=8

A 7 -Approximation Algorithm for MAX 3SAT? y Howard Karloff Uri Zwick

Abstract that the performance ratio of his algorithm is actually 2/3. Many years passed before Yannakakis [37] obtained a 3/4- We describe a randomized approximation algorithm which approximation algorithm for the problem. Goemans and takes an instance of MAX 3SAT as input. If the instanceÐa Williamson [17] then obtained a different and somewhat collection of clauses each of length at most threeÐis satis®- simpler 3/4-approximation algorithm. Their algorithm is

able, then the expected weight of the assignment found is at based on a linear programming relaxation of the problem.

least of optimal. We provide strong evidence (but not a In a major breakthrough, Goemans and Williamson [18] proof) that the algorithm performs equally well on arbitrary obtained a 0.878-approximation algorithm for MAX CUT MAX 3SAT instances. and MAX 2SAT, the version of MAX SAT in which each Our algorithm uses semide®nite programming and may be clause is of size at most two. Goemans and Williamson seen as a sequel to the MAX CUT algorithm of Goemans and used semide®nite relaxations of these problems. Feige and Williamson and the MAX 2SAT algorithm of Feige and Goe- Goemans [15] then obtained a 0.931-approximation algo- mans. Though the algorithm itself is fairly simple, its anal- rithm for MAX 2SAT. Using the MAX 2SAT algorithms of ysis is quite complicated as it involves the computation of Goemans and Williamson or of Feige and Goemans, slight volumes of spherical tetrahedra. improvements in the performance ratio for general MAX

SAT can be made. Goemans and Williamson [18] obtained

NP HÊastad has recently shown that, assuming P , no a 0.758 bound for MAX SAT. Asano [4] (following [5]) polynomial-time algorithm for MAX 3SAT can achieve a

slightly improved this bound to 0.770.

performance ratio exceeding , even when restricted to satis®able instances of the problem. Our algorithm is there- While semide®nite relaxations yield a huge improvement fore optimal in this sense. for MAX 2SAT (from 0.75 to 0.931), they give, so far, only a minor improvement for MAX SAT (from 0.75 to 0.770). We also describe a method of obtaining direct semide®nite The reason for this seems to be that the semide®nite relax- relaxations of any constraint satisfaction problem of the

ations used till now do not directly handle clauses of length

F F form MAX CSP , where is a ®nite family of Boolean three or more. functions. Our relaxations are the strongest possible within a natural class of semide®nite relaxations. An attempt to squeeze more from the MAX 2SAT algo- rithm of Feige and Goemans [15] was made by Trevisan, Sorkin, Sudan and Williamson [34]. They used an op- timal gadget, a concept formalized by Bellare, Goldre- 1 Introduction ich and Sudan [7], to reduce MAX 3SAT (the problem in which each clause has length at most three) to MAX MAX SAT is a central problem in theoretical computer sci- 2SAT, thereby obtaining a 0.801-approximation algorithm ence. As it is NP-hard and, in fact, MAX-SNP complete (Pa- for MAX 3SAT. Trevisan [33] recently obtained a 0.826- padimitriou and Yannakakis [30]), much attention has been approximation algorithm for satis®able instances of MAX devoted to approximating it. The ®rst approximation algo- 3SAT, and a 0.8-approximation algorithm for satis®able in- rithm for MAX SAT was proposed by Johnson [23]. John- stances of MAX SAT. son showed that the performance ratio of his algorithm is at

In another major breakthrough, following a long line of re-

least . Chen, Friesen and Zheng [12] recently showed search by many authors [16, 3, 2, 8, 7], HÊastad [20] recently

College of Computing, Georgia Institute of Technology, Atlanta, Geor- showed that MAX E3SAT, the version of the MAX SAT gia 30332-0280, U.S.A. Research supported in part by NSF grant CCR- problem in which each clause is of length exactly three, can-

9319106. E-mail address: [email protected]. y not be approximated in polynomial time to within a ratio

Department of Computer Science, Tel-Aviv University, Tel-Aviv

NP 69978, Israel. This work was done while this author was visiting ICSI and greater than 7/8, unless P . HÊastad shows, in fact, UC Berkeley. E-mail address: [email protected]. that no polynomial-time algorithm can have a performance

1 guarantee of more than 7/8 even when restricted to just sat-

is®able instances of the problem. HÊastad's result is easily X

Maximize w z ij k

seen to be tight: a random assignment satis®es, on average, ij k

ijk to

7/8 of the total weight of a MAX E3SAT instance. sub ject

v v
v v

In this paper we present a new approximation algorithm for

0 i j

k

z i j k n ij k

MAX 3SAT. The algorithm takes as input an instance of

v v
v v

0 j i

k

z i j k n ij k

MAX 3SAT and runs in time bounded by a polynomial in

v v
v v

0 i j

the length of this instance. If the instance is satis®able, the k

i j k n z ij k

expected weight of the assignment the algorithm returns is at

z i j k n

ij k

least of the weight of any assignment to the instance. We

n

v S i n

conjecture (but have only strong evidence, not a proof) that i

v v i n

i i the same performance is achieved on arbitrary MAX 3SAT n instances. Our algorithm can possibly be derandomized us- ing the techniques of Mahajan and Ramesh [28]. The novelty of our algorithm is that it uses a direct semidef- Figure 1. A direct semide®nite relaxation of a inite relaxation of MAX 3SAT. The algorithm itself is quite MAX 3SAT instance. simple. The analysis relies, however, on two inequali- two or one (or zero). ties involving the volume function of spherical tetrahedra (Lemma 4.4 and Conjecture 4.5 below). Proving these two inequalities seems, at least for now, extremely complicated. 3 The New Approximation Algorithm for We provide a computer-assisted proof of the ®rst inequality. MAX 3SAT For the second inequality we are able to present only strong numerical evidence. We are currently working on a simpler A direct semide®nite relaxation of a generic MAX 3SAT in- proof of the ®rst inequality and a complete, and hopefully stance is presented in Figure 1. In this relaxation, we attach

simple, proof of the second inequality. The ®rst inequality

v i n

a unit vector i to each Boolean variable, , and a

alone implies that the performance ratio of the algorithm for

z v scalar ij k to each clause. We also have a special vector

satis®able instances of MAX 3SAT is at least 7/8.

v v v

n

that corresponds to FALSE. The vectors are

n n R We also describe a method of obtaining direct semide®- vectors on the Euclidean unit sphere S in .

nite relaxations of any constraint satisfaction problem of the z

The scalar ij k is meant to get the value 1 if the clause is

F F F form MAX CSP or MIN CSP , where is a ®nite satis®ed and 0 if it is not satis®ed. De®ne

family of Boolean functions. Such problems were studied

relax v v v v

i j k

by Khanna, Sudan and Williamson [27] and by Khanna, Su-

dan and Trevisan [26]. Our relaxations are the strongest pos-

v v
v v v v
v v

0 i j 0 j i

k k

sible within a natural class of semide®nite relaxations. This

min

v v
v v

0 i j

class includes almost all the semide®nite relaxations pro- k

posed to date for these problems. z

The constraints on ij k in the relaxation are equivalent to the

We hope that the results presented here pave the way for sim-

z relax v v v v

i j k constraint ij k . ilar improvements for MAX SAT. It is fairly easy to see that program presented in Figure 1 is

equivalent to a semide®nite program. All we have to do is

v x

2 MAX 3SAT v

j ij

replace each inner product i by a scalar , add the

x x ij

constraints ii , and require that the matrix be pos-

v v

ni i

w

n itive semide®nite. The constraint is equivalent

An instance of MAX 3SAT in variables is an array ij k

v v z

i ni ij k

i j k n

of nonnegative weights, where . A valid to . (The 's can be assumed nonnega-

x x x

x tive but need not satisfy any semide®niteness constraints.)

n

assignment is a 0-1 vector such that

n

x f g

x x n hi j k i

x Why is it a relaxation of MAX 3SAT? Let

ni i

and , for . A clause is

v x

i i

x x x x x

x be a valid assignment. Let if

j k i j

satis®ed by iff i , i.e., if at least one of ,

v x v

i i

x and if . Let .

or k is assigned the value 1. The weight of a valid assign-

relax v v v v

i j k

ment is the sum of the weights of all the satis®ed clauses, It is easy to check (see Table 1) that

P

x x x

i j k

w x x x eight x

w The program described in Figure 1 is therefore

i j k

i.e., ij k The op- ijk

timal solution to the MAX 3SAT instance is an assignment a relaxation of MAX 3SAT.

x i j k

of maximum weight. Note that we always require . It is worthwhile noting that if and ,

i j k i This means that some of the clauses are effectively of length i.e., the clause h is in fact a clause of length one, then

2

v v v u u u

i j i j k

k always rational. Suppose that the performance ratio of the

0 0 0 algorithm when the semide®nite program is solved exactly

I m

1 1 1 is . Let be an instance of the problem of size . Let

I

1 1 1 opt be the value of an optimal solution to the semidef-

I

2 1 1 inite relaxation of the instance I . If for every instance , the

1 1 1 value of the approximate solution found for the relaxation is

1 2 1

opt I

at least , then the performance ratio of the al-

1 1 2

gorithm that uses approximate solutions is at least .

1 1 1

We can take to be an arbitrarily small constant, or even

m

W W

, where is the sum of the weights in the m

instance I , and is the number of clauses in the instance,

relax v v v v

i j k Table 1. The components of

and still have a polynomial-time algorithm.

v v v f g

j k

as a function of i , where

v There is, however, a simple way of getting a performance .

ratio of at least even when the semide®nite relaxation is

I

i j k i

the relaxation of the clause h simpli®es to the expres- not solved exactly. Let be an instance of MAX 3SAT in

v
v

0

k

x x x

relax v v v v i

n

k

sion and if but the variables . We may assume, w.l.o.g., that

j k hi j k i

j k and , i.e., the clause is of length appears both positively and negatively in the instance. Let

I I x

relax v v v v

j k

two, the relaxation simpli®es to be the instance obtained from by assigning the value

n o

v
v v
v v
v

0 j 0 j

I I x k k

0. Let be the instance obtained from by assigning the

min

For clauses of length

I I value 1. Solve both instances and using the approxima- one we get, therefore, exactly the MAX CUT relaxation of tion algorithm that achieves a performance ratio of at least

Goemans and Williamson [18]. For clauses of length one

W , with , compare the values of the two and two we get, almost exactly, the MAX 2SAT relaxation assignments obtained and return the better one. We claim used by Feige and Goemans [15].

that the performance ratio of this algorithm is at least . To

relax v v v v

i j k The expression includes 1 as one of its see this, assume, w.l.o.g., that there is an optimal assignment

terms. This is not needed for obtaining a relaxation. It is

I x A

of in which is assigned the value . Let be the total

used, however, in showing that the relaxation obtained is a

I x

weight of the clauses of in which appears negatively.

good one. Where does this relaxation comes from? This is I

Then the value of the assignment found by solving is at

explained in Section 5.

opt I A opt I

least A where

opt I W The semide®nite program described above can be solved in we have used the fact that A and . polynomial time. To be more precise, an (almost) feasible point whose cost is within an additive error of of optimal 4 The Performance Ratio of the Algorithm

can be found in time polynomial in the size of the problem

and log . (See Alizadeh [1], GrÈotschel et al. [19], Nesterov and Nemirovskii [29], Pataki [31] and Vaidya [35], and the All that remains is to analyze the performance ratio of the survey paper of Vandenberghe and Boyd [36].) It follows algorithm. We do this in two stages. We ®rst analyze the performance of the algorithm for clauses of size one of two. easily that if the value of the optimal solution is opt , then a

The analysis in this case is identical to the analysis of Goe-

v v n feasible solution, i.e., a sequence of unit vectors

mans and Williamson [18]. We describe the analysis here z and a set of scalars ij k , that satisfy all the constraints and for

P as a ªwarm upº for the much more complicated analysis for

w z opt ij k

which ij k can also be found in time ij k

clauses of length three.

polynomial in the size of the problem and log . We need to

v v n

ªroundº the vectors to truth values. We use the

Clauses of Length and

simple randomized rounding procedure introduced by Goe-

v v n

mans and Williamson [18] to round the vectors

v v

to truth values. We pick a random hyperplane that passes x

i

A clause i is relaxed into . The probability that

x

v v

through the origin. The Boolean variable i is assigned the i

a random hyperplane separates and is , where is

v

v v cos v v

value 1 if and only if the random hyperplane separates i

i i

the angle between the two vectors and , . v

from . The performance ratio of the algorithm for clauses of size

min

 

one is therefore at least

cos

In the next section we analyze the performance ratio of the

new approximation algorithm assuming that the semide®-

x x

nite relaxation given in Figure 1 can be solved exactly. Un- Consider now a clause of length two, e.g., . The

x x relax v v v

fortunately, the semide®nite relaxations cannot always be clause is relaxed into

v
v v
v v
v

0 1 0 2 1 2

solved exactly, at least because the optimal solution is not (We ignore here the minimum with 1

3

relax v v v v v v v r S

in the de®nition of .) Given three vectors We may assume, therefore, that , the

v v v R

and , what is the probability that the rounded assign- unit sphere in .

x x

v v v S

ment satis®es the clause ? This is exactly the prob- v

Let . The spherical tetrahedron

v

v v v v tetrav v v

ability that a random hyperplane separates at least one of tetra

is de®ned as follows:

P P

v v prob v v v

and from , which equals , where

v v j v f

i i i i i

i i

prob v v v

k

g

S A spherical tetrahedron is said to be nondegen-

v v v v

erate if are linearly independent. In this

v v v

k

lie on the same

v v v v

Pr

case, the vectors are said to be the vertices

side of a random hyperplane

tetrav v v v tetrav v v v

of . If is a non-

The performance ratio of the algorithm for clauses of length degenerate spherical tetrahedron, then its polar tetrahe-

0

prob v v v

tetra v v v v

0 1 2

dron, denoted by , is de®ned to be

n

min

v v v 2S

two is at least

0 1 2

relax v v v

0 1 2

tetrau u u u u u u u S

, where satisfy

prob v v v

What is ? There is a simple way

u v i u v

i i j

i , for , and , for

prob v v v

of working out this probability:

i j i j v v v v

, . As are linearly inde-

prob v jv prob v jv prob v jv

, where

u u u u

pendent, this determines uniquely. It is easy

prob v jv probv v

j i j i is the probability that a ran-

to see that the polar tetrahedron can be de®ned alternatively

v v j

dom hyperplane separates i and . We saw already that

0

tetra v v v v f r S j v r g

i

as Thus,

ij

prob v jv arccosv v

j ij i j

i , where is the angle

prob v v v v

is simply twice the probability that a

v v

j

between the vectors i and . Thus

S

random unit vector r falls into the polar tetrahedron

0

tetra v v v v

. This probability is proportional to the

volume of the tetrahedron. As the volume of S is (see

min

Berger [9], p. 261), we get

cos cos cos

 

ij

0

prob v v v v volume tetra v v v v

min

While computing areas of spherical triangles in S is a rela-

 

cos cos cos

ij

tively simple matter, Girard's formula (see [10, p. 278]) stat-

and it is easy to see, therefore, that . ing that the area of a spherical triangle with angles and

S

(on ) is , computing volumes of spherical

Clauses of Length tetrahedra is a much more complicated matter. This subject was investigated in the previous century by SchlÈa¯i [32] and

in the present one by Coxeter [13], BÈohm and Hertel [11]

x x x

Consider now a clause of length 3, say . The

performance ratio of the algorithm for clauses of length 3 and Hsiang [21]. Unfortunately, no closed-form formula for

prob v v v v

1 2 3

0 the volume is known. As the volume function is related to

n

min

v v v v 2S

is at least

0 1 2 3

relax v v v v

1 2 3 0 the dilogarithm function (see Kellerhals [25]), possibly no The performance ratio of the algorithm for satis®ed clauses

0 closed-form formula exists.

min prob v v v v

of length 3 is at least

n A spherical tetrahedron can be characterized by either

v v v v S

where the minimum is over such that

arccosv v

i j

the six angles ij between the

relax v v v v

. The simple way of evaluating

unit vectors that correspond to its verticesÐnote that ij

probv v v

cannot be used, unfortunately, for eval-

v v j

is also the distance between i and on the sphereÐ

probv v v v

uating . We have to use, therefore, a differentway that relies more heavily on spherical geometry. or by its six dihedral angles. The dihedral angle along an edge of the tetrahedron is the angle between the two A random hyperplane that passes through the origin is con- ªfacesº that meet at the edge; see Hsiang [22] for a for-

veniently chosen by choosing its normal vector r uniformly

Vol

n mal de®nition. Let be the at random in S . Note that

volume of a spherical tetrahedron with dihedral angles

prob v v v v

. While the volume itself may

Pr sgnv r sgnv r sgnv r sgnv r

be nonelementary, its partial derivatives have a surprisingly

Pr v r v r v r v r

simple form (see Appendix A). We need the following re-

lation between the lengths of the sides of a spherical tetra-

v r v r v

As we are only interested in the inner products , , hedron and the dihedral angles of the corresponding polar

r v r r

and , we are only interested in the projection of into tetrahedron.

v v v v

the 4-space spanned by and . It is not dif®cult to

0

r r

see that , the normalized projection of into this 4-space, Lemma 4.1 If are the dihedral angles and

is uniformly distributed on the unit sphere of this 4-space. are the side lengths of a spherical tetrahedron

4

0 0 0 0

T , and and are the dihedral angles then

0 T

and side lengths of the spherical tetrahedron T polar to ,

Vol

0 0

2 2

then and . (Other equalities

cos cos cos cos

0

such as follow by symmetry.)

This is a classical result in spherical geometry. For a proof, The evidence that we have in support of Conjecture 4.5 ap-

see Hsiang [22, eqn. (41)]. We now have pears in Appendix C. 2

0

Theorem 4.2

5 Semide®nite Relaxations of Constraint Sat-

Conjecture 4.3 isfaction Problems

0 0

It is easy to see that . Theupperboundon is

k

f g f g

Let f be a Boolean function. For con-

v v v v

obtained by considering the case in which and

venience we use differentencodings of the truth values at the

are all perpendicular (this is possible as we are in R ). It is

input and the output of f . At the input, we use to repre-

relax v v v v

easy to verify that in that case and

sent FALSE and 1 to represent TRUE. At the output, we use

prob v v v v

that .

0 to represent FALSE and 1 to represent TRUE.

v

Proof of Theorem 4.2: We have to show that for every

f

An instance of the problem MAX CSP is a collec-

m

v v v S relax v v v v prob v

with ,

f f y y w g

ik i

tion i of weighted constraints,

i

v v v prob v v v v

, or equivalently

y f x x x x g

n n

where ij . The goal

relax v v v v v v

Note that implies that

is to ®nd an assignment of values to the variables

v v v v v v v v v v

, ,

x x n

that maximizes the total weight of the clauses

0

arccosv v

j ij

. Let i . Let be the dihedral angles

ij

f that evaluate to 1. The MAX CSP problem is therefore

of the polar spherical tetrahedron. Now similar to the MAX SAT problem except that the constraints

f y y

ik

are now of the form i and not of the form

prob v v v v Vol

y y F F

ik

i . The problem MAX CSP , where is 0

Thus, using the relation between the 's (in the primal) and a ®nite collection of Boolean functions, can also be de®ned.

ij ij 's (in the polar) given by Lemma 4.1, we infer that it is There is no precise de®nition of the term ªrelaxation.º Here enough to prove the following: we propose a de®nition of a class of semide®nite relaxations

which we call ªstandard semide®nite relaxations.º

Lemma 4.4 Let be the six dihe- dral angles of a spherical tetrahedron. If De®nition 5.1 (standard semide®nite relaxations)

A standard semide®nite relaxation of an instance I of

cos cos cos cos

f

MAX CSP is a semide®nite program that has the follow-

cos cos cos cos

ing properties:

cos cos cos cos

v 2

1. The variables of the program are: a unit vector i cor-

Vol

then

x z j

responding to each variable i , and a scalar corre- v A sketch of the computer-assisted proof of Lemma 4.4 ap- sponding to each clause, and a unit vector (repre- senting FALSE).

pears in Appendix B. 2

P

m

w z j

2. The objective function to be maximized is j . Beginning of a possible proof of Conjecture 4.3: As j

we have proved Theorem 4.2, we may assume here that

f y y I

ik

3. For every clause i of , theprogramcon-

relax v v v v

. By symmetry, we may assume tains a set of linear inequalities involving inner prod-

v v
v v

0 2 1 3

relax v v v v

w.l.o.g. that

v v

j

ucts i of the vectors that correspond to the vari-

relax v v v v

Simple algebra and the fact that

y y z

ik ables i , and the scalar that corresponds to imply that to prove Conjecture 4.3, it suf®ces to prove the clause. All the constraints of the program are of this

form.

Conjecture 4.5 Let be the six

n

x x f g v

dihedral angles of a spherical tetrahedron. If

n i

4. For every , assign the vec-

x v

tor i , assign the vector ,

cos cos cos cos

z j

and assign j the value 1 if the th clause is satis®ed by

cos cos cos cos

this assignment and the value 0, otherwise. Then this

cos cos cos cos

assignment is a feasible point of the program.

5

k (k +1)

2

f R

Condition (4) insures that the program is indeed a relaxation Note that p olytop e is a polytope in I . We de-

k

f

and that the value of the program is at least the value of the ®ned p olytop e by giving its vertices. Alternatively,

I

f instance . Condition (3) says, in effect, that the program p olytop e , like any other polytope, can be de®ned as the

ªconsidersº the Boolean constraints one at a time. (Note the intersection of a ®nite number of halfspaces. In other words,

k k A

similarity of this to the gadgets of Bellare et al. [7] and Tre- there exists an m matrix and a vec-

k (k +1)

visan et al. [34].) Almost all semide®nite relaxations of

2

b p olytop e f f x R tor such that j

constraint satisfaction problems proposed to date are stan-

A b m b g Ax Let be such a pair with minimal. (If

dard, or can be made standard with only minor modi®cation.

f A p olytop e is full-dimensional, we can take the rows of Exceptions are the relaxations of MAX CUT proposed by

and entries of b to correspond to the facets of the polytope.)

Feige and Goemans (See Section 5 of [15]), that impose con-

A b

We refer to the pair as a set of de®ning hyperplanes k

straints involving up to k vertices, for varying values of .

f of p olytop e . Each de®ning hyperplane is an inequality

The de®nition of standard relaxations may be generalized P

x z x

ij ij

of the form , where ij is a vari- ij

appropriately. able corresponding to the position occupied by the product

OR x x x x

x z

Consider the Boolean functions , x j

i , and is a variable that corresponds to the last position

AND x x x x XOR x x x x

x f x

, , and in pro d .

NAE x x x x x x x

. With our input

p olytop e OR

As an example, the eight facets of are given

and output conventions, in Figure 2. The last four facets are exactly the four con-

x x x x x x x x

straints we used in the relaxation of MAX 3SAT given in

1 2 1 2 1 2 1 2

OR AND

Figure 1. The ®rst four facets give lower bounds, rather than

z

x x x x x x x x

upper bounds, on and they can therefore be ignored. The

1 2 1 3 2 3 1 2

NAE XOR

facets that give lower bounds on z can be eliminated by con- All these functions can be represented as degree-2 polyno- sidering the polytope

mials over the real numbers. Obtaining (standard) semidef-

0

p olytop e OR

inite relaxations for these functions is therefore relatively

x x z 2 p olytop e OR

01 23 3

0

x

straightforward. To each variable i , we attach a unit vector

x x z

z  z

v x x

i j

i . Every product is now replaced by the inner prod-

v v j uct i . To handle linear terms, we introduce another

whose 20 facets are given in Figure 3. The ®rst 16 facets

v x i unit vector, , that represents FALSE. A linear term is

are just the ªtriangle inequalitiesº used in the MAX 2SAT

v v i now replaced by the inner product . By doing so, we algorithm Feige and Goemans [15]. They are in fact facets obtain the semide®nite relaxations of MAX CUT and MAX of the cut polytope (see [6]). 2SAT used by Goemans and Williamson [18]. Are these the

best standard relaxations that we can obtain? We answer this We now de®ne canonical semide®nite relaxations for all in-

f question shortly. stances of MAX CSP . We later show that they are the strongest standard semide®nite relaxations possible.

To obtain a semide®nite relaxation of MAX 3SAT, we need

OR x x x

a semide®nite relaxation of the function . De®nition 5.2 (Canonical semide®nite relaxations) Unfortunately, this function cannot be represented as a

A canonical semide®nite relaxation of an instance I of

degree-2 polynomial and the simple approach described

f MAX CSP is a semide®nite program obtained in the fol- above cannot be used.

lowing way.

k

f g f g Let f be any Boolean function.

We now describe a way of obtaining the strongest standard v

1. The variables of the program are: a unit vector i cor-

f

semide®nite relaxations for instances of MAX CSP .

x z j

responding to each variable i , a scalar correspond-

k

x x x f g k

Given a vector , we let v

ing to each clause, and a unit vector representing

k (k +1)

2

pro dx x x x x x x f g

k k

, FALSE.

x P

where . De®ne

m

w z j

2. The objective function to be maximized is j .

j

k

p olytop e f convfpro dx f x j x f g g

f y y I k

3. For every clause of , and for every

P

x z

k k ij

de®ning hyperplane ij of

ij

x f x

where pro d denotes a vector of length

f

p olytop e , from a ®xed family of de®ning hyper-

x pro dx obtained by appending f to , and

planes, the program contains the inequality

X

X X

u u z

g v j conv fv v g f

ij i j

i i i i

ij

i i

6

size of the instance.

x x x z 

02 12

01 We now show that canonical semide®nite relaxations are the

x x x z 

03 13

01 strongest standard semide®nite relaxations possible. Given

x x x z 

02 03 23

opt I

an instance I , we let be the value of an optimal so-

x x x z 

12 13 23

P opt P

lution of I . Given a semide®nite program , we let

z 

be the value of an optimal solution of P .

x x x x z 

01 02 13 23

x x x x z 

01 12 03 23

f P

Theorem 5.4 Let I be an instance of MAX CSP . Let

x x x x z 

02 12 03 13 Q be a canonical semide®nite relaxation of I and let be

any standard semide®nite relaxation of I . Then, any feasi-

P Q

p olytop e OR

Figure 2. The facets of . ble point of is also a feasible point of . In particular,

I opt P opt Q

opt .

x x x  p v v z z

01 02 12

n m

Proof: Let be a feasible point

x x x 

01 02 12 p

of P . Suppose, for the sake of contradiction, that the point

x x x 

01 02 12

p Q

is not a feasible point of Q, so violates an inequality of .

x x x 

02 12

01 Assume w.l.o.g. that this inequality is one of the inequali-

x x x 

01 03 13

f x x k

ties attached to the Boolean constraint . Let

x x x 

01 03 13

0

p v v z p k

be the restriction of to the vectors

x x x 

03 13

01 and the scalar that appear in the inequalities attached to this

x x x 

01 03 13

00

p x x z v v v

k k

constraint. Let

x x x 

02 03 23

p P v v v z

k k

. As is a feasible point of , we infer

x x x 

02 03 23

00 00

p p olytop e f p

x x x 

that . Thus is a convex combination

02 03 23

00

p olytop e f p

x x x 

03 23

02 of the vertices of . On the other hand, vio-

x x x 

Q

13 23

12 lates at least one of the constraints of . Thus at least one

x x x 

12 13 23

f Q

of the vertices of p olytop e violates this constraint of

x x x 

13 23

12 and this contradicts condition 4 of the de®nition of standard

x x x 

12 13 23

semide®nite relaxations. 2

z 

x x x x z 

p olytop e XOR p olytop e OR

01 02 13 23

By computing and , we

x x x x z 

12 03 23

01 can infer that the semide®nite relaxation of MAX CUT

x x x x z 

12 03 13 02 given by Goemans and Williams [18] is a canonical semidef- inite relaxation of MAX CUT. Their relaxation of MAX

2SAT, however, is not canonical, as it does not include the

0

p olytop e OR Figure 3. The facets of . triangle inequalities. Feige and Goemans [15] include these inequalities in their relaxation and obtain a canonical re-

laxation. It is interesting to note that while the triangle in-

u v u

i Here , is the vector that corresponds to the

equalities help for MAX 2SAT, it seems that they are not re-

y y x u v y x

i j i j i j literal i (if , then , if , then

quired for getting an optimal 7/8 approximation algorithm

u v y u v u v

j i i i i , if , then , and if

for MAX 3SAT.

y z

i , and is the scalar corresponding to the clause.

p olytop e MAJ All the constraints of the program are of this form. The facets of can apparently

be used to obtain a 2/3-approximation algorithm for MAX

MAJ The simple proof of the following lemma is omitted. CSP . For this and some other results, see [38].

Lemma 5.3 A canonical semide®nite relaxation of an in- 6 Concluding Remarks

f stance I of MAX CSP is a standard semide®nite relax- ation.

While we described a way of getting the strongest semide®-

f k

The number of vertices of p olytop e is exponential in , nite relaxationsÐat least in some natural senseÐfor all con-

f

the number of variables of f . The number of facets, or straint satisfaction problems of the form MAX CSP , we

f de®ning hyperplanes, of p olytop e may be even larger. do not automatically get good approximation algorithms for

However, for every ®xed constraint satisfaction problem all of them. Rounding the optimal solutions of the semidef-

f k MAX CSP , is ®xed, and therefore the size of a canon- inite programs using a random hyperplane does not work

ical semide®nite relaxation of an instance I is linear in the well for all problems.

7 7 Acknowledgment [15] U. Feige and M. Goemans. Approximating the value of two- prover proof systems, with applications to MAX-2SAT and MAX-DICUT. In Proceedings of the 3nd Israel Symposium We would like to thank Wu-Yi Hsiang and Michael Loss for on Theory and Computing Systems, Tel Aviv, Israel, pages many helpful discussions on spherical tetrahedra. We would 182±189, 1995. also like to thank UC Berkeley and ICSI for their hospitality. [16] U. Feige, S. Goldwasser, L. LovÂasz, S. Safra, and M. Szegedy. Interactive proofs and the hardness of approximat- ing cliques. Journal of the ACM 43:268±292, 1996. References [17] M. Goemans and D. Williamson. New 3/4-approximation algorithms for the maximum satis®ability problem. SIAM Journal on Discrete Mathematics 7:656±666, 1994. [1] F. Alizadeh. Interior-point methods in semide®nite program- [18] M. Goemans and D. Williamson. Improved approxima- ming with applications to combinatorial optimization. SIAM tion algorithms for maximum cut and satis®ability prob- Journal on Optimization 5:13±51, 1995. lems using semide®nite programming. Journal of the ACM [2] S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. 42:1115±1145, 1995. Proof veri®cation and hardness of approximation problems. [19] M. GrÈotschel, L. LovÂasz, and A. Schrijver. Geometric al- In Proceedings of the 33rd Annual IEEE Symposium on gorithms and combinatorial optimization. Springer Verlag, Foundations of Computer Science, Pittsburgh, Pennsylva- 1988. nia, pages 14±23, 1992. [20] J. HÊastad. Some optimal inapproximability results. In Pro- [3] S. Arora and M. Safra. Probabilistic checking of proofs: a ceedings of the 28th Annual ACM Symposium on Theory of new characterization of NP. In Proceedings of the 33rd An- Computing, El Paso, Texas, pages 1±10, 1997. nual IEEE Symposium on Foundations of Computer Science, [21] W. Hsiang. On in®nitesimal symmetrization and volume Pittsburgh, Pennsylvania, pages 2±13, 1992. formula for spherical or hyperbolic tetrahedrons. Quarterly [4] T. Asano. Approximation algorithms for MAX SAT: Yan- Journal of Mathematics (Oxford) 39:463±468, 1988. nakakis vs. Goemans-Williamson. In Proceedings of the 3nd [22] W. Hsiang. On the development of quantitative geometry Israel Symposium on Theory and Computing Systems, Ramat from Pythagoras to Grassmann. In D.-Z. Du and F. Hwang, Gan, Israel, pages 24±37, 1997. editors, Computing in Euclidean Geometry, pages 1±21. [5] T. Asano, T. Ono, and T. Hirata. Approximation algorithms World Scienti®c Publishing Co., 1992. for the maximum satis®ability problem. In Proceedings of [23] D. Johnson. Approximation algorithms for combinatorical the 5th Scandinavian workshop on algorithm theory, pages problems. Journal of Computer and System Sciences, 9:256± 100±111, 1996. 278, 1974. [6] F. Barahona and A. Mahjoub. On the cut polytope. Mathe- [24] H. Karloff. How good is the Goemans-Williamson MAX matical Programming 36:157±173, 1986. CUT algorithm? In Proceedings of the 28th Annual ACM [7] M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCPs Symposium on Theory of Computing, Philadelphia, Pennsyl- and non-approximabilityÐtowards tight results. In Proceed- vania, pages 427±434, 1996. ings of the 36rd Annual IEEE Symposium on Foundations of [25] R. Kellerhals. The dilogarithm and volumes of hyperpolic Computer Science, Milwaukee, Wisconsin, pages 422±431, polytopes. In L. Lewin, editor, Structural Properties of Poly- 1995. Full version available as E-CCC Technical Report logarithms, Mathematical Surveys and Monographs, Vol- number TR95-024. ume 37, pages 301±336. American Mathematical Society, [8] M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Ef®- 1991. [26] S. Khanna, M. Sudan, and L. Trevisan. Constraint satisfac- cient probabilistically checkable proofs and applications to tion: the approximability of minimization problems. In Pro- approximation. In Proceedings of the 25rd Annual ACM ceedings of the 12th Annual IEEE Conference on Computa- Symposium on Theory of Computing, San Diego, California, tional Complexity, Ulm, Germany, pages 11±20, 1997. Full pages 294±304, 1993. See errata in STOC '94. version available as E-CCC Technical Report number TR96- [9] M. Berger. Geometry I. Springer-Verlag, 1987. 064. [10] M. Berger. Geometry II. Springer-Verlag, 1987. [27] S. Khanna, M. Sudan, and D. Williamson. A complete clas- [11] J. BÈohm and H. Hertel. Polyederge- si®cation of the approximability of maximization problems

ometrie in n-dimensionalen rÈaumen konstanter krÈummung. derived from Boolean constraint satisfaction. In Proceedings BirkhÈauser, 1981. of the 28th Annual ACM Symposium on Theory of Comput- [12] J. Chen, D. Friesen, and H. Zheng. Tight bound on John- ing, El Paso, Texas, 1997. Full version available as E-CCC son's algorithm for max-SAT. Proceedings of the 12th An- Technical Report number TR96-062. nual IEEE Conference on Computational Complexity, Ulm, [28] S. Mahajan and H. Ramesh. Derandomizing semide®nite Germany, 1997. programming-based approximation algorithms. In Proceed- [13] H. Coxeter. The functions of SchlÈa¯i and Lobatschef- ings of the 36rd Annual IEEE Symposium on Foundations of sky. Quarterly Journal of of Mathematics (Oxford) 6:13±29, Computer Science, Milwaukee, Wisconsin, pages 162±169, 1935. 1995. [14] H. Coxeter. Non-Euclidean Geometry. The University of [29] Y. Nesterov and A. Nemirovskii. Interior Point Polynomial Toronto Press, 1957. Methods in Convex Programming. SIAM, 1994.

8

Vol

[30] C. Papadimitriou and M. Yannakakis. Optimization, approx- ij

Theorem A.3 (Schla¯iÈ (1858) [32])

imation, and complexity classes. Journal of Computer and ij System Sciences 43:425±440, 1991. [31] G. Pataki. On cone-LP's and semide®nite programming: fa- cial structure, basic solutions and the simplex method. Tech- B Proof of Lemma 4.4 nical report, GSIA Carnegie Mellon University, Pittsburgh, PA, 1994.

R Here we provide a computer-assisted proof of Lemma 4.4.

n

dz

[32] L. SchlÈa¯i. On the multiple integral dx dy , whose

Vol

We have to prove that

p a x b y


h z p

1 1 1 2

limits are 1 ,

subject to

2 2 2

x y


z p

n , and . Quarterly

Journal of Mathematics (Oxford) 2:269±300, 1858. Contin-

cos cos cos cos

ued in Vol. 3 (1860), pp. 54±68 and pp. 97-108.

cos cos cos cos

[33] L. Trevisan. Approximating satis®able satis®ability prob-

cos cos cos lems. In Proceedings of the 5th European Symposium on Al- cos

gorithms, Graz, Austria, 1997.

[34] L. Trevisan, G. Sorkin, M. Sudan, and D. Williamson. Gad-

and subject to the condition that is a valid gets, approximation, and linear programming (extended ab- sequence of dihedral angles. It is easy to verify that

stract). In Proceedings of the 37rd Annual IEEE Symposium

on Foundations of Computer Science, Burlington, Vermont, is a valid sequence if and only if the following

pages 617±626, 1996. matrix is positive semide®nite:

[35] P. Vaidya. A new algorithm for minimizing convex func-

cos cos cos

tions over convex sets (extended abstract). In Proceed-

cos cos cos

ings of the 20th Annual IEEE Symposium on Foundations of

cos cos cos

Computer Science, Research Triangle Park, North Carolina,

cos cos cos

pages 338±343, 1989.

[36] L. Vandenberghe and S. Boyd. Semide®nite programming.

SIAM Review 38:49±95, 1996. If the minimal eigenvalue of this matrix is 0, then

[37] M. Yannakakis. On the approximation of maximum satis®-

v v v v

the matrix is singular, and the vectors and are ability. Journal of Algorithms 17:475±502, 1994. linearly dependent. They may assumed therefore to lie in

[38] U. Zwick. Approximation algorithms for constraint satis-

S

faction problems involving at most three variables per con- R . Instead of computing volumes in , we then have to

straint. Submitted for publication. compute areas in S . In this case, which is much easier

ol

than the general case, it is not dif®cult to show that V

, where is the performance ra-

A Volumes of Spherical Tetrahedra tio of the Goemans-Williamson MAX CUT algorithm. We

will henceforth assume that the matrix is positive-

v v v v S

Let be the vertices of a nondegenerate de®nite. By Theorem A.3, the volume is an increasing func-

arccosv v

i j

spherical tetrahedron. Let ij be the angle tion of the dihedral angles. Simple perturbation arguments

v v ij j

between i and , or, equivalently, the length of the edge , allow us, w.l.o.g., to reduce to the problem of proving that

Vol

as measured on the sphere. The dihedral angle ij is the an- if

gle between the two ªfacesº that meet at the edge ij .

cos cos cos cos

De®nition A.1 (high-dimensional inner prod-

cos cos cos cos

a a b

k

uct; see Hsiang [22, eqn. (1)]) Let and

cos cos cos cos

b

k be two sequences of vectors all of the same length. De-

h a a b b i detfa b g

k k i j

®ne , and

Symmetry arguments allow us to assume w.l.o.g. that

ja a j h a a a a i

k k k

.

v v v v S

Lemma A.2 Let be the vertices of a We de®ne the following sequence of three points:

nondegenerate spherical tetrahedron with dihedral angles

i j k

. Let be a permutation of .

Then

h v v v v v v i

i j k i j

cos

ij

jv v v j jv v v j

i j k i j

cos cos cos cos

where

Vol cos cos cos

Although the function , One can show

giving the volume of a spherical tetrahedron as a function that and are valid sequences of dihedral angles and of its six dihedral angles, is complicated, we have that they both satisfy the hypotheses of Lemma 4.4.

9

Vol Vol Vol

We next show that by C Evidence in Support of Conjecture 4.5

de®ning

A tuple satisfying the hypotheses of

0 00 0

0 00

t Volarccoscos tcos cos f

Lemma 4.4 is said to belong to case a. A tuple satisfying

the hypotheses of Conjecture 4.5 is said to belong to case

0 00 for two feasible points and , by showing that for

b. The evidence for Conjecture 4.5 is a sequence of com-

0

t f t f f t t i

prob v v v v

0 1 2 3

i , for and .

i1 i

i

puter runs testing whether The

0

relax v v v v

0 1 2 3

i In fact, it is enough to prove that f for ,

i conclusion of Conjecture 4.5 is equivalent to the statement

2

i

as i is just like any other point on the path from

Vol

01 23

cos cos cos cos

01 12 23 03

to i . (This part of the proof does not rely on a computer.)

f t

We calculate the derivative of i explicitly using Theo-

rem A.3 and Lemma A.2. Some calculus and lots of ugly 1. A systematic search over the space of case- b possibili-

0 ties with each of ®ve variables running from to , the

i computations eventually lead us to f for .

i sixth determined by the other ®ve, with a step size of

Vol Vol

This completes the proof that .

, yielded no ratio less that . (A simple perturba-

Vol Vol

Now let where tion argument shows that w.l.o.g. the minimum value

cos cos

01 23

arccos

It is easy to see that if

cos cos

of the ratio is attained at a point with

cos cos

then is a valid

cos cos

.) The only point in which

sequence of dihedral angles. It is hence enough to prove

was attained was .

Lemma B.1 For every such that 2. A systematic search of the space of 6-tuples

cos cos Vol a b

, . of both case and case using Matlab,

with each ij running from to in steps of , found



arccos cos

Proof: Let . We break no ratio less than and the only point in which a ratio

the proof into the following three cases: of was attained was .

cos cos Case 1 . This one is easy and the 3. A systematic search using Mathematica over all 6-

proof is omitted. tuples and with a step size of found no counterex-



cos cos

Case 2 and . The hard ample.

case. Sketch below. 

 Both the Matlab and Mathematica runs pruned the search

Case 3 or . Easy, based on the fact that

space by considering only tuples satisfying the

Vol prob v v v v

is just , the probabil-

triangle inequality constraints (the ®rst 16 in Figure 2).

v v v v

ity that and lie onthesamesideofarandomhy-

v v v

v Each of the three runs numerically integrated Hsiang's for-

perplane, and the probability that are not sep-

v v

v mula for the volume of a spherical tetrahedron. While Math-

arated is at most the probability that are not sepa-

 

ematica and Matlab have built-in numerical integration, the

rated. We then assume that or . Let us



®rst run, written in C, used 20-point Gaussian quadrature.

assume, w.l.o.g., that . It turns out to be enough



cos cos

to prove that if and , then As the ®rst systematic search suggests that the worst ratio

, which is proved by simple calculus. is obtained only near the point , we per-

We return to Case 2. Here it is suf®cient to prove that formed another systematic search in the neighborhood of

 this point. We enumerated again on ®ve angles, ranging

Vol

has no critical point with . It

from to in steps of . Again, no ra- is enough to prove

tio less than 7/8 was found.



cos

We also tried to ®nd the minimal ratio numerically using

Claim B.2 For every such that

sin Vol

01

2 constr

cos Matlab's constrained minimization function . No

, .

sin

01 counterexample was found.

Note that implies that

. In the statement of the claim, refers to this common value. To prove Claim B.2, we partition the feasible region into small squares, explicitly bound the derivative and, using

Mathematica with 50 digits of precision, show that the

2 derivative is bounded above by .

10