37 research outputs found
Approximation of non-boolean 2CSP
We develop a polynomial time
approximate algorithm for Max 2CSP-, the problem where we are given a
collection of constraints, each involving two variables, where each variable
ranges over a set of size , and we want to find an assignment to the
variables that maximizes the number of satisfied constraints. Assuming the
Unique Games Conjecture, this is the best possible approximation up to constant
factors.
Previously, a -approximate algorithm was known, based on linear
programming. Our algorithm is based on semidefinite programming (SDP) and on a
novel rounding technique. The SDP that we use has an almost-matching
integrality gap
Streaming Complexity of Approximating Max 2CSP and Max Acyclic Subgraph
We study the complexity of estimating the optimum value of a Boolean 2CSP (arity two constraint satisfaction problem) in the single-pass streaming setting, where the algorithm is presented the constraints in an arbitrary order. We give a streaming algorithm to estimate the optimum within a factor approaching 2/5 using logarithmic space, with high probability. This beats the trivial factor 1/4 estimate obtained by simply outputting 1/4-th of the total number of constraints.
The inspiration for our work is a lower bound of Kapralov, Khanna, and Sudan (SODA\u2715) who showed that a similar trivial estimate (of factor 1/2) is the best one can do for Max CUT. This lower bound implies that beating a factor 1/2 for Max DICUT (a special case of Max 2CSP), in particular, to distinguish between the case when the optimum is m/2 versus when it is at most (1/4+eps)m, where m is the total number of edges, requires polynomial space. We complement this hardness result by showing that for DICUT, one can distinguish between the case in which the optimum exceeds (1/2+eps)m and the case in which it is close to m/4.
We also prove that estimating the size of the maximum acyclic subgraph of a directed graph, when its edges are presented in a single-pass stream, within a factor better than 7/8 requires polynomial space
Streaming Hardness of Unique Games
We study the problem of approximating the value of a Unique Game instance in the streaming model. A simple count of the number of constraints divided by p, the alphabet size of the Unique Game, gives a trivial p-approximation that can be computed in O(log n) space. Meanwhile, with high probability, a sample of O~(n) constraints suffices to estimate the optimal value to (1+epsilon) accuracy. We prove that any single-pass streaming algorithm that achieves a (p-epsilon)-approximation requires Omega_epsilon(sqrt n) space. Our proof is via a reduction from lower bounds for a communication problem that is a p-ary variant of the Boolean Hidden Matching problem studied in the literature. Given the utility of Unique Games as a starting point for reduction to other optimization problems, our strong hardness for approximating Unique Games could lead to downstream hardness results for streaming approximability for other CSP-like problems
Improved Soundness for QMA with Multiple Provers
We present three contributions to the understanding of QMA with multiple
provers:
1) We give a tight soundness analysis of the protocol of [Blier and Tapp,
ICQNM '09], yielding a soundness gap Omega(1/N^2). Our improvement is achieved
without the use of an instance with a constant soundness gap (i.e., without
using a PCP).
2) We give a tight soundness analysis of the protocol of [Chen and Drucker,
ArXiV '10], thereby improving their result from a monolithic protocol where
Theta(sqrt(N)) provers are needed in order to have any soundness gap, to a
protocol with a smooth trade-off between the number of provers k and a
soundness gap Omega(k^2/N), as long as k>=Omega(log N). (And, when
k=Theta(sqrt(N)), we recover the original parameters of Chen and Drucker.)
3) We make progress towards an open question of [Aaronson et al., ToC '09]
about what kinds of NP-complete problems are amenable to sublinear
multiple-prover QMA protocols, by observing that a large class of such examples
can easily be derived from results already in the PCP literature - namely, at
least the languages recognized by a non-deterministic RAMs in quasilinear time.Comment: 24 pages; comments welcom
Hardness of Graph Pricing through Generalized Max-Dicut
The Graph Pricing problem is among the fundamental problems whose
approximability is not well-understood. While there is a simple combinatorial
1/4-approximation algorithm, the best hardness result remains at 1/2 assuming
the Unique Games Conjecture (UGC). We show that it is NP-hard to approximate
within a factor better than 1/4 under the UGC, so that the simple combinatorial
algorithm might be the best possible. We also prove that for any , there exists such that the integrality gap of
-rounds of the Sherali-Adams hierarchy of linear programming for
Graph Pricing is at most 1/2 + .
This work is based on the effort to view the Graph Pricing problem as a
Constraint Satisfaction Problem (CSP) simpler than the standard and complicated
formulation. We propose the problem called Generalized Max-Dicut(), which
has a domain size for every . Generalized Max-Dicut(1) is
well-known Max-Dicut. There is an approximation-preserving reduction from
Generalized Max-Dicut on directed acyclic graphs (DAGs) to Graph Pricing, and
both our results are achieved through this reduction. Besides its connection to
Graph Pricing, the hardness of Generalized Max-Dicut is interesting in its own
right since in most arity two CSPs studied in the literature, SDP-based
algorithms perform better than LP-based or combinatorial algorithms --- for
this arity two CSP, a simple combinatorial algorithm does the best.Comment: 28 page
The approximability of non-Boolean satisfiability problems and restricted integer programming
AbstractIn this paper we present improved approximation algorithms for two classes of maximization problems defined in Barland et al. (J. Comput. System Sci. 57(2) (1998) 144). Our factors of approximation substantially improve the previous known results and are close to the best possible. On the other hand, we show that the approximation results in the framework of Barland et al. hold also in the parallel setting, and thus we have a new common framework for both computational settings. We prove almost tight non-approximability results, thus solving a main open question of Barland et al.We obtain the results through the constraint satisfaction problem over multi-valued domains, for which we develop approximation algorithms and show non-approximability results. Our parallel approximation algorithms are based on linear programming and random rounding; they are better than previously known sequential algorithms. The non-approximability results are based on new recent progress in the fields of probabilistically checkable proofs and multi-prover one-round proof systems
Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?
In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of αGW + ∈, for all ∈ \u3e 0; here αGW ≈ .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 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)/q2 .
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 xi − xj = 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
Subexponential LPs Approximate Max-Cut
We show that for every , the degree-
Sherali-Adams linear program (with variables
and constraints) approximates the maximum cut problem within a factor of
, for some . Our
result provides a surprising converse to known lower bounds against all linear
programming relaxations of Max-Cut, and hence resolves the extension complexity
of approximate Max-Cut for approximation factors close to (up to
the function ). Previously, only semidefinite
programs and spectral methods were known to yield approximation factors better
than for Max-Cut in time . We also show that
constant-degree Sherali-Adams linear programs (with variables
and constraints) can solve Max-Cut with approximation factor close to on
graphs of small threshold rank: this is the first connection of which we are
aware between threshold rank and linear programming-based algorithms.
Our results separate the power of Sherali-Adams versus Lov\'asz-Schrijver
hierarchies for approximating Max-Cut, since it is known that
approximation of Max Cut requires
rounds in the Lov\'asz-Schrijver hierarchy.
We also provide a subexponential time approximation for Khot's Unique Games
problem: we show that for every the degree- Sherali-Adams linear program distinguishes instances of Unique Games
of value from instances of value , for
some , where is the alphabet size. Such
guarantees are qualitatively similar to those of previous subexponential-time
algorithms for Unique Games but our algorithm does not rely on semidefinite
programming or subspace enumeration techniques