361,293 research outputs found
Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder
Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat).
The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem
Optimization hardness as transient chaos in an analog approach to constraint satisfaction
Boolean satisfiability [1] (k-SAT) is one of the most studied optimization
problems, as an efficient (that is, polynomial-time) solution to k-SAT (for
) implies efficient solutions to a large number of hard optimization
problems [2,3]. Here we propose a mapping of k-SAT into a deterministic
continuous-time dynamical system with a unique correspondence between its
attractors and the k-SAT solution clusters. We show that beyond a constraint
density threshold, the analog trajectories become transiently chaotic [4-7],
and the boundaries between the basins of attraction [8] of the solution
clusters become fractal [7-9], signaling the appearance of optimization
hardness [10]. Analytical arguments and simulations indicate that the system
always finds solutions for satisfiable formulae even in the frozen regimes of
random 3-SAT [11] and of locked occupation problems [12] (considered among the
hardest algorithmic benchmarks); a property partly due to the system's
hyperbolic [4,13] character. The system finds solutions in polynomial
continuous-time, however, at the expense of exponential fluctuations in its
energy function.Comment: 27 pages, 14 figure
On the Complexity of Random Satisfiability Problems with Planted Solutions
The problem of identifying a planted assignment given a random -SAT
formula consistent with the assignment exhibits a large algorithmic gap: while
the planted solution becomes unique and can be identified given a formula with
clauses, there are distributions over clauses for which the best
known efficient algorithms require clauses. We propose and study a
unified model for planted -SAT, which captures well-known special cases. An
instance is described by a planted assignment and a distribution on
clauses with literals. We define its distribution complexity as the largest
for which the distribution is not -wise independent ( for
any distribution with a planted assignment).
Our main result is an unconditional lower bound, tight up to logarithmic
factors, for statistical (query) algorithms [Kearns 1998, Feldman et. al 2012],
matching known upper bounds, which, as we show, can be implemented using a
statistical algorithm. Since known approaches for problems over distributions
have statistical analogues (spectral, MCMC, gradient-based, convex optimization
etc.), this lower bound provides a rigorous explanation of the observed
algorithmic gap. The proof introduces a new general technique for the analysis
of statistical query algorithms. It also points to a geometric paring
phenomenon in the space of all planted assignments.
We describe consequences of our lower bounds to Feige's refutation hypothesis
[Feige 2002] and to lower bounds on general convex programs that solve planted
-SAT. Our bounds also extend to other planted -CSP models, and, in
particular, provide concrete evidence for the security of Goldreich's one-way
function and the associated pseudorandom generator when used with a
sufficiently hard predicate [Goldreich 2000].Comment: Extended abstract appeared in STOC 201
Introducing Structure to Expedite Quantum Search
We present a novel quantum algorithm for solving the unstructured search
problem with one marked element. Our algorithm allows generating quantum
circuits that use asymptotically fewer additional quantum gates than the famous
Grover's algorithm and may be successfully executed on NISQ devices. We prove
that our algorithm is optimal in the total number of elementary gates up to a
multiplicative constant. As many NP-hard problems are not in fact unstructured,
we also describe the \emph{partial uncompute} technique which exploits the
oracle structure and allows a significant reduction in the number of elementary
gates required to find the solution. Combining these results allows us to use
asymptotically smaller number of elementary gates than the Grover's algorithm
in various applications, keeping the number of queries to the oracle
essentially the same. We show how the results can be applied to solve hard
combinatorial problems, for example Unique k-SAT. Additionally, we show how to
asymptotically reduce the number of elementary gates required to solve the
unstructured search problem with multiple marked elements.Comment: 22 pages, 7 figure
Hypercontractivity, Sum-of-Squares Proofs, and their Applications
We study the computational complexity of approximating the 2->q norm of
linear operators (defined as ||A||_{2->q} = sup_v ||Av||_q/||v||_2), as well as
connections between this question and issues arising in quantum information
theory and the study of Khot's Unique Games Conjecture (UGC). We show the
following:
1. For any constant even integer q>=4, a graph is a "small-set expander"
if and only if the projector into the span of the top eigenvectors of G's
adjacency matrix has bounded 2->q norm. As a corollary, a good approximation to
the 2->q norm will refute the Small-Set Expansion Conjecture--a close variant
of the UGC. We also show that such a good approximation can be obtained in
exp(n^(2/q)) time, thus obtaining a different proof of the known subexponential
algorithm for Small Set Expansion.
2. Constant rounds of the "Sum of Squares" semidefinite programing hierarchy
certify an upper bound on the 2->4 norm of the projector to low-degree
polynomials over the Boolean cube, as well certify the unsatisfiability of the
"noisy cube" and "short code" based instances of Unique Games considered by
prior works. This improves on the previous upper bound of exp(poly log n)
rounds (for the "short code"), as well as separates the "Sum of
Squares"/"Lasserre" hierarchy from weaker hierarchies that were known to
require omega(1) rounds.
3. We show reductions between computing the 2->4 norm and computing the
injective tensor norm of a tensor, a problem with connections to quantum
information theory. Three corollaries are: (i) the 2->4 norm is NP-hard to
approximate to precision inverse-polynomial in the dimension, (ii) the 2->4
norm does not have a good approximation (in the sense above) unless 3-SAT can
be solved in time exp(sqrt(n) polylog(n)), and (iii) known algorithms for the
quantum separability problem imply a non-trivial additive approximation for the
2->4 norm.Comment: v1: 52 pages. v2: 53 pages, fixed small bugs in proofs of section 6
(on UG integrality gaps) and section 7 (on 2->4 norm of random matrices).
Added comments about real-vs-complex random matrices and about the
k-extendable vs k-extendable & PPT hierarchies. v3: fixed mistakes in random
matrix section. The result now holds only for matrices with random entries
instead of random column
Verification in Staged Tile Self-Assembly
We prove the unique assembly and unique shape verification problems,
benchmark measures of self-assembly model power, are
-hard and contained in (and in
for staged systems with stages). En route,
we prove that unique shape verification problem in the 2HAM is
-complete.Comment: An abstract version will appear in the proceedings of UCNC 201
Complexity classifications for different equivalence and audit problems for Boolean circuits
We study Boolean circuits as a representation of Boolean functions and
consider different equivalence, audit, and enumeration problems. For a number
of restricted sets of gate types (bases) we obtain efficient algorithms, while
for all other gate types we show these problems are at least NP-hard.Comment: 25 pages, 1 figur
The Complexity of Computing Minimal Unidirectional Covering Sets
Given a binary dominance relation on a set of alternatives, a common thread
in the social sciences is to identify subsets of alternatives that satisfy
certain notions of stability. Examples can be found in areas as diverse as
voting theory, game theory, and argumentation theory. Brandt and Fischer [BF08]
proved that it is NP-hard to decide whether an alternative is contained in some
inclusion-minimal upward or downward covering set. For both problems, we raise
this lower bound to the Theta_{2}^{p} level of the polynomial hierarchy and
provide a Sigma_{2}^{p} upper bound. Relatedly, we show that a variety of other
natural problems regarding minimal or minimum-size covering sets are hard or
complete for either of NP, coNP, and Theta_{2}^{p}. An important consequence of
our results is that neither minimal upward nor minimal downward covering sets
(even when guaranteed to exist) can be computed in polynomial time unless P=NP.
This sharply contrasts with Brandt and Fischer's result that minimal
bidirectional covering sets (i.e., sets that are both minimal upward and
minimal downward covering sets) are polynomial-time computable.Comment: 27 pages, 7 figure
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