1 Introduction and Results

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1 Discussiones Mathematicae Graph Theory 20 (2000 ) 7 79 SOME NEWS ABOUT THE INDEPENDENCE NUMBER OF A GRAPH Jochen Harant Department of Mathematics, Technical University of Ilmenau D Ilmenau, Germany Abstract For a finite undirected graph G on n vertices some continuous optimization problems taken over the n-dimensional cube are presented and it is proved that their optimum values equal the independence number of G. Keywords: graph, independence. 99 Mathematical Subject Classification: 05C35. Introduction and Results Let G be a finite simple and undirected graph on V (G) = {, 2,..., n} with its edge set E(G). A subset I of V (G), such that the subgraph of G induced by I is edgeless, is called an independent set of G, and the maximum cardinality of an independent set of G is named the independence number α(g) of G. N(i) and d i denote the set and the number of neighbours of i V (G) in G, respectively, and let (G) = max{d i i V (G)} and C n = {(x, x 2,..., x n ) 0, i =, 2,..., n}. For events A and B and for a random variable Z of an arbitrary random space, P (A), P (A B), and E(Z) denote the probability of A, the conditional probability of A given B, and the expectation of Z, respectively. Since the computation of α(g) is difficult (INDEPENDENT SET is an NP-complete problem; see [6]), much work was done to establish bounds on α(g) (e.g., see [, 3, 4, 5, 8, 0, 2, 3, 4, 5, 6, 7]), to find efficient algorithms forming a large independent set of G (e.g., see [2, 7, 8, 9, 0, 2]), or to replace the combinatorial optimization problem to determine α(g) by a continuous one

2 72 J. Harant (e.g., see [9, ]). The last approach leads to bounds on α(g) as well as to efficient algorithms (e.g., see [8, 9]). In the present paper some new continuous optimization problems taken over C n are presented and it is proved that their optimum values equal α(g). Theorem gives a remarkable result of T.S. Motzkin and E.G. Straus [] and Theorem 2 is proved in [9]. Theorem. α(g) = max (0,0,...,0) (x,x 2,...,x n ) C n Theorem 2. α(g) = ( ) 2. x 2 i +2 x j ij E(G) max ( ( x j )). (x,x 2,...,x n ) C n A classical lower bound on α(g) due to Y. Caro and V.K. Wei [3, 7] is given by the following theorem. Theorem 3. α(g) +d i. The next Theorems 4, 5, 6, and 7 are the main results of the present paper. Theorem 4. α(g) = max e G(x, x 2,..., x n ), where (x,x 2,...,x n ) C n e G (x, x 2,..., x n ) = ( ( ) ( x j ) + x j + ( x l ) l N(j)\(N(i) {i}) + Theorem 5. α(g) = max f G(x, x 2,..., x n ), where (x,x 2,...,x n) C n f G (x, x 2,..., x n ) = ( ( ) ( x j ) ) + + ( x l ) l N(j)\(N(i) {i}) ij E(G) ). x j. The following Theorem 6 looks more complicate, but it is stronger than Theorem 4 and Theorem 5 (see Remark ). Theorem 6. α(g) = max g G(x, x 2,..., x n ), where (x,x 2,...,x n ) C n g G (x, x 2,..., x n ) = ( ) x ( + i + ( x l ) l N(j)\(N(i) {i}) ( x j ))

3 Some News about the Independence Number i V ( ( x j )) 2 ( x j )+ { x j and V = i V (G) } x j > 0. A weaker (see Remark ), but a more transparent and (see Remark 2) an algorithmically realizable version of Theorem 5 is the following one. Theorem 7. α(g) = max h G(x, x 2,..., x n ), where (x,x 2,...,x n ) C n h G (x, x 2,..., x n ) = x j. ij E(G) 2 Proofs Throughout the proofs we will use the well-known fact that for a random subset M of a given finite set N, E( M ) = y N P (y M) = N kp ( M = k). Let I be a maximum independent set of G and let x i = if i I and x i = 0 if i / I. Since ( x i ) ( x j ) = 0 for i V (G) and = 0, we obtain ij E(G) x i x j Lemma. α(g) = e G (x, x 2,..., x n) = f G (x, x 2,..., x n) = g G (x, x 2,..., x n) = h G (x, x 2,..., x n). With Lemma, it is clear that Theorem 7 follows from Theorem 5. Now, let (x, x 2,..., x n ) be an arbitrary member of C n. We form a set X V (G) by random and independent choice of i V (G), where P (i X) =. Let H, H 2, and H 3 be the subgraph of G induced by the vertices of X, by the vertices i X with N(i) X, and by the vertices i / X with N(i) X =, respectively. Furthermore, let Y be a smallest subset of V (H 2 ) covering all edges of H 2, i.e., the graph induced by V (H 2 ) Y is edgeless, and let I and I 3 be a maximum independent set of H and H 3, respectively. It can be seen easily that Y = V (H 2 ) α(h 2 ), Y E(H 2 ) and that (X Y ) I 3 and I I 3 are independent sets of G. Because of these remarks and the property of the expectation to be an average value, we have Lemma 2 as follows.

4 74 J. Harant Lemma 2. α(g) E( X Y ) + E(α(H 3 )), α(g) E(α(H )) + E(α(H 3 )), E( X Y ) = E( X ) E( Y ) E( X ) E( E(H 2 ) ), and E( X Y ) = E( X ) E( V (H 2 ) ) + E(α(H 2 )). Lower bounds on E(α(H )), E(α(H 2 )), and E(α(H 3 )) are given in Lemma 3., + x j ( ( x j )) 2 i V ( x j )+ ( ) ( x j ). + ( x l ) l N(j)\(N(i) {i}) Lemma 3. E(α(H )) E(α(H 2 )) and E(α(H 3 )) x j, where V = {i V (G) x j > 0}, P roof. For i V (G) define the random variable Zi with Z i = +k if i X and N(i) X = k 0, and Zi = 0 if i / X. Using Theorem 3, E(α(H )) E( Zi ) = E(Zi ) = d i +k P (i X and N(i) X = k) = d i +k P (i X)P ( N(i) X = k) = d i +k P ( N(i) X = k)). For i V (G) we have d i P ( N(i) X = k) =. With Jensen s inequality m τ l φ(y l ) φ( m τ l y l ) for any convex function φ and any τ l 0 for l = l= l=, 2,..., m with m τ l =, l= E(α(H )) d i =. + kp ( N(i) X =k) + x j Now, let V = {i V (G) random variable with Zi 2 = +k = 0 otherwise. Then, Z 2 i x j > 0}. For i V (G) let Z 2 i be the if i X and N(i) X = k, and

5 Some News about the Independence Number E(α(H 2 )) E( = d i k= = d i k= d i k= = Z 2 i ) = E(Z 2 i ) +k P (i X and N(i) X = k) +k P (i X)P ( N(i) X = k) +k P ( N(i) X = k). P ( N(i) X = 0) + d i P ( N(i) X = k) = for i V (G) and with k= µ i = P ( N(i) X = 0) = ( x j ) and σ ik = P ( N(i) X = k), E(α(H 2 )) d i i V k= = For λ ik = d i k= Jensen s inequality, +k σ ik = +k σ ik = ( µ i ) di i V σ ik µ i we have λ ik 0, E(α(H 2 )) ( µ i ) i V d i d i,µ i < k= k= d i k= + kλ ik k= ( ( x j )) 2 i V d i ( x j )+ kp ( N(i) X =k) k= = σ ik (+k)( µ i ). +k σ ik λ ik = if i V, and again using ( ( x j )) 2. i V ( x j )+ x j = Finally, let us consider the random variable Zi 3 with Zi 3 = +k if i V (H 3) and N(i) V (H 3 ) = k 0, and Zi 3 = 0 if i / V (H 3). Then E(α(H 3 )) E( Zi 3) = E(Zi 3) = i V (G) d i = = d i +k P (i V (H 3) and N(i) V (H 3 ) = k) +k P (i V (H 3))P ( N(i) V (H 3 ) = k i V (H 3 )) (( ) ( x j ) d i +k P ( N(i) V (H 3) = k i V (H 3 )))

6 76 J. Harant = (( ) (( ) and Lemma 3 is proved. ( x j ) d i + kp ( N(i) V (H 3 ) =k i V (H 3 )) + ), ( x l ) ( x j ) l N(j)\(N(i) {i}) Theorem 4, 5, and 6 follow with E( X ) = = x j, E( V (H 2 ) ) = ij E(G) (, E( E(H 2 ) ) ( x j )), Lemma, 2, and 3. ) 3 Remarks For φ, ψ {e, f, g, h} define φ ψ if φ G (x, x 2,..., x n ) ψ G (x, x 2,..., x n ) for every graph G on n vertices and for every (x, x 2,..., x n ) C n. We write φ <> ψ if neither φ ψ nor ψ φ. Remark. h f g, e g and e <> f. P roof. We will use the following Lemma 4, which can be seen easily by induction on r. Lemma 4. For an integer r and a, a 2,..., a r [0, ], r a q + r ( a q ). q= q= The inequality h f is obvious. To see f g, first notice that x j = ( 2 x j ). If x j = 0 for ij E(G) an i V (G) then = ( ( x j )) = ( 2 x j ). Hence, with the abbreviation µ i = ( x j ) and ρ i = x j for i V (G) we have to show V = {i V (G) ( µ i)2 ( (µ i + i V µ i +ρ i )) ( ( i V 2 ρ i)), where again x j > 0}. Using Lemma 4, even µ i + ( µ i) 2 µ i +ρ i 2 ρ i for all i V. To prove e g, we have to show

7 Some News about the Independence Number Since + + x j +ρ i µ i + ( µ i) 2 µ i +ρ i if = ( x j ) if x j ( x j ) + ( ( x j )) 2 i V ( x j )+. x j x j = 0, it is sufficient to establish x j > 0, what is verified easily. For a cycle C n on n vertices e Cn ( 3, 3,..., 3 ) < f C n ( 3, 3,..., 3 ), e C n ( 2 3, 2 3,..., 2 3 ) > f Cn ( 2 3, 2 3,..., 2 3 ) and Remark is proved. With h f and e <> f it is clear that e h does not hold. It remains open, whether h <> e or h e. Theorems, 2, 4, 5, 6, and 7 are of that type that the independence number α(g) of a graph G on n vertices equals the optimum value of a continuous optimization problem O(G) to maximize a certain function φ G over C n. Hence, φ G (x, x 2,..., x n ) is a lower bound on α(g) for every (x, x 2,..., x n ) C n. Let (x, x 2,..., x n) C n be the solution of an arbitrary approximation algorithm for O(G). How to find an independent set I of G in polynomial time such that I φ G (x, x 2,..., x n)? In [8] and [9] efficient algorithms forming I with I φ G (x, x 2,..., x n) are given if O(G) is the optimization problem of Theorem or of Theorem 2. Remark 2 shows that this is also possible if we consider the case φ G = h G. In case φ G = e G, φ G = f G or φ G = g G the problem remains open, whether such an algorithm exists. Remark 2. There is an O( (G)n)-algorithm with INPUT: (x, x 2,..., x n ) C n, OUTPUT: an independent set I V (G) with I P roof. First we give the Algorithm:. For i = to n do if x j < then := else := 0. ij E(G) 2. For i = to n do if ( = and ( x j) = 0) then := I := {i V (G) = }. STOP x j. It is obvious that the algorithm in an O( (G)n)-algorithm. For the input vector (x, x 2,..., x n ) C n set x k x j = a.

8 78 J. Harant After step, the current (x, x 2,..., x n ) is a 0--vector and x k x j a because ( x k x j ) = x j, i.e., x k x j is multilinear. In step 2, ( x j ) = 0 if and only if there is at least one j N(i) such that x j =. With = 0 instead of = the sum x k decreases by and the sum x k x j decreases by at least, hence x k x j does not decrease. After step 2, x k x j = 0 for all kj E(G), I = and Remark 2 is proved. x k = x k x j a References [] E. Bertram, P. Horak, Lower bounds on the independence number, Geombinatorics, (V) 3 (996) [2] R. Boppana, M.M. Halldorsson, Approximating maximum independent sets by excluding subgraphs, BIT 32 (992) [3] Y. Caro, New results on the independence number (Technical Report, Tel-Aviv University, 979). [4] S. Fajtlowicz, On the size of independent sets in graphs, in: Proc. 9th S-E Conf. on Combinatorics, Graph Theory and Computing (Boca Raton 978) [5] S. Fajtlowicz, Independence, clique size and maximum degree, Combinatorica 4 (984) [6] M.R. Garey, D.S. Johnson, Computers and Intractability, A Guide to the Theory of N P -Completeness (W.H. Freeman and Company, San Francisco, 979).

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