1,633 research outputs found
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
A PTAS for the Classical Ising Spin Glass Problem on the Chimera Graph Structure
We present a polynomial time approximation scheme (PTAS) for the minimum
value of the classical Ising Hamiltonian with linear terms on the Chimera graph
structure as defined in the recent work of McGeoch and Wang. The result follows
from a direct application of the techniques used by Bansal, Bravyi and Terhal
who gave a PTAS for the same problem on planar and, in particular, grid graphs.
We also show that on Chimera graphs, the trivial lower bound is within a
constant factor of the optimum.Comment: 6 pages, corrected PTAS running tim
Work in progress: a quantitative study of effectiveness in group learning
It is generally assumed that group studies are more effective for students than individual studies. The objective of this work in progress is to quantitatively evaluate and analyze the effect of collaborative studies on individual student’s performance. This effort would help the student stimulate interest in group learning and collaboration along with exposing them towards multiple problem solving approaches while working individually or in groups. This way the students are challenged to use their existing knowledge and approach, and augment it further with the knowledge and approach provided by group partners. While there are several efforts that focus on developing new group learning techniques, we intend to study the efficacy of previously proposed techniques under various test settings for EE and CS courses without significantly diverting from the course framework
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