53 research outputs found
Super-polylogarithmic hypergraph coloring hardness via low-degree long codes
We prove improved inapproximability results for hypergraph coloring using the
low-degree polynomial code (aka, the 'short code' of Barak et. al. [FOCS 2012])
and the techniques proposed by Dinur and Guruswami [FOCS 2013] to incorporate
this code for inapproximability results. In particular, we prove
quasi-NP-hardness of the following problems on -vertex hyper-graphs:
* Coloring a 2-colorable 8-uniform hypergraph with
colors.
* Coloring a 4-colorable 4-uniform hypergraph with
colors.
* Coloring a 3-colorable 3-uniform hypergraph with colors.
In each of these cases, the hardness results obtained are (at least)
exponentially stronger than what was previously known for the respective cases.
In fact, prior to this result, polylog n colors was the strongest quantitative
bound on the number of colors ruled out by inapproximability results for
O(1)-colorable hypergraphs.
The fundamental bottleneck in obtaining coloring inapproximability results
using the low- degree long code was a multipartite structural restriction in
the PCP construction of Dinur-Guruswami. We are able to get around this
restriction by simulating the multipartite structure implicitly by querying
just one partition (albeit requiring 8 queries), which yields our result for
2-colorable 8-uniform hypergraphs. The result for 4-colorable 4-uniform
hypergraphs is obtained via a 'query doubling' method. For 3-colorable
3-uniform hypergraphs, we exploit the ternary domain to design a test with an
additive (as opposed to multiplicative) noise function, and analyze its
efficacy in killing high weight Fourier coefficients via the pseudorandom
properties of an associated quadratic form.Comment: 25 page
Hardness of Finding Independent Sets in 2-Colorable Hypergraphs and of Satisfiable CSPs
This work revisits the PCP Verifiers used in the works of Hastad [Has01],
Guruswami et al.[GHS02], Holmerin[Hol02] and Guruswami[Gur00] for satisfiable
Max-E3-SAT and Max-Ek-Set-Splitting, and independent set in 2-colorable
4-uniform hypergraphs. We provide simpler and more efficient PCP Verifiers to
prove the following improved hardness results: Assuming that NP\not\subseteq
DTIME(N^{O(loglog N)}),
There is no polynomial time algorithm that, given an n-vertex 2-colorable
4-uniform hypergraph, finds an independent set of n/(log n)^c vertices, for
some constant c > 0.
There is no polynomial time algorithm that satisfies 7/8 + 1/(log n)^c
fraction of the clauses of a satisfiable Max-E3-SAT instance of size n, for
some constant c > 0.
For any fixed k >= 4, there is no polynomial time algorithm that finds a
partition splitting (1 - 2^{-k+1}) + 1/(log n)^c fraction of the k-sets of a
satisfiable Max-Ek-Set-Splitting instance of size n, for some constant c > 0.
Our hardness factor for independent set in 2-colorable 4-uniform hypergraphs
is an exponential improvement over the previous results of Guruswami et
al.[GHS02] and Holmerin[Hol02]. Similarly, our inapproximability of (log
n)^{-c} beyond the random assignment threshold for Max-E3-SAT and
Max-Ek-Set-Splitting is an exponential improvement over the previous bounds
proved in [Has01], [Hol02] and [Gur00]. The PCP Verifiers used in our results
avoid the use of a variable bias parameter used in previous works, which leads
to the improved hardness thresholds in addition to simplifying the analysis
substantially. Apart from standard techniques from Fourier Analysis, for the
first mentioned result we use a mixing estimate of Markov Chains based on
uniform reverse hypercontractivity over general product spaces from the work of
Mossel et al.[MOS13].Comment: 23 Page
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
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be -colorable if its vertices can be colored
with colors so that no hyperedge is monochromatic. -colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a -colorable
-uniform hypergraph, it is NP-hard to find a -coloring miscoloring fewer
than a fraction of hyperedges (which is achieved by a random
-coloring), and the best algorithms to color the hypergraph properly require
colors, approaching the trivial bound of as
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a -coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than -colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ,
we give an algorithm to color the it with colors.
However, for the maximization version, we prove NP-hardness of finding a
-coloring miscoloring a smaller than (resp. )
fraction of the hyperedges when (resp. ). Assuming
the UGC, we improve the latter hardness factor to for almost
discrepancy- hypergraphs.
(B) Rainbow colorability: If the hypergraph has a -coloring such
that each hyperedge is polychromatic with all these colors, we give a
-coloring algorithm that miscolors at most of the
hyperedges when , and complement this with a matching UG
hardness result showing that when , it is hard to even beat the
bound achieved by a random coloring.Comment: Approx 201
Robust Multiplication-Based Tests for Reed-Muller Codes
We consider the following multiplication-based tests to check if a given function f: F^n_q -> F_q is the evaluation of a degree-d polynomial over F_q for q prime.
Test_{e,k}: Pick P_1,...,P_k independent random degree-e polynomials and accept iff the function f P_1 ... P_k is the evaluation of a degree-(d + ek) polynomial.
We prove the robust soundness of the above tests for large values of e, answering a question of Dinur and Guruswami (FOCS 2013). Previous soundness analyses of these tests were known only for the case when either e = 1 or k = 1. Even for the case k = 1 and e > 1, earlier soundness analyses were not robust.
We also analyze a derandomized version of this test, where (for example) the polynomials P_1 ,...P_k can be the same random polynomial P. This generalizes a result of Guruswami et al. (STOC 2014).
One of the key ingredients that go into the proof of this robust soundness is an extension of the standard Schwartz-Zippel lemma over general finite fields F_q, which may be of independent interest
- …