Approximating MIS over equilateral B 1 -VPG graphs
AApproximating MIS over equilateral B -VPGgraphs ∗ Abhiruk Lahiri † Joydeep Mukherjee ‡ C.R. Subramanian § December 18, 2019
Abstract
We present an approximation algorithm for the maximum indepen-dent set (MIS) problem over the class of equilateral B -VPG graphs.These are intersection graphs of L -shaped planar objects with botharms of each object being equal. We obtain a 36(log 2 d )-approximatealgorithm running in O ( n (log n ) ) time for this problem, where d isthe ratio d max /d min and d max and d min denote respectively the max-imum and minimum length of any arm in the input equilateral L -representation of the graph. In particular, we obtain O (1)-factor ap-proximation of MIS for B -VPG -graphs for which the ratio d is boundedby a constant. In fact, algorithm can be generalized to an O ( n (log n ) )time and a 36(log 2 d x )(log 2 d y )-approximate MIS algorithm over arbi-trary B -VPG graphs. Here, d x and d y denote respectively the ana-logues of d when restricted to only horizontal and vertical arms ofmembers of the input. This is an improvement over the previouslybest n (cid:15) -approximate algorithm [2] (for some fixed (cid:15) > d is exponentially large in n . In particular, O (1)-approximationof MIS is achieved for graphs with max { d x , d y } = O (1). Keywords : approximation algorithms, intersection graphs, independentsets
The problem of computing an approximation to a maximum independentset (
MIS ) of an arbitrary graph is notoriously hard. It is known [3] that, ∗ Journal version of a part of the preliminary version presented at COCOA-2015 [1]. † Indian Institute of Science, Bangalore, India; [email protected] ‡ The Institute of Mathematical Sciences, HBNI, Chennai, India; [email protected] § The Institute of Mathematical Sciences, HBNI, Chennai 600113, India; [email protected] a r X i v : . [ c s . D S ] D ec or every fixed (cid:15) >
0, MIS cannot be efficiently approximated (unless
N P = ZP P ) within a multiplicative factor of | V | − (cid:15) for an arbitrary G = ( V, E ).Naturally, there have been algorithmic studies of this problem on specialclasses of graphs. One such graph class is denoted by B -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. [4].For a member of this class of graphs, its vertices represent paths joininggrid-points on a rectangular planar grid and two such vertices are adjacentif and only if the corresponding paths intersect in a grid-point. In particular, B -VPG graphs denotes the class of intersection graphs of paths on a gridwhere each path has one of the following shapes: (cid:120) , (cid:112) , (cid:113) and (cid:121) , commonlyreferred to as an l . By an arm of an l , we mean either a horizontal or avertical line segment associated with l . An l is said to be equilateral if itsvertical and horizontal arms are of equal length le ( l ). In this paper, wefocus on equilateral B -VPG graphs formed by equilateral l ’s. For a set L of equilateral l ’s, we denote by d max ( L ) and d min ( L ) the maximum andminimum values of le ( l ) over l ∈ L .VPG graphs are a special type of string graphs , which are intersectiongraphs of simple curves in the plane [4]. The best known approximationalgorithm for MIS on string graphs is only known to have a guarantee of n (cid:15) ,for some (cid:15) > B -EPG, etc., classes which admit efficient compu-tation of either a MIS or a constant factor approximation of a MIS [5, 6, 7].Recently, in [8], we presented a O ( n (log n ) )-time, 4(log n ) -approximationalgorithm for MIS over B -VPG graphs on n vertices. Our Results :
In this paper, we present new approximation algorithmsfor the class of equilateral B -VPG graphs. The decision version of thisproblem is NP-complete even if restricted to instances where the arms of ofall l ’s are of equal length (see Theorem 5 in [1]). Throughout the paper, weassume that the input is in the form of a set L of equilateral l ’s. Precisely,we obtain the following results. Theorem 1.
There exists an O ( n (log n ) ) -time and d ) -approximationalgorithm for MIS restricted to equilateral B -VPG graphs, where d = d ( L ) = d max ( L ) /d min ( L ) . In particular, for all equilateral L with d ( L ) = O (1), the algorithm ofTheorem 1 yields an O (1)-approximation of MIS. Also, when this is re-sult combined with the approximation algorithm of [8], we obtain a slightly2lower O ( n (log n ) )-time and min { n ) , d ) } -approximation algo-rithm for MIS over equilateral B -VPG graphs.When all members have a uniformly common arm length, the follow-ing corollary can be inferred by slightly modifying the proof arguments ofTheorem 1. We provide a brief sketch of of this modification in Section 5. Corollary 1.
There exists an O ( n (log n ) ) -time and -approximation al-gorithm for computing a MIS over equilateral B -VPG graphs formed by setsof l ’s of uniformly common arm lengths. It is easy to see from the description of the algorithm and its analysisthat the algorithm can be suitably generalized to obtain an approximate MISalgorithm over arbitrary B -VPG graphs. Precisely, we obtain the followingtheorem. A brief sketch of its proof is provided in Section 3. Theorem 2.
There exists an O ( n (log n ) ) -time and d x )(log 2 d y ) ap-proximation algorithm for MIS restricted to B -VPG graphs, where d x = d xmax ( L ) /d xmin ( L ) , d y = d ymax ( L ) /d ymin ( L ) . d xmax ( L ) is the maximum lengthof the horizontal arm of any member of L . d xmin , d ymax and d ymin are similarlydefined. When combined with the approximation algorithm of [8], this yields an O ( n (log n ) )-time and min { n ) , d x )(log 2 d y ) } approximation al-gorithm for MIS over arbitrary B -VPG graphs. In particular, for B -VPGgraphs having d x , d y = O (1), we obtain an efficient, O (1)-approximationalgorithm for MIS. To the best of our knowledge, no such O (1)-factor ap-proximation of MIS is known for any class of B -VPG graphs. We alsoinfer the following corollary by slightly modifying the proof arguments ofTheorem 2. A sketch of this modification is provided in Section 5 Corollary 2.
There exists an O ( n (log n ) ) -time and -approximation al-gorithm for computing a MIS over arbitrary B -VPG graphs formed by setsof l ’s having a uniformly common arm length for each of the horizontal andvertical arms of all members L . The vertical and horizontal arm lengthsmay however differ for any l in the input L . We introduce some conventions and notations in Section 2. In Section 3,we present the MIS approximation algorithm and its analysis for equilateral B -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 toarbitrary B -VPG graphs and its analysis. This constitutes the proof ofTheorem 2. In Section 5, we provide a sketch of the proof arguments ofCorollaries 1 and 2. In Section 6, we conclude with some remarks.3 Preliminaries
We work with geometric shapes (cid:120) , (cid:112) , (cid:113) and (cid:121) . For ease of further discussion,we refer to them as follows. L refers to the shape (cid:120) , L refers to (cid:112) , L refersto (cid:113) and L to (cid:121) . Henceforth, we use l to denote a geometric object withone of the four shapes L , L , L and L .The corner of an l is defined to be the point where the two arms meetand is denoted by c l , the tip of the horizontal arm is denoted by h l and thatof the vertical arm is denoted by v l . For an object l , we use ( x c , y c , x h , y v ) todenote respectively the x - and y - coordinates of c l , the x - coordinate of h l and the y - coordinate of v l . This 4-tuple completely and uniquely describes l . The set of points constituting l is denoted by P l and is given by (when l is of shape L ) P l = { ( x, y c ) : x c ≤ x ≤ x h } ∪ { ( x c , y ) : y c ≤ y ≤ y v } . We say that two distinct objects l and l intersect if P l ∩ P l (cid:54) = ∅ . Given a set L of l ’s, the intersection graph G formed by L is defined to be G = ( L , E )where E consists of all those unordered pairs ( l , l ) such that l and l intersect. A set of l ’s such that no two of them form an intersecting pair issaid to be an independent set.For each 1 ≤ i ≤
4, we refer to an intersection graph formed by objectseach of shape L i as a L i -graph. For ease of description, we refer to a L -graph as a L -graph. By symmetry (based on rotations), one can adapt anyefficeint and exact/approximate MIS algorithm for L -graphs to a similaralgorithm (with same time and approximation guarantee) for L i -graphs, forevery i . This enables us to focus only on L -graphs (at the cost of increasingthe approximation guarantee by a multiplicative factor of 4) as is establishedby the following Claim 1. The claim is essentially a formal statement (inthe context of equilateral B -VPG graphs) of an approach that has beenemployed in [6] for B -EPG graphs. Let α ( G ) denote the size of a MIS ofgraph G and let A ( G ) denote the size of an independent set of G returnedby an algorithm A . Claim 1.
If there is an efficient algorithm A which approximates MIS overequilateral L -graphs within a multiplicative factor of c ( d ( L )) ( for some in-creasing function c ( d )) , then there is an efficient algorithm B which approx-imates MIS over equilateral B -VPG graphs within a multiplicative factor of c ( d ( L )) . Throughout, L denotes the input to the corresponding algorithm. Hence, from now on, we focus only on the subclass of equilateral L -graphs. 4ll logarithms used below are with respect to base 2. We denote a set { , , . . . , n } by [ n ]. A permutation over [ n ] is any bijection π : [ n ] → [ n ]. Aninversion of π is any unordered pair ( i, j ) satisfying ( i − j )( π − ( i ) − π − ( j )) <
0. The graph G π associated with π is defined as G π = ([ n ] , E ) where E isset of all inversions of π . A permutation graph on n vertices is any graphwhich is isomorphic to G π for some permutation π . In this section, we prove Theorem 1 by designing an approximation algo-rithm for the following problem and then applying Claim 1.
Maximum Independent Set over equilateral L -graphs Input : A set L of L -shaped equilateral l ’s Output : a set I ⊂ L such that I is independent and | I | is maximized.Before we present an approximation algorithm for this problem, we presentthe following claim (proved and employed in [8]). Claim 2.
Without loss of generality, we can assume that : l .x c (cid:54) = l .x c and l .y c (cid:54) = l .y c for any pair of distinct l , l ∈ S , where L isthe input set of L -shaped l ’s. We will also be making use of the following lemma (Lemma 1 of [8]) inthe design of new approximation algorithms.
Lemma 1. ([8]) Suppose S (cid:48) is a set of l ’s, each being of type L . Supposethere exist a horizontal line y = b and a vertical line x = a such that each l ∈ S (cid:48) intersects both y = b and x = a . Then, the intersection graph ofmembers of S (cid:48) is a permutation graph. We will also be making use of the following fact from [9].
Theorem 3. [9] Given an arbitrary permutation graph G (in the form oftwo permutations defining G ), a MIS of G can be computed in O ( n (log n )) time where n = | V ( G ) | . We will also be making use of the following claim.
Claim 3.
Without loss of generality, assume that input L satisfies d min = 2 .Proof. (sketch:) Rescale the the coordinates of x -axis and y -axis by stretch-ing both of them by a multiplicative factor of 2 /d min .5he algorithm begins by dividing the input set L into disjoint sets S , S ,. . . , S (cid:98) log 2 d (cid:99) where S i = { l ∈ S | i ≤ le ( l ) < i +1 } , ∀ i ∈ [ (cid:98) log 2 d (cid:99) ]. Thissplit is to exploit the fact that d ( S i ) ≤
2, for any i . We further partitioneach S i into nine subsets as follows.For the i th set, we do the following. We place a sufficiently large butfinite grid structure on the plane covering all members of S i . The grid ischosen in such a way so that grid-length in each of the x and y directionsis 2 i . What we get is a rectangular array of square boxes of side length 2 i each. We number the rows of boxes from bottom and the columns of boxesfrom left, with numbers 0 , , . . . .We denote a box by ( r (cid:48) , c (cid:48) ) if it is in the intersection of r (cid:48) th row and c (cid:48) th column. We say l lies inside a box if its corner c l lies either in the interioror on the boundary of the box, except that it should not lie either on its tophorizontal boundary or on its right vertical boundary. If l lies inside a box( r (cid:48) , c (cid:48) ) we denote it by l ∈ ( r (cid:48) , c (cid:48) ).For every k r , k c ∈ { , , } , define S i,k r ,k c = { l ∈ ( r (cid:48) , c (cid:48) ) | r (cid:48) ∼ = k r mod 3 , c (cid:48) ∼ = k c mod 3 } . For a pictorial representation of the partition of S i into 9 subsets, see thefigure below. row cell column Figure 1: The grid is for l ’s of type 1 whose lengths lie within the range[2 i , i +1 ).Thus, we partition in the input into induced subgraphs G , . . . , G D where D = 9( (cid:98) log 2 d (cid:99) ). Each G j is a subgraph induced by S i,k r ,k c for some i ∈ [ (cid:98) log 2 d (cid:99) ] and k r , k c ∈ { , , } . In Lemma 2, we establish that a MIS canbe computed in efficiently for the intersection graph G ( S , , ) induced by6 , , and hence, by symmetry, a MIS can be computed for each of the9( (cid:98) log 2 d (cid:99) ) induced subgraphs. More precisely, for each of the 9( (cid:98) log 2 d (cid:99) )induced subgraphs, a MIS can be computed in O ( p (log p ) ) time, where p represents the number of vertices in the respective induced subgraph,leading to an O ( n (log n ) ) overall running time. We choose the largestof these 9( (cid:98) log 2 d (cid:99) ) independent sets and return it as the output. Since α ( G ) ≤ (cid:80) ≤ i ≤ D α ( G i ), we deduce that the algorithm just outlined abovereturns an independent set of size at least α ( G ) / (cid:98) log 2 d (cid:99) ) for an arbitraryequilateral L -graph G . Now, combining this observation with Claim 1, wededuce Theorem 1.It now remains only to prove that a MIS can be efficiently computed for G ( S , , ), as stated in Lemma 2 below. Lemma 2.
A MIS can be computed in O ( p (log p ) ) time for G ( S , , ) . Here, p = | S , , | .Proof. Recall that S , , = { l ∈ S : l ∈ ( r (cid:48) , c (cid:48) ) , r (cid:48) , c (cid:48) ∼ = 0 mod 3 } . From thedefinitions of boxes given above, the following can be seen immediately : if r , r , c , c ∼ = 0 mod 3 and ( r , c ) (cid:54) = ( r , c ), then for any l ∈ ( r , c ) andany l ∈ ( r , c ), l and l are independent. Hence, computing a MIS for H = G ( S , , ) reduces to computing, for each ( r, c ) such that r, c ∼ = 0 mod 3,a MIS for the subgraph of H induced by those l ∈ ( r, c ). Since, for any such( r, c ), each l ∈ ( r, c ) intersects the vertical line forming the left-border ofthe ( c + 1)-th column of boxes as well as the horizontal line forming thebottom-border of the ( r + 1)-th row of boxes, we can deduce, by applyingLemma 1 of [8] stated before, that the subgraph of H induced by l ∈ ( r, c )forms a permutation graph.Hence, by applying Theorem 3, it follows that a MIS can be computedin O ( q (log q ) ) time where q = |{ l ∈ S : l ∈ ( r, c ) }| , provided the subgraphinduced by l ∈ ( r, c ) is specified in the form of two permutations defining it.It follows from the proof of Lemma 1 that the two permutations specifyingthe input are the two increasing orders formed by the l.x c and l.y c values ofits members. These two orders can be computed in O ( q (log q )) time. B -VPG graphs . Proof of Theorem 2 :
The broad approach is the same as for equilateral graphs except thatwe partition the members based on their lengths in each of the vertical and7orizontal directions independently. This independent partitioning was notneeded for the equilateral case since lengths are the same for both arms ofany (cid:96) .As earlier, we assume (without loss of generality) that d xmin = 2 and d ymin = 2. We divide horizontal lengths lying in [ d xmin , d xmax ] into groups[2 i , i +1 ) for i ∈ [ (cid:98) log 2 d x (cid:99) ] and also divide vertical lengths lying in [ d ymin , d ymax ]into groups [2 i , i +1 ) for j ∈ [ (cid:98) log 2 d y (cid:99) ]. Using these two partitions, we di-vide members of L into sets S i,j with each S i,j consisting of those members of L whose horizontal and vertical lengths lie in groups [2 i , i +1 ) and [2 j , j +1 )respectively.We further divide each of these ( (cid:98) log 2 d x (cid:99) )( (cid:98) log 2 d y (cid:99) ) groups S i,j into9 smaller groups S i,j,k r ,k c for k r , k c ∈ { , , } by imposing a rectangulargrid structure with grid-points being separated by lengths 2 i and 2 j in thehorizontal and vertical directions respectively. The remaining details are asbefore leading to an algorithm running in O ( n (log n ) ) time and producinga 36(log 2 d x )(log 2 d y ) MIS-approximation. Proof sketch of Corollary 1 :
The algorithm is essentially the same as the one described in the proofof Theorem 1 except for the following changes. When all equilateral l ’s inthe input L have uniformly the same arm length a , we have only one groupinstead of ( (cid:98) log 2 d (cid:99) ) groups we had in the proof of Theorem 1. In this case, itsuffices to impose a finite grid structure whose side lengths (both horizontaland vertical) are a . After numbering the rows and columns of boxes, wepartion the input into 4 subsets S k r ,k c with k r , k c ∈ { , } where, as before, S k r ,k c = { l ∈ ( r (cid:48) , c (cid:48) ) | r (cid:48) ∼ = k r mod 2 , c (cid:48) ∼ = k c mod 2 } . For each k r , k c ∈ { , } , MIS can be computed exactly for the subgraphinduced by S k r ,k c . The reason is the same as before : for any ( r , c ) (cid:54) =( r , c ) satisfying r , r ∼ = k r mod 2 , c , c ∼ = k c mod 2 and for any l ∈ ( r , c ) and l ∈ ( r , c ), l and l are independent and hence it reduces tocomputing, for each such ( r, c ), exactly an MIS for the subgraph inducedby those l ∈ ( r, c ) and this can be realized in O ( p (log p ) ) time as explainedbefore, where p = |{ l ∈ ( r, c ) }| . The largest of the 4 MIS’s (one for each S k r ,k c ) is then returned as the output yielding a 4-approximation of MIS for8he subgraph induced by (cid:96) ’s of Type 1. When combined with Claim 1, weobtain Corollary 1. Proof sketch of Corollary 2 :
When all equilateral l ’s in the input L have uniformly the same verticalarm length a and horizontal arm length b , we have only one group instead of( (cid:98) log 2 d x (cid:99) )( (cid:98) log 2 d y (cid:99) ) groups we had in the proof of Theorem 2. In this case,it suffices to impose a finite grid structure whose vertical and horizontal sidelengths are a and b respectively. After numbering the rows and columns ofboxes, we partion the input into 4 subsets S k r ,k c with k r , k c ∈ { , } where,as before, S k r ,k c = { l ∈ ( r (cid:48) , c (cid:48) ) | r (cid:48) ∼ = k r mod 2 , c (cid:48) ∼ = k c mod 2 } . As explained before, for each k r , k c ∈ { , } , MIS can be computed exactlyfor the subgraph induced by S k r ,k c leading to a 16-approximation of MIS inpolynomial time for arbitrary B -VPG graphs having a uniformly commonvertical and horizontal arm lengths. This establishes Corollary 2. In further works [10], we have obtained further improvements on MIS ap-proximation of B -VPG graphs and also for improved MIS approximationalgorithms B -VPG graphs. It would be interesting to establish some inap-proximability results for the MIS problem over equilateral B -VPG graphs.Also the question of obtaining better approximations in terms of ratios oflengths would be worth pursuing. Acknowledgements :
We thank an anonymous referee (of a related sub-mission) for pointing out that a special type of input studied here actuallyinduces a permutation graph, thereby leading to improved running timebounds.
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