8,158 research outputs found
A Hypergraph Dictatorship Test with Perfect Completeness
A hypergraph dictatorship test is first introduced by Samorodnitsky and
Trevisan and serves as a key component in their unique games based \PCP
construction. Such a test has oracle access to a collection of functions and
determines whether all the functions are the same dictatorship, or all their
low degree influences are Their test makes queries and has
amortized query complexity but has an inherent loss of
perfect completeness. In this paper we give an adaptive hypergraph dictatorship
test that achieves both perfect completeness and amortized query complexity
.Comment: Some minor correction
On the Usefulness of Predicates
Motivated by the pervasiveness of strong inapproximability results for
Max-CSPs, we introduce a relaxed notion of an approximate solution of a
Max-CSP. In this relaxed version, loosely speaking, the algorithm is allowed to
replace the constraints of an instance by some other (possibly real-valued)
constraints, and then only needs to satisfy as many of the new constraints as
possible.
To be more precise, we introduce the following notion of a predicate
being \emph{useful} for a (real-valued) objective : given an almost
satisfiable Max- instance, there is an algorithm that beats a random
assignment on the corresponding Max- instance applied to the same sets of
literals. The standard notion of a nontrivial approximation algorithm for a
Max-CSP with predicate is exactly the same as saying that is useful for
itself.
We say that is useless if it is not useful for any . This turns out to
be equivalent to the following pseudo-randomness property: given an almost
satisfiable instance of Max- it is hard to find an assignment such that the
induced distribution on -bit strings defined by the instance is not
essentially uniform.
Under the Unique Games Conjecture, we give a complete and simple
characterization of useful Max-CSPs defined by a predicate: such a Max-CSP is
useless if and only if there is a pairwise independent distribution supported
on the satisfying assignments of the predicate. It is natural to also consider
the case when no negations are allowed in the CSP instance, and we derive a
similar complete characterization (under the UGC) there as well.
Finally, we also include some results and examples shedding additional light
on the approximability of certain Max-CSPs
An Improved Dictatorship Test with Perfect Completeness
A Boolean function f:{0,1}^n->{0,1} is called a dictator if it depends on exactly one variable i.e f(x_1, x_2, ..., x_n) = x_i for some i in [n]. In this work, we study a k-query dictatorship test. Dictatorship tests are central in proving many hardness results for constraint satisfaction problems.
The dictatorship test is said to have perfect completeness if it accepts any dictator function. The soundness of a test is the maximum probability with which it accepts any function far from a dictator. Our main result is a k-query dictatorship test with perfect completeness and soundness (2k + 1)/(2^k), where k is of the form 2^t -1 for any integer t > 2. This improves upon the result of [Tamaki-Yoshida, Random Structures & Algorithms, 2015] which gave a dictatorship test with soundness (2k + 3)/(2^k)
Ignorance is Almost Bliss: Near-Optimal Stochastic Matching With Few Queries
The stochastic matching problem deals with finding a maximum matching in a
graph whose edges are unknown but can be accessed via queries. This is a
special case of stochastic -set packing, where the problem is to find a
maximum packing of sets, each of which exists with some probability. In this
paper, we provide edge and set query algorithms for these two problems,
respectively, that provably achieve some fraction of the omniscient optimal
solution.
Our main theoretical result for the stochastic matching (i.e., -set
packing) problem is the design of an \emph{adaptive} algorithm that queries
only a constant number of edges per vertex and achieves a
fraction of the omniscient optimal solution, for an arbitrarily small
. Moreover, this adaptive algorithm performs the queries in only a
constant number of rounds. We complement this result with a \emph{non-adaptive}
(i.e., one round of queries) algorithm that achieves a
fraction of the omniscient optimum. We also extend both our results to
stochastic -set packing by designing an adaptive algorithm that achieves a
fraction of the omniscient optimal solution, again
with only queries per element. This guarantee is close to the best known
polynomial-time approximation ratio of for the
\emph{deterministic} -set packing problem [Furer and Yu, 2013]
We empirically explore the application of (adaptations of) these algorithms
to the kidney exchange problem, where patients with end-stage renal failure
swap willing but incompatible donors. We show on both generated data and on
real data from the first 169 match runs of the UNOS nationwide kidney exchange
that even a very small number of non-adaptive edge queries per vertex results
in large gains in expected successful matches
Improved Inapproximability Results for Maximum k-Colorable Subgraph
We study the maximization version of the fundamental graph coloring problem.
Here the goal is to color the vertices of a k-colorable graph with k colors so
that a maximum fraction of edges are properly colored (i.e. their endpoints
receive different colors). A random k-coloring properly colors an expected
fraction 1-1/k of edges. We prove that given a graph promised to be
k-colorable, it is NP-hard to find a k-coloring that properly colors more than
a fraction ~1-O(1/k} of edges. Previously, only a hardness factor of 1-O(1/k^2)
was known. Our result pins down the correct asymptotic dependence of the
approximation factor on k. Along the way, we prove that approximating the
Maximum 3-colorable subgraph problem within a factor greater than 32/33 is
NP-hard. Using semidefinite programming, it is known that one can do better
than a random coloring and properly color a fraction 1-1/k +2 ln k/k^2 of edges
in polynomial time. We show that, assuming the 2-to-1 conjecture, it is hard to
properly color (using k colors) more than a fraction 1-1/k + O(ln k/ k^2) of
edges of a k-colorable graph.Comment: 16 pages, 2 figure
Average-Case Complexity
We survey the average-case complexity of problems in NP.
We discuss various notions of good-on-average algorithms, and present
completeness results due to Impagliazzo and Levin. Such completeness results
establish the fact that if a certain specific (but somewhat artificial) NP
problem is easy-on-average with respect to the uniform distribution, then all
problems in NP are easy-on-average with respect to all samplable distributions.
Applying the theory to natural distributional problems remain an outstanding
open question. We review some natural distributional problems whose
average-case complexity is of particular interest and that do not yet fit into
this theory.
A major open question whether the existence of hard-on-average problems in NP
can be based on the PNP assumption or on related worst-case assumptions.
We review negative results showing that certain proof techniques cannot prove
such a result. While the relation between worst-case and average-case
complexity for general NP problems remains open, there has been progress in
understanding the relation between different ``degrees'' of average-case
complexity. We discuss some of these ``hardness amplification'' results
Perfect zero knowledge for quantum multiprover interactive proofs
In this work we consider the interplay between multiprover interactive
proofs, quantum entanglement, and zero knowledge proofs - notions that are
central pillars of complexity theory, quantum information and cryptography. In
particular, we study the relationship between the complexity class MIP, the
set of languages decidable by multiprover interactive proofs with quantumly
entangled provers, and the class PZKMIP, which is the set of languages
decidable by MIP protocols that furthermore possess the perfect zero
knowledge property.
Our main result is that the two classes are equal, i.e., MIP
PZKMIP. This result provides a quantum analogue of the celebrated result of
Ben-Or, Goldwasser, Kilian, and Wigderson (STOC 1988) who show that MIP
PZKMIP (in other words, all classical multiprover interactive protocols can be
made zero knowledge). We prove our result by showing that every MIP
protocol can be efficiently transformed into an equivalent zero knowledge
MIP protocol in a manner that preserves the completeness-soundness gap.
Combining our transformation with previous results by Slofstra (Forum of
Mathematics, Pi 2019) and Fitzsimons, Ji, Vidick and Yuen (STOC 2019), we
obtain the corollary that all co-recursively enumerable languages (which
include undecidable problems as well as all decidable problems) have zero
knowledge MIP protocols with vanishing promise gap
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