1,252 research outputs found
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]
Conditional Hardness for Approximate Coloring
We study the AprxColoring(q,Q) problem: Given a graph G, decide whether Χ(G) ≤ q or Χ(G)≥Q. We present hardness results for this problem for any constants 3 ≤ q \u3c Q. For q ≥ 4, our result is base on Khot\u27s 2-to-1 label cover, which is conjectured to be NP-hard [S. Khot, Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767–775]. For q=3, we base our hardness result on a certain “⋉-shaped variant of his conjecture. Previously no hardness result was known for q = 3 and Q ≥ 6. At the heart of our proof are tight bounds on generalized noise-stability quantities, which extend the recent work of Mossel, O\u27Donnell, and Oleszkiewicz [ Noise stability of functions with low influences: Invariance and optimality, Ann. of Math. (2), to appear] and should have wider applicability
On Coloring Resilient Graphs
We introduce a new notion of resilience for constraint satisfaction problems,
with the goal of more precisely determining the boundary between NP-hardness
and the existence of efficient algorithms for resilient instances. In
particular, we study -resiliently -colorable graphs, which are those
-colorable graphs that remain -colorable even after the addition of any
new edges. We prove lower bounds on the NP-hardness of coloring resiliently
colorable graphs, and provide an algorithm that colors sufficiently resilient
graphs. We also analyze the corresponding notion of resilience for -SAT.
This notion of resilience suggests an array of open questions for graph
coloring and other combinatorial problems.Comment: Appearing in MFCS 201
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
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
Approximation for Maximum Surjective Constraint Satisfaction Problems
Maximum surjective constraint satisfaction problems (Max-Sur-CSPs) are
computational problems where we are given a set of variables denoting values
from a finite domain B and a set of constraints on the variables. A solution to
such a problem is a surjective mapping from the set of variables to B such that
the number of satisfied constraints is maximized. We study the approximation
performance that can be acccchieved by algorithms for these problems, mainly by
investigating their relation with Max-CSPs (which are the corresponding
problems without the surjectivity requirement). Our work gives a complexity
dichotomy for Max-Sur-CSP(B) between PTAS and APX-complete, under the
assumption that there is a complexity dichotomy for Max-CSP(B) between PO and
APX-complete, which has already been proved on the Boolean domain and 3-element
domains
Improved Hardness of Approximating Chromatic Number
We prove that for sufficiently large K, it is NP-hard to color K-colorable
graphs with less than 2^{K^{1/3}} colors. This improves the previous result of
K versus K^{O(log K)} in Khot [14]
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
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