74 research outputs found
Derandomized Parallel Repetition via Structured PCPs
A PCP is a proof system for NP in which the proof can be checked by a
probabilistic verifier. The verifier is only allowed to read a very small
portion of the proof, and in return is allowed to err with some bounded
probability. The probability that the verifier accepts a false proof is called
the soundness error, and is an important parameter of a PCP system that one
seeks to minimize. Constructing PCPs with sub-constant soundness error and, at
the same time, a minimal number of queries into the proof (namely two) is
especially important due to applications for inapproximability.
In this work we construct such PCP verifiers, i.e., PCPs that make only two
queries and have sub-constant soundness error. Our construction can be viewed
as a combinatorial alternative to the "manifold vs. point" construction, which
is the only construction in the literature for this parameter range. The
"manifold vs. point" PCP is based on a low degree test, while our construction
is based on a direct product test. We also extend our construction to yield a
decodable PCP (dPCP) with the same parameters. By plugging in this dPCP into
the scheme of Dinur and Harsha (FOCS 2009) one gets an alternative construction
of the result of Moshkovitz and Raz (FOCS 2008), namely: a construction of
two-query PCPs with small soundness error and small alphabet size.
Our construction of a PCP is based on extending the derandomized direct
product test of Impagliazzo, Kabanets and Wigderson (STOC 09) to a derandomized
parallel repetition theorem. More accurately, our PCP construction is obtained
in two steps. We first prove a derandomized parallel repetition theorem for
specially structured PCPs. Then, we show that any PCP can be transformed into
one that has the required structure, by embedding it on a de-Bruijn graph
Conditional Hardness for Approximate Coloring
We study the coloring problem: Given a graph G, decide whether
or , where c(G) is the chromatic number of G. We derive conditional
hardness for this problem for any constant . For , our
result is based on Khot's 2-to-1 conjecture [Khot'02]. For , we base our
hardness result on a certain `fish shaped' variant of his conjecture.
We also prove that the problem almost coloring is hard for any constant
\eps>0, assuming Khot's Unique Games conjecture. This is the problem of
deciding for a given graph, between the case where one can 3-color all but a
\eps fraction of the vertices without monochromatic edges, and the case where
the graph contains no independent set of relative size at least \eps.
Our result is based on bounding various generalized noise-stability
quantities using the invariance principle of Mossel et al [MOO'05]
Toward the KRW Composition Conjecture: Cubic Formula Lower Bounds via Communication Complexity
One of the major challenges of the research in circuit complexity is proving super-polynomial lower bounds for de-Morgan formulas. Karchmer, Raz, and Wigderson suggested to approach this problem by proving that formula complexity behaves "as expected" with respect to the composition of functions f * g. They showed that this conjecture, if proved, would imply super-polynomial formula lower bounds.
The first step toward proving the KRW conjecture was made by Edmonds et al., who proved an analogue of the conjecture for the composition of "universal relations". In this work, we extend the argument of Edmonds et al. further to f * g where f is an arbitrary function and g is the parity function.
While this special case of the KRW conjecture was already proved implicitly in Hastad\u27s work on random restrictions, our proof seems more likely to be generalizable to other cases of the conjecture. In particular, our proof uses an entirely different approach, based on communication complexity technique of Karchmer and Wigderson. In addition, our proof gives a new structural result, which roughly says that the naive way for computing f * g is the only optimal way. Along the way, we obtain a new proof of the state-of-the-art formula lower bound of n^{3-o(1)} due to Hastad
ETH-Hardness of Approximating 2-CSPs and Directed Steiner Network
We study the 2-ary constraint satisfaction problems (2-CSPs), which can be
stated as follows: given a constraint graph , an alphabet set
and, for each , a constraint , the goal is to find an assignment
that satisfies as many constraints as possible, where a constraint is
satisfied if .
While the approximability of 2-CSPs is quite well understood when
is constant, many problems are still open when becomes super
constant. One such problem is whether it is hard to approximate 2-CSPs to
within a polynomial factor of . Bellare et al. (1993) suggested
that the answer to this question might be positive. Alas, despite efforts to
resolve this conjecture, it remains open to this day.
In this work, we separate and and ask a related but weaker
question: is it hard to approximate 2-CSPs to within a polynomial factor of
(while may be super-polynomial in )? Assuming the
exponential time hypothesis (ETH), we answer this question positively by
showing that no polynomial time algorithm can approximate 2-CSPs to within a
factor of . Note that our ratio is almost linear, which is
almost optimal as a trivial algorithm gives a -approximation for 2-CSPs.
Thanks to a known reduction, our result implies an ETH-hardness of
approximating Directed Steiner Network with ratio where is
the number of demand pairs. The ratio is roughly the square root of the best
known ratio achieved by polynomial time algorithms (Chekuri et al., 2011;
Feldman et al., 2012).
Additionally, under Gap-ETH, our reduction for 2-CSPs not only rules out
polynomial time algorithms, but also FPT algorithms parameterized by .
Similar statement applies for DSN parameterized by .Comment: 36 pages. A preliminary version appeared in ITCS'1
Derandomized Graph Product Results using the Low Degree Long Code
In this paper, we address the question of whether the recent derandomization
results obtained by the use of the low-degree long code can be extended to
other product settings. We consider two settings: (1) the graph product results
of Alon, Dinur, Friedgut and Sudakov [GAFA, 2004] and (2) the "majority is
stablest" type of result obtained by Dinur, Mossel and Regev [SICOMP, 2009] and
Dinur and Shinkar [In Proc. APPROX, 2010] while studying the hardness of
approximate graph coloring.
In our first result, we show that there exists a considerably smaller
subgraph of which exhibits the following property (shown for
by Alon et al.): independent sets close in size to the
maximum independent set are well approximated by dictators.
The "majority is stablest" type of result of Dinur et al. and Dinur and
Shinkar shows that if there exist two sets of vertices and in
with very few edges with one endpoint in and another in
, then it must be the case that the two sets and share a single
influential coordinate. In our second result, we show that a similar "majority
is stablest" statement holds good for a considerably smaller subgraph of
. Furthermore using this result, we give a more efficient
reduction from Unique Games to the graph coloring problem, leading to improved
hardness of approximation results for coloring
A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover
Given a -uniform hyper-graph, the E-Vertex-Cover problem is to find the
smallest subset of vertices that intersects every hyper-edge. We present a new
multilayered PCP construction that extends the Raz verifier. This enables us to
prove that E-Vertex-Cover is NP-hard to approximate within factor
for any and any . The result is
essentially tight as this problem can be easily approximated within factor .
Our construction makes use of the biased Long-Code and is analyzed using
combinatorial properties of -wise -intersecting families of subsets
Exponentially Small Soundness for the Direct Product Z-Test
Given a function f:[N]^k->[M]^k, the Z-test is a three query test for checking if a function f is a direct product, namely if there are functions g_1,...g_k:[N]->[M] such that f(x_1,...,x_k)=(g_1(x_1),...,g_k(x_k)) for every input x in [N]^k.
This test was introduced by Impagliazzo et. al. (SICOMP 2012), who showed that if the test passes with probability epsilon > exp(-sqrt k) then f is Omega(epsilon) close to a direct product function in some precise sense. It remained an open question whether the soundness of this test can be pushed all the way down to exp(-k) (which would be optimal). This is our main result: we show that whenever f passes the Z test with probability epsilon > exp(-k), there must be a global reason for this: namely, f must be close to a product function on some Omega(epsilon) fraction of its domain.
Towards proving our result we analyze the related (two-query) V-test, and prove a "restricted global structure" theorem for it. Such theorems were also proven in previous works on direct product testing in the small soundness regime. The most recent work, by Dinur and Steurer (CCC 2014), analyzed the V test in the exponentially small soundness regime. We strengthen their conclusion of that theorem by moving from an "in expectation" statement to a stronger "concentration of measure" type of statement, which we prove using hyper-contractivity. This stronger statement allows us to proceed to analyze the Z test.
We analyze two variants of direct product tests. One for functions on ordered tuples, as above, and another for functions on sets of size k. The work of Impagliazzo et al. was actually focused only on functions of the latter type, i.e. on sets. We prove exponentially small soundness for the Z-test for both variants. Although the two appear very similar, the analysis for tuples is more tricky and requires some additional ideas
Every set in P is strongly testable under a suitable encoding
We show that every set in P is strongly testable under a suitable encoding. By "strongly testable" we mean having a (proximity oblivious) tester that makes a constant number of queries and rejects with probability that is proportional to the distance of the tested object from the property. By a "suitable encoding" we mean one that is polynomial-time computable and invertible. This result stands in contrast to the known fact that some sets in P are extremely hard to test, providing another demonstration of the crucial role of representation in the context of property testing. The testing result is proved by showing that any set in P has a strong canonical PCP, where canonical means that (for yes-instances) there exists a single proof that is accepted with probability 1 by the system, whereas all other potential proofs are rejected with probability proportional to their distance from this proof. In fact, we show that UP equals the class of sets having strong canonical PCPs (of logarithmic randomness), whereas the class of sets having strong canonical PCPs with polynomial proof length equals "unambiguous- MA". Actually, for the testing result, we use a PCP-of-Proximity version of the foregoing notion and an analogous positive result (i.e., strong canonical PCPPs of logarithmic randomness for any set in UP)
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