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Optimal Inapproximability Results for MAX-CUT and Other Optimal Inapproximability Results for MAX-CUT and Other
2-Variable CSPs? 2-Variable CSPs?
S. Khot
G. Kindler
Elchanan Mossel
University of Pennsylvania
R. O'Donnell
Follow this and additional works at: https://repository.upenn.edu/statistics_papers
Part of the Computer Sciences Commons, and the Statistics and Probability Commons
Recommended Citation Recommended Citation
Khot, S., Kindler, G., Mossel, E., & O'Donnell, R. (2004). Optimal Inapproximability Results for MAX-CUT and
Other 2-Variable CSPs?.
45th Annual IEEE Symposium on Foundations of Computer Science,
146-154.
http://dx.doi.org/10.1109/FOCS.2004.49
This paper is posted at ScholarlyCommons. https://repository.upenn.edu/statistics_papers/395
For more information, please contact repository@pobox.upenn.edu.
Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs? Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?
Abstract Abstract
In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-
CUT to within a factor of α
GW
+ ∈, for all ∈ > 0; here α
GW
≈ .878567 denotes the approximation ratio
achieved by the Goemans-Williamson algorithm [26]. This implies that if the Unique Games Conjecture of
Khot [37] holds then the Goemans-Williamson approximation algorithm is optimal. Our result indicates
that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT
problem.
Our reduction relies on a theorem we call Majority Is Stablest. This was introduced as a conjecture in the
original version of this paper, and was subsequently conQrmed in [45]. A stronger version of this
conjecture called Plurality Is Stablest is still open, although [45] contains a proof of an asymptotic version
of it.
Our techniques extend to several other two-variable constraint satisfaction problems. In particular,
subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX-2SAT,
MAX-q-CUT, and MAX-2LIN(
q
).
For MAX-2SAT we show approximation hardness up to a factor of roughly .943. This nearly matches the
.940 approximation algorithm of Lewin, Livnat, and Zwick [41]. Furthermore, we show that our .943...
factor is actually tight for a slightly restricted version of MAX-2SAT. For MAX-
q
-CUT we show a hardness
factor which asymptotically (for large q) matches the approximation factor achieved by Frieze and Jerrum
[25], namely 1 − 1/
q
+ 2(ln
q
)/
q
2
.
For MAX-2LIN(
q
) we show hardness of distinguishing between instances which are (1−∈)-satisQable and
those which are not even, roughly, (q
−∈/2
)-satisQable. These parameters almost match those achieved by
the recent algorithm of Charikar, Makarychev, and Makarychev [10]. The hardness result holds even for
instances in which all equations are of the form x
i
− x
j
= c. At a more qualitative level, this result also
implies that 1 − ∈ vs. ∈ hardness for MAX-2LIN(
q
) is equivalent to the Unique Games Conjecture.
Keywords Keywords
approximation theory, communicating sequential processes, computability, computational complexity,
game theory, graph theory, optimisation, CSP, Goemans-Williamson algorithm, MAX-2CSP problem,
MAX-2SAT problem, MAX-CUT, NP-hard, approximation algorithm, games conjecture, majority is stablest
conjecture, nonBoolean domains, optimal inapproximability, additive noise, approximation algorithms,
bipartite graph, Boolean functions, computer science, labeling, mathematics, stability, statistics
Disciplines Disciplines
Computer Sciences | Statistics and Probability
This conference paper is available at ScholarlyCommons: https://repository.upenn.edu/statistics_papers/395
Optimal Inapproximability Results for MAX-CUT
and Other 2-Variable CSPs?
Subhash Khot
∗
College of Computing
Georgia Tech
khot@cc.gatech.edu
Guy Kindler
†
Faculty of Mathematics and Computer Science
Weizmann Institute
gkindler@weizmann.ac.il
Elchanan Mossel
‡
Department of Statistics
U.C. Berkeley
mossel@stat.berkeley.edu
Ryan O’Donnell
∗
Department of Computer Science
Carnegie Mellon University
odonnell@cs.cmu.edu
February 7, 2007
Abstract
In this paper we show a reduction from the Unique Games problem to the problem of approximating
MAX-CUT to within a factor of α
GW
+ , for all > 0; here α
GW
≈ .878567 denotes the approxima-
tion ratio achieved by the Goemans-Williamson algorithm [26]. This implies that if the Unique Games
Conjecture of Khot [37] holds then the Goemans-Williamson approximation algorithm is optimal. Our
result indicates that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the
MAX-CUT problem.
Our reduction relies on a theorem we call Majority Is Stablest. This was introduced as a conjecture
in the original version of this paper, and was subsequently confirmed in [45]. A stronger version of
this conjecture called Plurality Is Stablest is still open, although [45] contains a proof of an asymptotic
version of it.
Our techniques extend to several other two-variable constraint satisfaction problems. In particular,
subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX-2SAT,
MAX-q-CUT, and MAX-2LIN(q).
For MAX-2SAT we show approximation hardness up to a factor of roughly .943. This nearly matches
the .940 approximation algorithm of Lewin, Livnat, and Zwick [41]. Furthermore, we show that our
.943... factor is actually tight for a slightly restricted version of MAX-2SAT. For MAX-q-CUT we show
a hardness factor which asymptotically (for large q) matches the approximation factor achieved by Frieze
and Jerrum [25], namely 1 − 1/q + 2(ln q)/q
2
.
For MAX-2LIN(q) we show hardness of distinguishing between instances which are (1−)-satisfiable
and those which are not even, roughly, (q
−/2
)-satisfiable. These parameters almost match those achieved
by the recent algorithm of Charikar, Makarychev, and Makarychev [10]. The hardness result holds even
for instances in which all equations are of the form x
i
− x
j
= c. At a more qualitative level, this result
also implies that 1 − vs. hardness for MAX-2LIN(q) is equivalent to the Unique Games Conjecture.
∗
Work performed while the author was at the Institute for Advanced Study. This material is based upon work supported by the
National Science Foundation under agreement Nos. DMS-0111298 and CCR-0324906 respectively. Any opinions, findings and
conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the
National Science Foundation.
†
Work performed while the author was at DIMACS and at the Institute for Advanced Study, Princeton, and was partly supported
by CCR grant N CCR-0324906 and N DMS-0111298.
‡
Supported by a Miller fellowship in Computer Science and Statistics, U.C. Berkeley
1 Introduction
The main result in this paper is a bound on the approximability of the MAX-CUT problem which matches
the approximation ratio achieved by the well-known Goemans-Williamson algorithm [26]. The proof of this
hardness result relies on the Unique Games Conjecture of Khot [37]. We also rely critically on a theorem
we call Majority Is Stablest, which was introduced as a conjecture in the original version of this paper. For
the convenience of the reader we will now briefly describe these two tools; formal statements appear in
Sections 3 and 4.
Unique Games Conjecture (roughly): Given a bipartite graph G, a large constant size set of labels [M],
and a permutation of [M] written on each edge, consider the problem of trying to find a labeling of the
vertices of G from [M] so that each edge permutation is ‘satisfied;’ i.e., is consistent with the labeling. The
conjecture is that if M is a large enough constant then it is NP-hard to distinguish instances which are 99%
satisfiable from instances which are 1% satisfiable.
Majority Is Stablest Theorem (roughly): Let f be a boolean function which is equally often 0 or 1.
Suppose the string x is picked uniformly at random and the string y is formed by flipping each bit of x
independently with probability η; we call Pr[f (x) = f(y)] the noise stability of f . The theorem states
that among all f in which each coordinate has o(1) ‘influence,’ the Majority function has the highest noise
stability, up to an additive o(1).
We add in passing that the name Majority Is Stablest is a bit of a misnomer in that almost all balanced
boolean (weighted) threshold functions are equally noise stable (see Theorem 5). We also note that the
Majority Is Stablest theorem has interesting applications outside of this work — to the economic theory
of social choice [34] for example — and has already proven useful for other PCP-based inapproximability
results [14]. In Section 6.3 we mention interesting generalizations of the Majority Is Stablest theorem for
q-ary functions, q > 2, which are relevant for hardness of approximation and are not resolved in full.
Despite the fact that our hardness result for MAX-CUT relies on the unproven Unique Games Conjec-
ture, we feel it is interesting for several reasons. First, in our opinion it is remarkable that the Unique Games
Conjecture should yield a tight hardness of approximation ratio for MAX-CUT, and that indeed the best
factor should be the peculiar number α
GW
. It is intriguing that the precise quantity α
GW
should arise from a
noise stability property of the Majority function, and certainly there was previously little evidence to suggest
that the Goemans-Williamson algorithm might be optimal.
Another reason we believe our result is interesting is related to this last point. Since the Goemans-
Williamson algorithm was published a decade ago there has been no algorithmic progress on approximating
MAX-CUT. Since H
˚
astad’s classic inapproximability paper [32] from two years later there has been no
progress on the hardness of approximating MAX-CUT, except for the creation of a better reduction gad-
get [55]. As one of the most natural and simple problems to have resisted matching approximability bounds,
we feel MAX-CUT deserves further investigation and analysis. In particular, we think that regardless of the
truth of the Unique Games Conjecture, this paper gives interesting insight into the geometric nature of MAX-
CUT. Indeed, insights we have gleaned from studying the MAX-CUT problem in this light have motivated
us to give new positive approximation results for variants of other 2-variable CSPs such as MAX-2SAT; see
Section 9.
Finally, instead of viewing our result as relying on the unproven Unique Games Conjecture, we can
view it as being an investigation into the truth of UGC. Indeed our hardness results for both MAX-CUT
and for two-variable linear equations modulo q provide explicit parameters for which the Unique Games
Conjecture, if true, must hold. (Note that both problems are Unique Games themselves.) Thus our work
gives a target for algorithmic attacks on the Unique Games Conjecture, which if passed will refute it.
2
Indeed, works subsequent to the original version of this paper have provided approximation algorithms
for the Unique Games problem [54, 29, 10] improving on Khot’s original algorithm [37]. In particular,
in [10] Charikar, Makarychev, and Makarychev gave a semidefinite programming-based approximation
algorithm for Unique Games whose approximation factor nearly matches our hardness bound for MAX-
2LIN(q). The current situation is therefore that any improvement in the approximation factors for either
MAX-CUT or for the more general MAX-2LIN(q) will refute the Unique Games Conjecture.
1.1 Overview of the paper
In Section 2 we describe the MAX-CUT problem and discuss its history. We then state the Unique Games
Conjecture in Section 3 and discuss very recent algorithm results for the problem. The Majority Is Stablest
problem is discussed in Section 4, along with its generalization to q-ary domains, q ≥ 2. We discuss the
geometric aspects of MAX-CUT and their connection with Majority Is Stablest result and the Goemans-
Williamson approximation algorithm in Section 5. Our main results are stated in Section 6. Section 7 is
devoted to some technical definitions, preliminaries, and Fourier analytic formulas. In Section 8 we prove
our main theorem on the hardness of approximating MAX-CUT, based on the Unique Games Conjecture.
In Section 9 we investigate the approximability of other binary 2-CSPs, such as MAX-2SAT. In Section 10
we prove some special cases of the Majority Is Stablest theorem that are of independent interest, with proofs
simpler than those in [45]. Finally, Section 11 is devoted to extending our techniques to the q-ary domain; we
prove some results about noise stability in this domain and then prove our Unique Games-hardness results
for MAX-q-CUT and MAX-2LIN(q) and MAX-q-CUT.
2 About MAX-CUT
The MAX-CUT problem is a classic and simple combinatorial optimization problem: Given a graph G, find
the size of the largest cut in G. By a cut we mean a partition of the vertices of G into two sets; the size of the
cut is the number of edges with one vertex on either side of the partition. One can also consider a weighted
version of the problem in which each edge is assigned a nonnegative weight and the goal is to cut as much
weight as possible.
MAX-CUT is NP-complete (indeed, it is one of Karp’s original NP-complete problems [36]) and so
it is of interest to try to find polynomial time approximation algorithms. For maximization problems such
as MAX-CUT we say an algorithm gives an α-approximation if it always returns an answer which is at
least α times the optimal value; we also often relax this definition to allow randomized algorithms which
in expectation give α-approximations. Crescenzi, Silvestri, and Trevisan [11] have shown that the weighted
and unweighted versions of MAX-CUT have equal optimal approximation factors (up to an additive o(1))
and so we pass freely between the two problems in this paper.
The trivial randomized algorithm for MAX-CUT — put each vertex on either side of the partition in-
dependently with equal probability — is a 1/2-approximation, and this algorithm is easy to derandom-
ize; Sahni and Gonzalez [48] gave the first 1/2-approximation algorithm in 1976. Following this some
(1/2 + o(1))-approximation algorithms were given, but no real progress was made until the breakthrough
1994 paper of Goemans and Williamson [26]. This remarkable work used semidefinite programming to
achieve an α
GW
-approximation algorithm, where the constant α
GW
≈ .878567 is the trigonometric quantity
α
GW
= min
0<θ<π
θ/π
(1 −cos θ)/2
.
The minimizing choice of θ here is the solution of θ = tan(θ/2), namely θ
∗
≈ 2.33 ≈ 134
◦
, and α
GW
=
2
π sin θ
∗
. The geometric nature of Goemans and Williamson’s algorithm might be considered surprising, but
as we shall see, this geometry seems to be an inherent part of the MAX-CUT problem.
3