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The question of whether all problems in NP class are also in P class is generally considered one of the most important open questions in mathematics and theoretical computer science as it has far-reaching consequences to other problems in mathematics, computer science, biology, philosophy and cryptography. There are intensive research on proving `NP not equal to P and `NP equals to P. However, none of the `proved results is commonly accepted by the research community up to date. In this paper, motived by approximability of traveling salesman problem (TSP) in polynomial time, we aim to provide a new perspective: showing that NP=P from polynomial time approximation-bounded solutions of TSP in Euclidean space.
We give an asymptotic approximation scheme (APTAS) for the problem of packing a set of circles into a minimum number of unit square bins. To obtain rational solutions, we use augmented bins of height $1+gamma$, for some arbitrarily small number $gamm
Given a graph $G=(V,E)$ with two distinguished vertices $s,tin V$ and an integer $L$, an {em $L$-bounded flow} is a flow between $s$ and $t$ that can be decomposed into paths of length at most $L$. In the {em maximum $L$-bounded flow problem} the tas
We give polynomial-time approximation schemes for monotone maximization problems expressible in terms of distances (up to a fixed upper bound) and efficiently solvable in graphs of bounded treewidth. These schemes apply in all fractionally treewidth-
Using the probability theory-based approach, this paper reveals the equivalence of an arbitrary NP-complete problem to a problem of checking whether a level set of a specifically constructed harmonic cost function (with all diagonal entries of its He
Dealing with NP-hard problems, kernelization is a fundamental notion for polynomial-time data reduction with performance guarantees: in polynomial time, a problem instance is reduced to an equivalent instance with size upper-bounded by a function of