Vol. 2(1997): nr 4. Tight Lower Bounds on the. Approximability of Some. Peter Jonsson. Linkoping University

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1 Linkoping Electronic Articles in Computer and Information Science Vol. 2(1997): nr 4 Tight Lower Bounds on the Approximability of Some NPO PB-Complete Problems Peter Jonsson Department of Computer and Information Science Linkoping University Linkoping, Sweden Linkoping University Electronic Press Linkoping, Sweden http: /

2 Published on April 11, 1997 by Linkoping University Electronic Press Linkoping, Sweden Linkoping Electronic Articles in Computer and Information Science ISSN Series editor: Erik Sandewall c1997 Peter Jonsson Typeset by the author using La TE X Formatted using etendu style Recommended citation: <Author>. <Title>. Linkoping Electronic Articles in Computer and Information Science, Vol. 2(1997): nr 4. http: / April 11, This URL will also contain a link to the author's home page. The publishers will keep this article on-line on the Internet (or its possible replacement network in the future) for a period of 25 years from the date of publication, barring exceptional circumstances as described separately. The on-line availability of the article implies a permanent permission for anyone to read the article on-line, to print out single copies of it, and to use it unchanged for any non-commercial research and educational purpose, including making copies for classroom use. This permission can not be revoked by subsequent transfers of copyright. All other uses of the article are conditional on the consent of the copyright owner. The publication of the article on the date stated above included also the production of a limited number of copies on paper, which were archived in Swedish university libraries like all other written works published in Sweden. The publisher has taken technical and administrative measures to assure that the on-line version of the article will be permanently accessible using the URL stated above, unchanged, and permanently equal to the archived printed copies at least until the expiration of the publication period. For additional information about the Linkoping University Electronic Press and its procedures for publication and for assurance of document integrity, please refer to its WWW home page: http: / or by conventional mail to the address stated above.

3 Abstract We investigate the approximability of the NPO PB-complete problems Min Ones, Min Dones, Max Ones, Max Dones and Max PB 0/1 Programming. We show that, unless P = NP, these problems are not approximable within n 1?" for any " > 0 where n is the number of variables in the input. Since all of these problems are approximable within n, this lower bound is close to optimal.

4 1 1 Introduction Approximation of NP-complete optimization problems has become a very active area of research. It is now well-known that the approximability of such problems ranges from problems that are approximable within every constant in polynomial time (e.g., Knapsack [Ibarra and Kim, 1975]) to problems that are not approximable within n " for any " > 0, where n is the size of the input instance (e.g., Maximum Weighted Satisfiability [Ausiello et al., 1981]). It is well known that even NP optimization problems whose objective function is bounded by some polynomial in the size of the input may be hard to approximate. This class of problems, called NPO PB, can be divided into two classes, Max PB and Min PB, containing maximization and minimization problems respectively. Both Max PB and Min PB contains complete problems under approximation preserving reductions [Berman and Schnitger, 1992; Kann, 1994]. Furthermore, Crescenzi et al. [1995] has proven that any Min PB-complete problem is NPO PB-complete and that any Max PBcomplete problem is NPO PB-complete. Thus, the classes of Min PB-complete, Max PB-complete and NPO PB-complete problems coincide. Tight (or even optimal) lower bounds on the approximability of several NPO PB-complete problems have appeared in the literature. For instance, Kann [1994] has presented optimal bounds on Min # Sat and Min PB 0-1 Programming. However, there are still many problems for which tight lower bounds are not known. In this paper we provide such lower bounds for ve NPO PB-complete problems; Min Ones, Min Dones, Max Ones, Max Dones and Max PB 0/1 Programming. For each of these problems we show that they cannot be approximated within n 1?" for any " > 0 where n is the number of variables. Since all of these problems can trivially be approximated within n, this bound is close to optimal. 2 Preliminaries We begin by giving some standard denitions in the eld of approximation and optimization theory. Denition 2.1 An NPO problem A is a quadruple hi; sol; m; goali such that 1. I is the set of instances of A and it is recognizable in polynomial time. 2. Given an instance x 2 I, sol(x) denotes the set of feasible solutions of x. These solutions are short, that is, a polynomial p exists such that for any y 2 sol(x), jyj p(jxj). Moreover, it is decidable in polynomial time whether, for any y such that jyj p(jxj), y 2 sol(x).

5 2 3. Given an instance x and a feasible solution y of x, m(x; y) denotes the positive integer measure of y. The function m must be computable in polynomial time. 4. goal 2 fmax; ming. The class NPO is the set of all NPO problems. The goal of an NPO problem with respect to an instance x is to nd an optimum solution, i.e., a feasible solution y such that m(x; y) = goalfm(x; y 0 ) ; y 0 2 sol(x)g. An NPO problem is said to be polynomially bounded if a polynomial q exists such that, for any instance x and for any solution y of x, m(x; y) q(jxj). The class NPO PB is the set of polynomially bounded NPO problems. In the following opt will denote the function mapping an NPO instance x to the measure of an optimal solution. Denition 2.2 Given an NPO problem A and a function from N to (1; 1), we say that a polynomial-time algorithm P is an - approximation algorithm for A i for every instance x of A of size n, P produces a solution in the range [opt(x)=(n); (n)opt(x)]. We say that A is approximable within a factor if such an algorithm exists. Proposition 2.3 Let : N! (1; 1), ` : N! N be functions and A be an NP maximization problem. A is not approximable within a factor if there exists a polynomial-time reduction T from an NPcomplete decision problem F to A satisfying the following: 1. For an arbitrary instance I F of F of size n, the size of T (I F ) is `(n). 2. For any two instances I F, IF 0 of size n of F such that I F 2 F and IF 0 62 F, opt(t (I F ))=opt(t (IF 0 )) (`(n)). Similarly, if A is a minimization problem then A is not approximable within a factor if 1. For an arbitrary instance I F of F of size n, the size of T (I F ) is `(n). 2. For any two instances I F, I 0 F of size n of F such that I F 2 F and I 0 F 62 F, opt(t (I 0 F ))=opt(t (I F )) (`(n)). The nonapproximability of problems is described as a function of the size of the problem instance, or more often, as a function of some size parameter such as the number of variables in the instance. For all problems considered in this paper, we will assume that j j returns the number of variables in the given instance. It is often demanded that there exists a trivial solution triv(x) for each input x so that it can be ensured that an approximation

6 3 algorithm always nds a feasible solution. Throughout this paper we assume the existence of trivial solutions. We assume that the measure of the trivial solution is 1 for the minimization problems and n (the number of variables) for the maximization problems. We continue by giving formal denitions of the problems that we will investigate. We also give the previously known best lower bounds on the approximability of these problems. The rst approximation problem to be dened is Min Dones: Instance: Disjoint sets X; Z of variables and a 3CNF formula F over X [ Z. Solution: Truth assignment satisfying F. Measure: The number of variables in Z that are set to true in the assignment. Min Dones is NPO PB-complete and cannot be approximated within jzj 1?" for any " > 0 [Kann, 1994]. Min Ones, the variation in which all variables are distinguished, i.e. X =?, is also NPO PB-complete and not approximable within jzj 1=2?" for any " > 0 [Kann, 1994]. The maximization version of Min Dones (which we denote Max Dones) is NPO PB-complete [Kann, 1992] and is not approximable within jzj 1?" and not within (jxj + jzj) 1=2?" for any " > 0 [Kann, 1995]. Let Max Ones denote the variation of Max Dones where X =?. Max Ones is NPO PB-complete [Kann, 1992] and not approximable within jzj 1=3?" [Kann, 1995]. Finally, we present the problem Max PB 0-1 Programming: Instance: Integer m n matrix A, integer m-vector b and nonnegative binary n-vector c 2 f0; 1g n. Solution: A binary n-vector x 2 f0; 1g n such that Ax b. Measure: The scalar product of c and x, i.e., P n i=1 c ix i. Berman and Schnitger [1992] have proved NPO PB-completeness of this problem and Kann [1995] has shown that it cannot be approximated within n 1=2?" where n is the number of variables. A tight lower bound on the minimization version of Max PB 0-1 Programming has previously been established by Kann [1994]. By assuming the existence of trivial solutions, it is easy to see that these problems can be approximated within n where n is the number of variables. 3 Minimization Problems The rst problem that we will consider is Min Ones. Before we begin, we need some basic terminology of propositional logic. In this logic, a proposition (also called a variable) x can be either true or false. A truth assignment is an assignment of \true" (T) or \false" (F) to every proposition. A literal is an proposition x or its negation

7 4 :x. A clause is a disjunction of literals and it is satised by a given truth assignment i at least one of its literals is true. A formula is in conjunctive normal form (CNF) i it is a conjunction of clauses. Such a formula is satised by a truth assignment i every clause is satised. A model is a truth assignment which satises a given formula. We dene the computational problem Satisfiability (Sat) as usual: Instance: A CNF formula F. Question: Is there a satisfying truth assignment for F? Sat is NP-complete [Cook, 1971]. We say that a propositional formula is 3CNF i it is the conjunction of clauses with at most three literals per clause. The Sat problem restricted to 3CNF formulae (3Sat) is also NP-complete [Cook, 1971]. Lemma 3.1 Let Min1 denote the function from satisable 3CNF formulae to N that returns the minimum number of propositions that must be assigned T in order to satisfy the given formula. Let c > 4 be a xed integer. There exists a polynomial-time tranformation T c on 3CNF formulae satisfying the following: for an arbitrary 3CNF formula F, 1. jt c (F )j = jfj c ; 2. Min1(T c (F )) jfj 4 if F is satisable; 3. Min1(T c (F )) jfj c? 8jFj 3? jfj otherwise. Proof: Let F be an arbitrary 3CNF formula over the propositions P = fp1; : : :; p k g. We begin by describing T c (F ). There exists less than 8k 3 distinct 3-clauses over the propositional symbols in P. We can thus assume that F = C1 ^ : : : ^ C s where s 8k 3. Let w be a fresh proposition and Q = fq1; : : :; q kc?k?sg, R = fr1; : : :; r s g be sets of fresh propositions. Consider the clause C i = (l 1 i _l 2 i _l 3 i ), 1 i s. Since the clauses in F contain at most three literals, we do not require that l 1 i, l2 i and l 3 i are distinct. Let K i = f(r i, (l 1 i _ l 2 i )); (l3 i _ r i _ w)g: where, denotes logical equivalence, i.e.,, i (:_)^(_:). Dene T c (F ) as follows: T c (F ) = [ fk i j 1 i sg [ f(:w _ q j ) j 1 j k c? k? sg: Note that jt c (F )j k c and let t = k c?jt c (F )j. By adding t clauses of the type (:o i ), 1 i t, where o i, 1 i t, are fresh propositions, we can assume that jt c (F )j = k c. Let O = fo i j 1 i tg.

8 5 Now, observe that T c (F ) is not a 3CNF formula since it contains the, symbol. However, we can easily show that T c (F ) can be written as a 3CNF formula without introducing any extra variables. This boils down to showing that the expression (r i, (l 1 i _ l 2 i )) can be transformed to an equivalent 3CNF formula without extra variables: (r i, (l 1 i _ l 2 i )) (:r i _ l 1 i _ l 2 i ) ^ (r i _ :(l 1 i _ l 2 i )) (:r i _ l 1 i _ l 2 i ) ^ (r i _ (:l 1 i ^ :l 2 i )) (:r i _ l 1 i _ l 2 i ) ^ (r i _ :l 1 i ) ^ (r i _ :l 2 i ) Finally, it is obvious that T c (F ) can be computed in polynomial time since c is xed. We continue by showing that T c satises the given requirements. Let F 0 = T c (F ). We begin by showing that Min1(F 0 ) k 4 if F is satisable. Let M be a satisfying assignment for F. We extend M to also act on literals in the obvious way: M(:p) = F if M(p) = T and vice versa. Construct a model M 0 for F 0 as follows: M 0 (p) = M(p) p 2 P ; M 0 (w) = F; M 0 (q) = F( q 2 Q; F if M(l 1 M 0 (r i ) = i ) = M(l 2 i ) = F 1 i s; T otherwise M 0 (o) = F o 2 O: In this case, Min1(F 0 ) k + s k + 8k 3 k 4. Assume to the contrary that F is not satisable, that is, no truth assignment can satisfy all clauses in F simultaneously. Note that a clause of the type (l 3 i _ r i _ w), 1 i s, is logically equivalent to the clause (l 1 i _ l 2 i _ l 3 i _ w). This follows from the fact that, by the construction of F 0, r i is logically equivalent to (l 1 i _l 2 i ). Consequently, the clauses (l 3 i _ r i _ w), 1 i s, in F 0 cannot be simultaneously satised unless w is assigned T. But then q j, 1 j k c?k?s, must also be assigned T in order to satisfy F 0. Hence, a model M 0 for F 0 which assigns T to as few propositions as possible has the following appearance: M 0 (p) = F p 2 P ; M 0 (w) = T; M 0 (q) = T q 2 Q; M 0 (r) = F r 2 R; M 0 (o) = F o 2 O:

9 6 Thus, Min1(F 0 ) 1 + k c? k? s k c? k? 8k 3 since s 8k 3. 2 Theorem 3.2 Min Ones is not approximable within n 1?" for any " > 0 where n is the number of variables, unless P = NP. Proof: Arbitrarily choose an " > 0 and let c > 5 satisfy 1?5=c ". Dene `(k) = k c and (n) = n 1?5=c. By Lemma 3.1, there exists a polynomial-time reduction T c from 3Sat to MinOnes with the following properties: 1. For an arbitrary instance I of 3Sat containing k variables, T c (I) contains `(k) variables. 2. Let I and I 0 be 3Sat instances with k variables such that I is satisable and I 0 is not satisable. Then, opt(t c (I 0 )) opt(t c (I)) kc? 8k 3? k k 4 kc?1 k 4 = k c?5 = (`(k)): Hence, Min Ones is not approximable within (n) = n 1?5=c. Since 1? 5=c " and " was arbitrarily chosen, the theorem follows. 2 The second minimization problem that we will consider is Min Dones. Kann [1994] has shown that Min Dones is not approximable within n 1?" for any " > 0 where n is the number of distinguished variables. We strengthen this result in the next corollary. Corollary 3.3 Min Dones is not approximable within n 1?" for any " > 0, where n is the number of variables (i.e., n = jxj + jzj). Proof: The proof is by a cost-preserving polynomial-time transformation from Min Ones. Let F be an arbitrary, satisable 3CNF formula over the variables Z = fz1; : : :; z n g. Let X =? and let Z; X; F be an instance of Min Dones where Z denotes the distinguished variables. This is a polynomial-time transformation and the optimal value of this instance of Min Dones equals Min1(F ). Thus, if there exists an " > 0 such that Min Dones is approximable within (jxj+jzj) 1?", then Min Ones would be approximable within n 1?" which contradicts Theorem Maximization Problems To show the nonapproximability of Max Ones, we use a construction similar to the construction in Lemma 3.1. Lemma 4.1 Dene the function Max 1 from satisable 3CNF formulae to N such that Max1(F ) equals the maximum number of propositions that can be assigned T in an satisfying assignment of F. Let c > 4 be a xed integer. There exists a polynomial-time tranformation U c on 3CNF formulae satisfying the following: for an arbitrary 3CNF formula F,

10 7 1. ju c (F )j = jfj c ; 2. Max1(U c (F )) jfj 4 if F is not satisable; 3. Max1(U c (F )) jfj c? 8jFj 3? jfj otherwise. Proof: Arbitrarily choose a 3CNF formula over the propositions P = fp1; : : :; p k g. There exists less than 8k 3 distinct 3-clauses over the propositions in P. We can thus assume that F = C1 ^ : : : ^ C s where s 8k 3. Let w be a fresh proposition and Q = fq1; : : :; q kc?k?sg, R = fr1; : : :; r s g be sets of fresh propositions. For arbitrary clauses C i = (l 1 i _ l 2 i _ l 3 i ), 1 i s, let K i = f(r i, (l 1 i _ l 2 i )); (l 3 i _ r i _ :w)g: Dene U c (F ) as follows: U c (F ) = [ fk i j 1 i sg [ f(w _ :q j ) j 1 j k c? k? sg Note that ju c (F )j k c and let t = k c? ju c (F )j. By adding t clauses of the type (:o i ), 1 i t, where o i, 1 i t, are fresh propositions, we can assume that ju c (F )j = k c. Let O be the set of these propositions. We begin by showing that Max 1(F 0 ) k c? 8k 3? k if F is satis- able. Let M be a satisfying assignment for F. Construct a model M 0 for F 0 as follows: M 0 (p) = M(p) p 2 P ; M 0 (w) = T; M 0 (q) = T( q 2 Q; F if M(l 1 M 0 (r i ) = i ) = M(l 2 i ) = F 1 i s; T otherwise M 0 (o) = F o 2 O: In this case, Max1(F 0 ) k c? k? s k c? k? 8k 3. Assume to the contrary that F is not satisable. Then there exists a clause of the form (l 3 i _ r i _ :w), 1 i s, which cannot be satised unless w is assigned F. But then q j, 1 j k c?k?s, must also be assigned F. We can dene a model M 0 for F 0 as follows: M 0 (p) = T p 2 P ; M 0 (w) = F; M 0 (q) = F q 2 Q; M 0 (r) = T r 2 R; M 0 (o) = F o 2 O: Thus, Max 1(F 0 ) k + s k 4. By using the transformation used in Lemma 3.1, we know that F 0 can be written as a 3CNF formula

11 8 without introducing any extra variables. Hence, the lemma follows. 2 The proofs of the following theorem and corollary are omitted since they are similar to the proofs of Theorem 3.2 and Corollary 3.3. Theorem 4.2 Max Ones is not approximable within n 1?" for any " > 0 where n is the number of variables, unless P = NP. Corollary 4.3 Max Dones is not approximable within n 1?" for any " > 0, where n is the number of variables (i.e., n = jxj + jzj). Next, we show that Max PB 0/1 Programming is as hard to approximate as Max Ones. To do so, we will use a well-known connection between propositional logic and zero/one programming, established by Dantzig [1963]. Given a truth assignment M on a set of propositions x1; : : :; x n, we dene the corresponding 0/1-vector y = (y1; : : :; y n ) such that y i = 1 if M(x i ) = T and y i = 0 otherwise. A truth assignment satises a set of S clauses: _ j2p i x j _ _ j2n i :x j 1 A for 1 i S. i the corresponding 0/1-vector y satises the following system of inequalitites: X y j? X y j 1? jn i j for all 1 i S. j2p i j2n i For a 0, 1 matrix A, denote by n(a) the vector whose ith component n i (A) is the number of?1's in the ith row of A. Let 1 stand for the all-one vector of appropriate dimension. Hence, the above system of inequalities can be written on the form Ay 1? n(a): () Sat is thus equivalent to the problem of nding a zero/one solution y to () or show that none exists. Given a 3CNF formula F, let IE(F ) denote the corresponding set of inequalitites. Theorem 4.4 Max PB 0/1 Programming is not approximable within n 1?" for any " > 0 where n is the number of variables. Proof: The proof is by a cost-preserving polynomial-time transformation from Max Ones. Let F be an arbitrary, satisable 3CNF formula with n variables and m clauses. Consider the following instance I of Max PB 0/1 Programming: max nx y i over IE(F ): i=1

12 9 Observe that the optimal value of I equals Max Ones(F ) and I contains exactly as many variables as the given 3CNF formula. By Corollary 4.2, MaxOnes is not approximable within n 1?" for any " > 0. Since the transformation can be carried out in polynomial time without introducing any extra variables, the theorem follows. 2 By Theorem 4.4, Max PB 0/1 Programming is hard to approximate even if the constraint matrix A is a 0, 1 matrix with at most three non-zero entries per row and the right-hand side vector b is restricted to take its values from f1; 0;?1;?2g. References [Ausiello et al., 1981] G Ausiello, A D'Atri, and M Protasi. Lattice theoretic ordering properties for NP-complete optimization problems. Annales Societatis Mathematicae Polonae, 4:83{94, [Berman and Schnitger, 1992] P Berman and G Schnitger. On the complexity of approximating the independent set problem. Information and Computation, 96:77{94, [Cook, 1971] S A Cook. The complexity of theorem-proving procedures. In Proceedings of the 3rd ACM Symposium on Theory of Computing, pages 151{158, [Crescenzi et al., 1995] P Crescenzi, V Kann, R Silvestri, and L Trevisan. Structure in approximation classes. In Proceedings of First Annual International Conference on Computing and Combinatorics, pages 539{548. Springer-Verlag LNCS 959, [Dantzig, 1963] G B Dantzig. Linear Programming and Extensions. Princeton University Press, Princeton, New Jersey, [Ibarra and Kim, 1975] O H Ibarra and C E Kim. Fast approximation for the knapsack and sum of subset problems. Journal of the ACM, 22(4):463{468, [Kann, 1992] V Kann. On the approximability of NP-complete optimization problems. PhD thesis, Royal Institute of Technology, Stockholm, [Kann, 1994] V Kann. Polynomially bounded minimization problems that are hard to approximate. Nordic Journal of Computing, 1:317{ 331, [Kann, 1995] V Kann. Strong lower bounds on the approximability of some NPO PB-complete maximization problems. In Proceedings of the 20th International Symposium on Mathematical Foundations of Computer Science, pages 227{236. Springer-Verlag, 1995.