- Email: [email protected]
Near-optimal nonapproximability results for some Npo PB-complete problems
Inform+tion ~~;gwY Information
Processing
Letters 68 (1998) 249-253
Near-optimal nonapproximability results for some NPO PB-complete problems ’ Department of Computer and Information Science, Linkiipings Universitet, S-581 83 Link&ping, Sweden Received 26 September 1997; received in revised form 4 September Communicated by L.A. Hemaspaandra
1998
Abstract We show that a number of NPO PB-complete problems, including MIN ONES and MAX ONES, are hard to approximate within n’-’ for arbitrary E > 0. 0 1998 Elsevier Science B.V. All rights reserved. Keywords: Approximation;
Computational
complexity;
Combinatorial
problems
1. Introduction
2. Preliminaries
NPO PB is the class of NP optimization problems whose objective function is bounded by some polynomial in the size of the input. It is well known that many problems that are complete for NPO PB are notoriously hard to approximate and near-optimal lower bounds on the approximability of NPO PB-complete problems such as MIN # SAT and MIN PB O-l PROGRAMMING have appeared in the literature [5]. However, there are still many problems for which tight bounds are not known. In this paper we provide such bounds for five NPO PB-complete problems: MIN ONES, MIN DONE& MAX ONES, MAX DONES and MAX PB O-l PROGRAMMING. For each of these problems we show that they cannot be approximated within n I-’ for any E > 0 where n is the number of variables. Since all of these problems can trivially be approximated within n, this bound is essentially optimal.
Throughout this paper, we assume that P # NP. Solving an NP optimization problem A given the input instance x means finding a solution y such that the value of the objective function m A (x, y) is maximum or minimum depending on if A is a maximization or minimization problem. As usual, we assume that mA is polynomial-time computable. Let the optimal value function optA return the optimal value of mA for arbitrary instances x of A. For technical reasons, we follow Berman and Schnitger’s [l] convention that whenever the optimal value function returns value 0, this value is to be redefined to I instead. The class NPO is the set of all NP optimization problems. An NPO problem A is said to be polynomially bounded if a polynomial q exists such that for any instance x, optA (x) < q (Ix I). The class NPO PB is the set of polynomially bounded NPO problems.
’ Email: [email protected].
Definition 2.1. Given an NPO problem A and a function (2 : N + (1, co), we say that a polynomial-time algorithm P is an a-approximation algorithm for A iff
0020-0190/98/$ - see front matter 0 1998 Elsevier Science B.V. All rights reserved. PII: SOO20-0190(98)00170-7
250
f! Jonsson /Information
Processing Letters 68 (1998) 249-253
for every instance x of A of size n, P produces a solution in the range [o&x)/a(n), a(n)op We say that A is upproximable within a factor a! if such an algorithm exists. Lemma 2.2. Let a, b : N + (1,~) be functions such that given natural numbers i, j, it can be decided in polynomial time whether a(i) < j or not. Let A be an NP maximization problem and T a polynomialtime computable transformation from an NP-complete decision problem F to A. Assume that for every instance I E F, IT(Z)1 = [((Zl) for some function L :N + N. Then, A is not approximable within a factor a(n) = b(n)/a(n) tf T has the following two properties: (1) if1 E F then optA(T(Z)) > b(lT(Z)I); and (2) ifZ 4 F then optA(T(Z)) < a(lT(Z)I). Similarly, if A is a minimization problem then A is not approximable within a factor w(n) = b(n)/a(n) tf T satisfies the following: (1) ifZ E F then optA(T(Z)) < a(lT(Z)I); and (2) ifZ q! F then optA(T(Z)) > b(lT(Z)I). Proof. We prove the maximization case; the minimization case is analogous. Assume the existence of an polynomial-time algorithm K that can approximate A within a. Then, for an arbitrary instance x of A,
ac’x&~~‘x)
Arbitrarily select an instance Z of F and let I’ = T(Z) and m = ]I’). Note that the value of m does not depend on whether Z is in F or not. If Z E F then optA
> b(m)
and
< u(m)
and
INSTANCE: Disjoint sets X, Z of variables 3CNF formula F over X U Z. SOLUTION: Truth assignment
satisfying
and a
F.
MEASURE: The number of variables in Z that are set to true in the assignment. Arbitrarily choose 6 > 0. MIN DONES cannot be approximated within ]ZI ‘-’ [5]. MIN ONES, the variation in which all variables are distinguished, i.e., X = 0, is not approximable within ]ZI ‘/*-’ [5]. The maximization version of MIN DONES (denoted MAX DONES) is not approximable within IZ( ‘-” and not within (1x1 + ]Zl) ‘i2-’ [6]. Let MAX ONES denote the MAX DONES problem restricted to X = 0. MAX ONES is not approximable within ]Z]‘/3-E [6]. Finally, we present the problem MAX PB O-l PROGRAMMING: INSTANCE: Integer m x n matrix A, integer m-vector b and nonnegative binary n-vector c E (0, I}“.
a(m) -CmA(Z’, K(Z’)). SOLUTION: A binary n-vector Ax>b.
Assume to the contrary that I’ $ F. Then optA
Further, we assume that there exists a trivial solution triv(x) for each input x so that it can be ensured that an approximation algorithm always finds a feasible solution. We continue by giving formal definitions of the problems that we will investigate. Proofs of NPO PBcompleteness of these problems can be found in [ 1, 4,5]. We say that a propositional formula is 3CNF iff it is the conjunction of clauses with at most three literals per clause. The first problem we consider is MIN DONES:
mA(Z’, K(Z’))
Hence, Z E F iff u(m) < mA(Z’, K(Z’)). condition can be decided in polynomial lemma follows. 0
Since this time, the
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 ) . 1 returns the number of variables in the given instance.
x E (0, 1)” such that
MEASURE: The scalar product of c and x, i.e.,
2
CiXi.
i=l
Kann [6] has shown that this problem cannot be approximated within n 1/2-E. A strong lower bound on the minimization version of MAX PB 0- 1 PROGRAMMING has been established by Kann 151. Since the optimal value cannot equal 0 by definition, these problems can trivially be approximated within n
P:Jonsson / hfomationProcessing Letters 68 (1998) 249-253 (where n is the number of variables) since we have assumed the existence of trivial solutions. It is easy to see that if MAX ONES is not approximable within (II then MAX DONES is not approximable within CX.Using the well-known connection between propositional logic and zero/one programming [7], it is also easy to see that MAX PB O-l PROGRAMMING cannot be approximated within a. Similarly, nonapproximability of MIN ONES carryover to MIN DONES and MIN PB 0- 1 PROGRAMMING.
3. Minimization
problems
The first problem that we will consider is MIN ONES. Define the NP-complete decision problem 3sAT as usual [2]: Given a 3CNF formula F, is there a satisfying truth assignment for F? Lemma 3.1. Let Mini denote the optimal valuefunction for MIN ONES. Let c > 4 be a fixed integer. There exists a polynomial-time
transformation
(1) IT,(F)1 = IFY; (2) Min(T,(F)) < 1F14 tf F is satisjiable; (3) Min(T,(F))
> \FIC - 8(F13 - IFJ otherwise.
Proof. Let F be an arbitrary 3CNF formula over the propositions P = (PI, . . . , pk}. We begin by describing T,(F). There exists less than 8k3 distinct clauses of length 3 over the propositional symbols in P. We can thus assume that F = Cl A . . . A C,Ywhere s < 8k3. Let w be a fresh proposition and Q = (91, . . . , qk~_k-s_l }, R = {rl, . . . , rs} be sets of fresh propositions. Consider an arbitrary clause Ci = (1: v 1; v I?), 1 < i < s. Since the clauses in F contain at most three literals, we do not require that 1:) 1; and l! are distinct. Let Ki = (Yi eS (If
V
12)) A (13 V
ri V W),
where + denotes logical equivalence, a+/3
iff
i.e.,
(~~Vfi)A((21v~#J).
Define Tc( F) as follows: T,(F)=
A 1
Ki A
A ('Wvqj). l
Notethat]T,(F)\=s+k+l+k’-k-s-l=k” and that Tc( F) is not a 3CNF formula since it contains the + symbol. However, it is easy to show that Tc( F) can be written as a 3CNF formula without introducing any extra variables. This boils down to showing that the expression (ri ($ (1; v 1;)) can be transformed to an equivalent 3CNF formula without extra variables. This is a simple exercise. As a final observation, T,,(F) can be computed in polynomial time since c is fixed. We continue by showing that T, satisfies the given requirements. Let F’ = Tc( F). We begin by showing that Min(F’) < k4 if F is satisfiable. 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’ for F’ as follows: M’(p)
= M(p),
M’(w)
= F;
M’(q)
= F,
Tc on
3CNF formulae satisfying the following: for an arbitrary 3CNF formula F,
251
F M’(ri)
=
i T
p E P;
4 E Q; if M(l!) = M(lf) l,
= F,
In this case, Mint (F’) < k + s < k + 8k3 6 k4 for k sufficiently large. Assume to the contrary that F is not satisfiable, that is, no truth assignment can satisfy all clauses in F simultaneously. Note that a clause of the type (1; v rt v w), 1 < i < s, is logically equivalent to the clause (1; v 1; v l,? v w). This follows from the fact that, by the construction of F’, ri is logically equivalent to (I! v Zf).Consequently, the clauses (1” v riv w), 1 < i < s, in F’ cannot be simultaneously satisfied unless w is assigned T. But then qj, 1 < j < kc - k s - 1, must also be assigned T in order to satisfy F’. Hence, a model M’ for F’ which assigns T to as few propositions as possible has the following appearance: M’(p)
= F,
M’(w)
= T;
M’(q) = T, M’(r)
= F,
p E P;
q E Q; r E R.
Thus, Mint (F’) 3 I+ k’ - k - s - 1 > kc - k - 8k3 since s 6 8k3. q
t? Jonsson /Information Processing Letters 68 (1998) 249-253
252
n’-’
for any
E
MIN ONES is not approximable within 0 where n is the number of variables.
>
Proof. Arbitrarily choose an E > 0 and let the natural number c > 6 satisfy 1 - 6/c 3 E. Define a(n) = $1’ and b(n) = n ’ - I/‘. Given arbitrary natural numbers i, j, we can decide (in polynomial time) whether a(i) < j since a(i) < j iff i5 < j’. By Lemma 3.1, there exists a polynomial-time reduction Tc from 3 SAT to MIN ONES with the following properties: for an arbitrary instance I of 3s~~ containing k variables, (1) T,(Z) contains kc variables. (2) Minl(T,(Z)) < k5 = IT,(Z)J’/’ = a(lT,(Z)I) if Z is satisfiable. (3) Minl(T,(Z)) > kc-* = IT,(Z)l’-‘j“ = b(JT,(Z)I) if Z is not satisfiable. Hence, MIN ONES is not approximable within a(n) = b(n)/a(n)
= rzle6/’
by Lemma 2.2. Since 1-6/c 3 E and q chosen, the theorem follows.
E
was arbitrarily
Corollary 3.3. MIN DONES is not approximable within n’-’ for any F > 0, where n is the number of variables.
4. Maximization
problems
To show the nonapproximability of MAX ONES, similar to the construction in Lemma 3.1.
For arbitrary clauses Ci = (1; v 1: v l:), 1 < i < s, let Ki = (ri + (1: V 1’)) A (1” V ri V Define V,(F) UC(F)=
A
7~).
as follows: Ki A
I
A
(w v ‘4j).
l
Showing that U, satisfies the stated requirements is similar to the proof of Lemma 3.1. Proving the nonapproximability result is analogous to the proof of Theorem 3.2. q Corollary 4.2. MAX DONES and MAX PB 0- 1 PROGRAMMING are not approximable within n'-' for any E > 0, where n is the number of variables. Under the additional assumption that NP # ZPP, it is possible to show that MAX ONES cannot be approximated within n ’ --E even if restricted to 2CNF formulae. H&tad [3] has shown that the CLIQUE problem cannot be approximated within n 1--E (where n denotes the number of nodes in the graph), unless NP = ZPP. The reduction of CLIQUE to MAX ONES is straightforward; for each missing edge in the given graph G (say, between node x and node y), add a clause of the form (lx v -y). Obviously, the optimal value of the resulting MAX ONES instance equals the size of the largest clique in G.
we use a construction
Theorem 4.1. MAX ONES is notapproximable within n’-” for any E > 0 where n is the number of variables. Proof (Sketch). Let Max1 denote the optimal value function for MAX ONES. Let c > 4 be a fixed integer. We give a polynomial-time transformation U, on 3CNF formulae satisfying the following: for an arbitrary 3CNF formula F, (1) IUc(WI = IW; (2) Max] (U,(F)) < )F14 if F is not satisfiable; (3) Max1 (U,(F)) > IFIG - 81Fj3 - IFI otherwise. Arbitrarily choose a 3CNF formula over the propositions P = (pi,. . . , pk}. As noted earlier, we can safely assume that F = Cl A . . . A C, where s < 8k3. Let w be a fresh proposition and Q = {ql, . . . , qkc-k-s-l), R = {rl, . . . , rs} be sets of fresh propositions.
Acknowledgements Thanks to Viggo Kann, Thomas Drakengren the anonymous referee for helpful discussions comments.
and and
References [l] P. Berman, G. Schnitger, On the complexity of approximating the independent set problem, Inform. and Comput. 96 (1992) 77-94. [2] S.A. Cook, The complexity of theorem-proving procedures, in: Proc. 3rd Annual ACM Symp. on Theory of Computing, 1971, pp. 151-158. [3] J. HBstad, Clique is hard to approximate within n’-‘, in: Electronic Colloquium on Computational Complexity, Report TR97-038, 1997. See also an early version in: Proc. 37th Ann. IEEE Symp. on Foundations of Computer Science, 1996, pp. 627-636.
F! Jonsson /Information Processing Letters 68 (1998) 249-253
[4] V. Kann, On the approximability of NP-complete optimization problems, PhD Thesis, Royal Institute of Technology, Stockholm, 1992. [5] V. Kann, Polynomially bounded minimization problems that are hard to approximate, Nordic J. Comput. 1 (1994) 317-33 1. [6] V. Kann, Strong lower bounds on the approximability of some NPO PB-complete maximization problems, in: Proc. 20th
253
Intemat. Symp. on Mathematical Foundations of Computer Science, 1995, pp. 227-236. [7] C.H. Papadimitriou, K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, NJ, 1982.
Processing
Letters 68 (1998) 249-253
Near-optimal nonapproximability results for some NPO PB-complete problems ’ Department of Computer and Information Science, Linkiipings Universitet, S-581 83 Link&ping, Sweden Received 26 September 1997; received in revised form 4 September Communicated by L.A. Hemaspaandra
1998
Abstract We show that a number of NPO PB-complete problems, including MIN ONES and MAX ONES, are hard to approximate within n’-’ for arbitrary E > 0. 0 1998 Elsevier Science B.V. All rights reserved. Keywords: Approximation;
Computational
complexity;
Combinatorial
problems
1. Introduction
2. Preliminaries
NPO PB is the class of NP optimization problems whose objective function is bounded by some polynomial in the size of the input. It is well known that many problems that are complete for NPO PB are notoriously hard to approximate and near-optimal lower bounds on the approximability of NPO PB-complete problems such as MIN # SAT and MIN PB O-l PROGRAMMING have appeared in the literature [5]. However, there are still many problems for which tight bounds are not known. In this paper we provide such bounds for five NPO PB-complete problems: MIN ONES, MIN DONE& MAX ONES, MAX DONES and MAX PB O-l PROGRAMMING. For each of these problems we show that they cannot be approximated within n I-’ for any E > 0 where n is the number of variables. Since all of these problems can trivially be approximated within n, this bound is essentially optimal.
Throughout this paper, we assume that P # NP. Solving an NP optimization problem A given the input instance x means finding a solution y such that the value of the objective function m A (x, y) is maximum or minimum depending on if A is a maximization or minimization problem. As usual, we assume that mA is polynomial-time computable. Let the optimal value function optA return the optimal value of mA for arbitrary instances x of A. For technical reasons, we follow Berman and Schnitger’s [l] convention that whenever the optimal value function returns value 0, this value is to be redefined to I instead. The class NPO is the set of all NP optimization problems. An NPO problem A is said to be polynomially bounded if a polynomial q exists such that for any instance x, optA (x) < q (Ix I). The class NPO PB is the set of polynomially bounded NPO problems.
’ Email: [email protected].
Definition 2.1. Given an NPO problem A and a function (2 : N + (1, co), we say that a polynomial-time algorithm P is an a-approximation algorithm for A iff
0020-0190/98/$ - see front matter 0 1998 Elsevier Science B.V. All rights reserved. PII: SOO20-0190(98)00170-7
250
f! Jonsson /Information
Processing Letters 68 (1998) 249-253
for every instance x of A of size n, P produces a solution in the range [o&x)/a(n), a(n)op We say that A is upproximable within a factor a! if such an algorithm exists. Lemma 2.2. Let a, b : N + (1,~) be functions such that given natural numbers i, j, it can be decided in polynomial time whether a(i) < j or not. Let A be an NP maximization problem and T a polynomialtime computable transformation from an NP-complete decision problem F to A. Assume that for every instance I E F, IT(Z)1 = [((Zl) for some function L :N + N. Then, A is not approximable within a factor a(n) = b(n)/a(n) tf T has the following two properties: (1) if1 E F then optA(T(Z)) > b(lT(Z)I); and (2) ifZ 4 F then optA(T(Z)) < a(lT(Z)I). Similarly, if A is a minimization problem then A is not approximable within a factor w(n) = b(n)/a(n) tf T satisfies the following: (1) ifZ E F then optA(T(Z)) < a(lT(Z)I); and (2) ifZ q! F then optA(T(Z)) > b(lT(Z)I). Proof. We prove the maximization case; the minimization case is analogous. Assume the existence of an polynomial-time algorithm K that can approximate A within a. Then, for an arbitrary instance x of A,
ac’x&~~‘x)
Arbitrarily select an instance Z of F and let I’ = T(Z) and m = ]I’). Note that the value of m does not depend on whether Z is in F or not. If Z E F then optA
> b(m)
and
< u(m)
and
INSTANCE: Disjoint sets X, Z of variables 3CNF formula F over X U Z. SOLUTION: Truth assignment
satisfying
and a
F.
MEASURE: The number of variables in Z that are set to true in the assignment. Arbitrarily choose 6 > 0. MIN DONES cannot be approximated within ]ZI ‘-’ [5]. MIN ONES, the variation in which all variables are distinguished, i.e., X = 0, is not approximable within ]ZI ‘/*-’ [5]. The maximization version of MIN DONES (denoted MAX DONES) is not approximable within IZ( ‘-” and not within (1x1 + ]Zl) ‘i2-’ [6]. Let MAX ONES denote the MAX DONES problem restricted to X = 0. MAX ONES is not approximable within ]Z]‘/3-E [6]. Finally, we present the problem MAX PB O-l PROGRAMMING: INSTANCE: Integer m x n matrix A, integer m-vector b and nonnegative binary n-vector c E (0, I}“.
a(m) -CmA(Z’, K(Z’)). SOLUTION: A binary n-vector Ax>b.
Assume to the contrary that I’ $ F. Then optA
Further, we assume that there exists a trivial solution triv(x) for each input x so that it can be ensured that an approximation algorithm always finds a feasible solution. We continue by giving formal definitions of the problems that we will investigate. Proofs of NPO PBcompleteness of these problems can be found in [ 1, 4,5]. We say that a propositional formula is 3CNF iff it is the conjunction of clauses with at most three literals per clause. The first problem we consider is MIN DONES:
mA(Z’, K(Z’))
Hence, Z E F iff u(m) < mA(Z’, K(Z’)). condition can be decided in polynomial lemma follows. 0
Since this time, the
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 ) . 1 returns the number of variables in the given instance.
x E (0, 1)” such that
MEASURE: The scalar product of c and x, i.e.,
2
CiXi.
i=l
Kann [6] has shown that this problem cannot be approximated within n 1/2-E. A strong lower bound on the minimization version of MAX PB 0- 1 PROGRAMMING has been established by Kann 151. Since the optimal value cannot equal 0 by definition, these problems can trivially be approximated within n
P:Jonsson / hfomationProcessing Letters 68 (1998) 249-253 (where n is the number of variables) since we have assumed the existence of trivial solutions. It is easy to see that if MAX ONES is not approximable within (II then MAX DONES is not approximable within CX.Using the well-known connection between propositional logic and zero/one programming [7], it is also easy to see that MAX PB O-l PROGRAMMING cannot be approximated within a. Similarly, nonapproximability of MIN ONES carryover to MIN DONES and MIN PB 0- 1 PROGRAMMING.
3. Minimization
problems
The first problem that we will consider is MIN ONES. Define the NP-complete decision problem 3sAT as usual [2]: Given a 3CNF formula F, is there a satisfying truth assignment for F? Lemma 3.1. Let Mini denote the optimal valuefunction for MIN ONES. Let c > 4 be a fixed integer. There exists a polynomial-time
transformation
(1) IT,(F)1 = IFY; (2) Min(T,(F)) < 1F14 tf F is satisjiable; (3) Min(T,(F))
> \FIC - 8(F13 - IFJ otherwise.
Proof. Let F be an arbitrary 3CNF formula over the propositions P = (PI, . . . , pk}. We begin by describing T,(F). There exists less than 8k3 distinct clauses of length 3 over the propositional symbols in P. We can thus assume that F = Cl A . . . A C,Ywhere s < 8k3. Let w be a fresh proposition and Q = (91, . . . , qk~_k-s_l }, R = {rl, . . . , rs} be sets of fresh propositions. Consider an arbitrary clause Ci = (1: v 1; v I?), 1 < i < s. Since the clauses in F contain at most three literals, we do not require that 1:) 1; and l! are distinct. Let Ki = (Yi eS (If
V
12)) A (13 V
ri V W),
where + denotes logical equivalence, a+/3
iff
i.e.,
(~~Vfi)A((21v~#J).
Define Tc( F) as follows: T,(F)=
A 1
Ki A
A ('Wvqj). l
Notethat]T,(F)\=s+k+l+k’-k-s-l=k” and that Tc( F) is not a 3CNF formula since it contains the + symbol. However, it is easy to show that Tc( F) can be written as a 3CNF formula without introducing any extra variables. This boils down to showing that the expression (ri ($ (1; v 1;)) can be transformed to an equivalent 3CNF formula without extra variables. This is a simple exercise. As a final observation, T,,(F) can be computed in polynomial time since c is fixed. We continue by showing that T, satisfies the given requirements. Let F’ = Tc( F). We begin by showing that Min(F’) < k4 if F is satisfiable. 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’ for F’ as follows: M’(p)
= M(p),
M’(w)
= F;
M’(q)
= F,
Tc on
3CNF formulae satisfying the following: for an arbitrary 3CNF formula F,
251
F M’(ri)
=
i T
p E P;
4 E Q; if M(l!) = M(lf) l,
= F,
In this case, Mint (F’) < k + s < k + 8k3 6 k4 for k sufficiently large. Assume to the contrary that F is not satisfiable, that is, no truth assignment can satisfy all clauses in F simultaneously. Note that a clause of the type (1; v rt v w), 1 < i < s, is logically equivalent to the clause (1; v 1; v l,? v w). This follows from the fact that, by the construction of F’, ri is logically equivalent to (I! v Zf).Consequently, the clauses (1” v riv w), 1 < i < s, in F’ cannot be simultaneously satisfied unless w is assigned T. But then qj, 1 < j < kc - k s - 1, must also be assigned T in order to satisfy F’. Hence, a model M’ for F’ which assigns T to as few propositions as possible has the following appearance: M’(p)
= F,
M’(w)
= T;
M’(q) = T, M’(r)
= F,
p E P;
q E Q; r E R.
Thus, Mint (F’) 3 I+ k’ - k - s - 1 > kc - k - 8k3 since s 6 8k3. q
t? Jonsson /Information Processing Letters 68 (1998) 249-253
252
n’-’
for any
E
MIN ONES is not approximable within 0 where n is the number of variables.
>
Proof. Arbitrarily choose an E > 0 and let the natural number c > 6 satisfy 1 - 6/c 3 E. Define a(n) = $1’ and b(n) = n ’ - I/‘. Given arbitrary natural numbers i, j, we can decide (in polynomial time) whether a(i) < j since a(i) < j iff i5 < j’. By Lemma 3.1, there exists a polynomial-time reduction Tc from 3 SAT to MIN ONES with the following properties: for an arbitrary instance I of 3s~~ containing k variables, (1) T,(Z) contains kc variables. (2) Minl(T,(Z)) < k5 = IT,(Z)J’/’ = a(lT,(Z)I) if Z is satisfiable. (3) Minl(T,(Z)) > kc-* = IT,(Z)l’-‘j“ = b(JT,(Z)I) if Z is not satisfiable. Hence, MIN ONES is not approximable within a(n) = b(n)/a(n)
= rzle6/’
by Lemma 2.2. Since 1-6/c 3 E and q chosen, the theorem follows.
E
was arbitrarily
Corollary 3.3. MIN DONES is not approximable within n’-’ for any F > 0, where n is the number of variables.
4. Maximization
problems
To show the nonapproximability of MAX ONES, similar to the construction in Lemma 3.1.
For arbitrary clauses Ci = (1; v 1: v l:), 1 < i < s, let Ki = (ri + (1: V 1’)) A (1” V ri V Define V,(F) UC(F)=
A
7~).
as follows: Ki A
I
A
(w v ‘4j).
l
Showing that U, satisfies the stated requirements is similar to the proof of Lemma 3.1. Proving the nonapproximability result is analogous to the proof of Theorem 3.2. q Corollary 4.2. MAX DONES and MAX PB 0- 1 PROGRAMMING are not approximable within n'-' for any E > 0, where n is the number of variables. Under the additional assumption that NP # ZPP, it is possible to show that MAX ONES cannot be approximated within n ’ --E even if restricted to 2CNF formulae. H&tad [3] has shown that the CLIQUE problem cannot be approximated within n 1--E (where n denotes the number of nodes in the graph), unless NP = ZPP. The reduction of CLIQUE to MAX ONES is straightforward; for each missing edge in the given graph G (say, between node x and node y), add a clause of the form (lx v -y). Obviously, the optimal value of the resulting MAX ONES instance equals the size of the largest clique in G.
we use a construction
Theorem 4.1. MAX ONES is notapproximable within n’-” for any E > 0 where n is the number of variables. Proof (Sketch). Let Max1 denote the optimal value function for MAX ONES. Let c > 4 be a fixed integer. We give a polynomial-time transformation U, on 3CNF formulae satisfying the following: for an arbitrary 3CNF formula F, (1) IUc(WI = IW; (2) Max] (U,(F)) < )F14 if F is not satisfiable; (3) Max1 (U,(F)) > IFIG - 81Fj3 - IFI otherwise. Arbitrarily choose a 3CNF formula over the propositions P = (pi,. . . , pk}. As noted earlier, we can safely assume that F = Cl A . . . A C, where s < 8k3. Let w be a fresh proposition and Q = {ql, . . . , qkc-k-s-l), R = {rl, . . . , rs} be sets of fresh propositions.
Acknowledgements Thanks to Viggo Kann, Thomas Drakengren the anonymous referee for helpful discussions comments.
and and
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F! Jonsson /Information Processing Letters 68 (1998) 249-253
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