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An Approximation Algorithm For The Maximum Independent Set Problem In Cubic Planar Graphs Elarbi Choukhmane Case Western Reserve University, Department of Computer Engineering and Science, Cleveland, 0 H 44 106 John Franco Indiana University, Department of Computer Science, Bloomington, IN 4 7405 A polynomial time approximation algorithm A for the problem of finding a maximal independent set for cubic planar graphs is presented. It is shown that MA > 6/7 in the case of cubic planar graphs and MA = 7/8 in the case of triangle free cubic planar graphs where MA is the worst-case ratio of the size of the independent set found by A to the size of the maximum independent set for the graph input to A. 1. INTRODUCTION Given a graph G = (V, E), a maximum independent set for G is the maximum cardinality subset V' C V such that no pair of vertices in V' is connected by an edge in E. The problem of finding a maximum independent set (MIS) is NP- complete [3]. Let W be any algorithm that finds an independent set for any graph and let M, be the worst case ratio of the size of the independent set found by W to the size of the independent set for the graph input to W. Let M be the maximum of Mw over all W such that W always finds an independent set in polynomial time. It is not known whether M > E for some E > 0; if this is the case then there is a polynomial time approximation scheme for MIS [3]. MIS remains NP-complete in the case of planar graphs and cubic planar graphs [2] but effective approximation algorithms for those kinds of graphs exist: a polynomial time approximation algorithm W with Mw = 1/2 for MIS on planar Supported in part by the Air Force Office of Scientific Research under Grant AFOSR- 82-033 1. NETWORKS, Vol. 16 (1986) 349-356 (C 1986 John Wiley & Sons, Inc. CCC 0028-30451861040349-08$04.00

An approximation algorithm for the maximum independent set problem in cubic planar graphs

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Page 1: An approximation algorithm for the maximum independent set problem in cubic planar graphs

An Approximation Algorithm For The Maximum Independent Set Problem In Cubic Planar Graphs Elarbi Choukhmane Case Western Reserve University, Department of Computer Engineering and Science, Cleveland, 0 H 44 106 John Franco Indiana University, Department of Computer Science, Bloomington, IN 4 7405

A polynomial time approximation algorithm A for the problem of finding a maximal independent set for cubic planar graphs is presented. It is shown that M A > 6/7 in the case of cubic planar graphs and MA = 7/8 in the case of triangle free cubic planar graphs where M A is the worst-case ratio of the size of the independent set found by A to the size of the maximum independent set for the graph input to A.

1. INTRODUCTION

Given a graph G = ( V , E ) , a maximum independent set for G is the maximum cardinality subset V' C V such that no pair of vertices in V' is connected by an edge in E. The problem of finding a maximum independent set (MIS) is NP- complete [3]. Let W be any algorithm that finds an independent set for any graph and let M , be the worst case ratio of the size of the independent set found by W to the size of the independent set for the graph input to W. Let M be the maximum of Mw over all W such that W always finds an independent set in polynomial time. It is not known whether M > E for some E > 0; if this is the case then there is a polynomial time approximation scheme for MIS [3].

MIS remains NP-complete in the case of planar graphs and cubic planar graphs [2] but effective approximation algorithms for those kinds of graphs exist: a polynomial time approximation algorithm W with M w = 1/2 for MIS on planar

Supported in part by the Air Force Office of Scientific Research under Grant AFOSR- 82-033 1.

NETWORKS, Vol. 16 (1986) 349-356 (C 1986 John Wiley & Sons, Inc. CCC 0028-30451861040349-08$04.00

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350 CHOUKHMANE AND FRANC0

graphs is presented in. [l] and, in this paper, a polynomial time approximation algorithm A with M A > 6/7 in the case of cubic planar graphs and M A = 7/8 in the case of cubic planar graphs which are triangle free is presented. Unfortunately, the technique used in [3] cannot be used to construct a polynomial time approx- imation scheme from either of these algorithms since the technique involves repeated applications of the composition operator and this generally destroys planarity.

2. PRELIMINARIES

Three facts and some terminology which is used in the statement of the algo- rithm and the analysis are presented in this section.

Fact 1. set for G contains at most n / 2 vertices.

If G = ( V , E ) is a cubic graph with I Vl = n then a maximum independent

Proof. Let OPT( G) be the size of a maximum independent set S for G. The total number of edges incident to vertices in S is 3 . OPT(G) . The total number of edges incident to vertices in V - S is

3 . IV - SI = 3 * OPT(G) + N ( V - S)

where N ( V - S) is the number of edges connecting vertices in V - S . Hence

OPT(G) I IV - SI = n - OPT(G)

so

n

2 OPT(G) 5 - rn

The algorithm of section 3 requires the construction of a maximum bipartite subgraph of a given graph. Let G = ( V , E ) be a graph with vertex set r/‘ and edge set E. Let G’ = (X, Y, E ’ ) be a bipartite graph with vertex set X U Y such that X f l Y = r j ~ and edge set E ’ such that all edges in E‘ have one endpoint in X and one endpoint in Y. G’ is a bipartite subgraph of G if V = X U Y and E’ E. G’ is a maximum bipartite subgraph of G if for any other bipartite subgraph G“ = (X’, Y’, E”) of G IE‘I 2 1E’’I.

Fact 2. For any planar graph G a maximum bipartite subgraph of G may be found in polynomial time.

Proof. The algorithm is due to Hadlock [4].

Fact 3. If G” = ( X , Y , E ’ ) is a maximum bipartite subgraph of a cubic graph G = ( V , E ) then every vertex in X U Y is an endpoint of at most one edge in ( E - E’) .

Page 3: An approximation algorithm for the maximum independent set problem in cubic planar graphs

MAXIMUM INDEPENDENT SET PROBLEM 351

Proof. Suppose there is a vertex in X, say, that is an endpoint of more than one edge in ( E - E ' ) . Then moving x to Y would add at least one edge to E' contradicting the fact that E' is of maximum cardinality. H

3. THE ALGORITHM

The algorithm finds an independent set for G = ( V , E ) from a maximum bipartite subgraph of G. The maximum bipartite subgraph G' = (X, Y, E ' ) of G is constructed using the algorithm in [4]. Two possible independent sets for G ' are X and Y and each would be an independent set for G except that edges in ( E - E ' ) require certain pairs of vertices in both X and Y not to be in the same independent set for G. The algorithm removes from X and from Y enough vertices to insure that what is left in X and Y are independent sets for G. The sizes of the two reduced sets are compared and the larger is chosen as the approximate maximum independent set I for G.

Algorithm A

Given a cubic planar graph G = ( V , E ) , A returns an independent set I C of G.

1: Find a maximum bipartite subgraph G' = (X, Y, E ' ) of G. 2: For every edge e = ( x , x ' ) in ( E - E ' ) such that x , x ' E X do X + X -

3: For every edge e = ( y , y ' ) in ( E - E ' ) such that y , y ' E Y do Y + Y -

4: If 1x1 > 1 YI then I t X else I +- Y.

{x ) .

cv). Algorithm A returns an independent set I of G in polynomial time since, from fact 2, step 1 requires polynomial time and steps 2, 3, and 4 require time linear in IEl.

4. THE ANALYSIS

We now show that MA > 617 when G is cubic planar and MA > 718 when G is cubic planar and triangle free. These results depend on the following theorem and lemma. Theorem 1 is due to Hopkins and Staton [5,6].

Theorem 1. (a) Let G = ( V , E ) be a cubic planar graph not containing a tetrahedron. Suppose IVJ = n. Then G has a bipartite subgraph with at least 716n edges.

(b) Let G = ( V , E ) be a cubic planar graph not containing a triangle. Suppose IVI = n. Then G has a bipartite subgraph with at least 615n edges.

Lemma 1. Let G = ( V , E ) be a cubic planar graph with IVI = n and let G' = (X, Y, E ' ) be a maximum bipartite subgraph of G. Also, let A (G) be the number

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352 CHOUKHMANE AND FRANC0

of vertices in the independent set returned by A given input G. Then, for any constant a

n - a then A ( G ) 2 - 2

iflE - E’/ 5 a

Proof. From fact 3 every vertex in V is an endpoint of at most one edge in ( E - E‘) . Let xo be the number of vertices in X which are endpoints of no edge in ( E - I?’)--, yo be the number of vertices in Y which are endpoints of no edge in ( E - E ‘ ) , x, be the number of vertices in X which are endpoints of one edge in ( E - E ‘ ) and y, be the number of vertices in Y which are endpoints of one edge in ( E - E’) . Then

xo + x , + yo + y , = n

and

Suppose

then

so

This implies

but then

x1 Y l a < - + - - - , 2 2

and this contradicts the hypothesis that E - E’I 5 a. Hence

This implies A (G) 2 ( n - a) /2 .

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MAXIMUM INDEPENDENT SET PROBLEM 353

The next two lemmas bound A (G) /OPT( G) in terms of n and a constant k for the case that G is a tetrahedron free cubic planar graph and the case that G is a triangle free cubic planar graph.

Lemma 2. Let G = ( V , E ) be a tetrahedron free cubic planar graph with 1V1 = n. Then there is an integer k , 0 I k I n / 6 , such that

2 max { 3 ( n '" 2k)' %}' OPT( G ) A ( G )

Proof. Since G is cubic we have IEl = 3/21VI = 3 /2n and, from fact 1 , OPT(G) 5 n / 2 . Hence there is an integer k 2 0 such that OPT(G) = n / 2 - k . We can create a bipartite subgraph G"(X ' , Y' , E") of G as follows: let X' be a maximum independent set for G and let Y' = V - X ' ; let E" be the subset of edges in E connecting vertices in X ' with vertices in Y ' . Since G is cubic and since vertices in X' are an independent set for G, then ( a ) (X'I = OPT(G) , ( b ) no pair of vertices in X' is connected by an edge in E, and (c ) each vertex in X' has degree three. Thus, IE"I = 3 . O P T ( G ) . Since the maximum bipartite subgraphof G obtained in step 1 of algorithm A has more edges than any other bipartite sub- graph of G, IE'I ? [,??I. Then IE'I ? 3 * OPT(G) = 3 /2n - 3k . Hence IE - E'I I 3/2n - (3 /2n - 3 k ) = 3k. From Lemma 1 , then, A ( G ) ? ( n - 3 k ) / 2 . Hence

A ( G ) n - 3k OPT(G) n - 2k'

2-

From Theorem l a we know that IE'I 2 7 / 6 n . So IE - E'I I 3 / 2 n - 716n = n / 3 . Thus, from Lemma 1 with a = nl3 , we have A ( G ) 2 n13 and

A ( G ) 2n OPT(G) ? 3(n - 2k)'

It now is only necessary to show that k 5 n / 6 . But, OPT(G) 2 A ( G ) 2 n / 3 so n / 2 - k nl3 and this implies k 5 n16. H

Lemma 3. suppose I VI = n . Then there is an integer k, 0 5 k I 3n/20 , such that

Let G = ( V , E ) be a cubic planar graph which is triangle free and

7n 2 max

OPT( G ) 10(n - 2k)' n - 2k

Proof. Following the argument of Lemma 2 , A(G)IOPT(G) 2 ( n - 3 k ) / ( n - 2 k ) . From Theorem l b , IE'I ? 6 / 5 n . Following the argument of Lemma 2 , using this bound for IE'I gives A ( G ) 2 7n/20 and A(G)IOPT(G) 2

7n/10(n - 2 k ) . We have k I 3n/20 from OPT(G) = n / 2 - k ? 7n/20 . We now prove the main results.

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354 CHOUKHMANE AND FRANC0

Theorem 2. If G = ( V , E ) is a cubic planar graph then

Proof, We consider the part of G containing tetrahedrons (G') and the part not containing tetrahedrons ( G h ) separately. Clearly, A finds a maximum inde- pendent set for all tetrahedrons in G' so A(Gh)/OPT(Gh) 2 6/7 implies A(G)IOPT(G) L 617. From Lemma 2

2 max { 3(n 2 k ) , s} for some k , 0 5 k 5 -. n OPT( G h ) 6

Suppose k I nl9. Then

2n it - 3k 5-

3(n - 2k) n - 2k'

But, then

A ( G h ) ,n - 3k 6 ?' OPT(Gh) - n - 2k

Suppose k > nl9. Then

2n n - 3k n - 2k

2- 3(n - 2k)

so

A ( G h ) > 2n 6 7' OPT(G") - 3(n - 2k)

Theorem 3. If G = ( V , E ) is a cubic planar graph which is triangle free then

A ( G ) 7 OPT(G) 8

Proof. From Lemma 3

7n 3n for some k, 0 I k I -.

20 "-'"I 10(n - 2k)' n - 2k A ( G ) z m a x

OPT( G )

Suppose k I n110. Then

7n n - 3k n - 2k

5- 10(n - 2k)

so

A ( G ) n - 3k 7 2-2 -

OPT(G) n - 2k 8'

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MAXIMUM INDEPENDENT SET PROBLEM 355

IS

FIG. 1 . A triangle free cubic planar graph and its maximum bipartite subgraph.

Suppose k > n/10. Then

7n n - 3k n - 2k

2- 10(n - 2 k )

so A ( G ) 7n 7

OPT(G) 10(n - 2 k ) s An example showing that the bound of Theorem 3 is tight is given in Figure 1.

In Figure 1 the cubic planar graph shown on the left has a maximum independent set consisting of eight vertices which are shown shaded. On the right is a maximum bipartite subgraph of that graph: the solid lines represent the subgraph and the dotted lines represent edges in the graph but not in the subgraph. Execution of steps 2 , 3 , and 4 of algorithm A on the subgraph results in an independent set, depicted by the shaded vertices in the subgraph, returned as output. This inde- pendent set contains seven vertices so, in this case, A(G)/OPT(G) = 7/8. The graph on the right side of Figure 1 may be duplicated rn times and the duplicates combined to form a large planar graph of 18rn vertices by removing the edge connecting vertices 1 and 2 in each duplicate and adding an edge between vertex 2 of the irh duplicate and vertex 1 of the i + 1'' duplicate for 1 5 i 5 rn - 1 and adding an edge connecting vertex 2 of the rnrh duplicate with vertex 1 of the first duplicate. For this graph A ( G ) / O P T ( G ) = 7/8 also.

5. CONCLUSION

We have shown the existence of good approximation algorithms for the NP- complete problem of finding the maximum independent set for a cubic planar

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356 CHOUKHMANE AND FRANC0

graph. This result is interesting in light of the earlier result that either the max- imum independent set problem can be solved with a polynomial approximation scheme, or else there is no polynomial time approximation algorithm W for it that satisfies Mw > E for some E > 0.

References

[l] N. Chiba, T. Nishizeki, and N. Saito, An approximation algorithm for the maximum independent set problem on planar graphs. SIAM J . Comput. 11 (1982) 663-675.

[2] M. Garey, D. Johnson, and L. Stockmeyer, Some simplified NP-complete graph problems. Theoret. Compuf. Sci. 1 (1976) 237-267.

[3] M. Garey and D. Johnson, Computers and Intractability: A guide to the Theory of NP-Completeness. Freeman, San Francisco (1979).

[4] F. Hadlock, Findng a maximum cut of a planar graph in polynomial time. SIAM J . Comput. 4 (1975) 221-225.

[5] G . Hopkins and W. Staton, Extremal bipartite subgraphs of cubic triangle free graphs. J . Graph Theory 6 (1982) 115-121.

[6] W. Staton, Edge deletions and the chromatic number. Ars Cornbinatoria 10 (1980) 103-106.