontal line forming the bottom-border of the -th row of boxes, we can deduce, by applying Lemma 1 of [vpgipl17] stated before, that the subgraph of induced by forms a permutation graph.
Hence, by applying Theorem 3, it follows that a MIS can be computed in time where , provided the subgraph induced by is specified in the form of two permutations defining it. It follows from the proof of Lemma 1 that the two permutations specifying the input are the two increasing orders formed by the and values of its members. These two orders can be computed in time. ∎
4 Generalization to arbitrary -VPG graphs
Proof of Theorem 2 :
The broad approach is the same as for equilateral graphs except that we partition the members based on their lengths in each of the vertical and horizontal directions independently. This independent partitioning was not needed for the equilateral case since lengths are the same for both arms of any .
As earlier, we assume (without loss of generality) that and . We divide horizontal lengths lying in into groups for and also divide vertical lengths lying in into groups for . Using these two partitions, we divide members of into sets with each consisting of those members of whose horizontal and vertical lengths lie in groups and respectively.
We further divide each of these groups into 9 smaller groups for by imposing a rectangular grid structure with grid-points being separated by lengths and in the horizontal and vertical directions respectively. The remaining details are as before leading to an algorithm running in time and producing a MIS-approximation.
Proof sketch of Corollary 1 :
The algorithm is essentially the same as the one described in the proof of Theorem 1 except for the following changes. When all equilateral ’s in the input have uniformly the same arm length , we have only one group instead of groups we had in the proof of Theorem 1. In this case, it suffices to impose a finite grid structure whose side lengths (both horizontal and vertical) are . After numbering the rows and columns of boxes, we partion the input into 4 subsets with where, as before,
For each , MIS can be computed exactly for the subgraph induced by . The reason is the same as before : for any satisfying and for any and , and are independent and hence it reduces to computing, for each such , exactly an MIS for the subgraph induced by those and this can be realized in time as explained before, where . The largest of the 4 MIS’s (one for each ) is then returned as the output yielding a 4-approximation of MIS for the subgraph induced by ’s of Type 1. When combined with Claim 1, we obtain Corollary 1.
Proof sketch of Corollary 2 :
When all equilateral ’s in the input have uniformly the same vertical arm length and horizontal arm length , we have only one group instead of groups we had in the proof of Theorem 2. In this case, it suffices to impose a finite grid structure whose vertical and horizontal side lengths are and respectively. After numbering the rows and columns of boxes, we partion the input into 4 subsets with where, as before,
As explained before, for each , MIS can be computed exactly for the subgraph induced by leading to a 16-approximation of MIS in polynomial time for arbitrary -VPG graphs having a uniformly common vertical and horizontal arm lengths. This establishes Corollary 2.
In further works [crsb2vpgsep16], we have obtained further improvements on MIS approximation of -VPG graphs and also for improved MIS approximation algorithms -VPG graphs. It would be interesting to establish some inapproximability results for the MIS problem over equilateral -VPG graphs. Also the question of obtaining better approximations in terms of ratios of lengths would be worth pursuing.
Acknowledgements : We thank an anonymous referee (of a related submission) for pointing out that a special type of input studied here actually induces a permutation graph, thereby leading to improved running time bounds.