# 数学代考|计算复杂性理论代写computational complexity theory代考|Polynomial-Time Isomorphism

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## 数学代考|计算复杂性理论代写computatiknal complexity theory代考|Polynomial-Time Isomorphism

Recall the decision problems IS and CLIQUE from Chapter 2:
INDEPENDENT SET (IS): Given a graph $G$ and an integer k>0, determine whether G has an independent set of size at least k.

CLIQUE: Given a graph G and an integer k>0, determine whether G has a clique of size at least k.
For any graph G=(V, E), let the complement graph G^{c} of G be the graph that has the same vertex set V but has the edge set $E^{\prime}={{u, v}: u, v \in G,{u, v} \notin E}$. It is clear that a subset $A$ of vertices is an independent set of a graph $G=(V, E)$ if and only if the complement graph $G^{c}$ of $G$ has a clique on subset $A$. Thus, a graph $G$ has an independent set of size k if and only if its complement $G^{c}$ has a clique of size k. That is, there is a bijection (i.e., a one-to-one and onto mapping) f between the inputs of these problems: f(G, k)=\left(G^{c}, k\right) such that IS $\leq_{m}^{P}$ CLIQUE via f and CLIQUE $\leq_{m}^{P}$ IS via $f^{-1}$. In other words, these two problems are not only reducible to each other by the polynomial-time many-one reductions but are also equivalent to each other through polynomial-time computable bijections. We say that they are polynomial-time isomorphic to each other.

We now give the formal definition of the notion of polynomial-time isomorphism.

We have seen, in the last section, an example of improving the $\leq_{m}^{P}$ reductions to $\leq_{i n v, l i}^{P}$-reductions, which allows us to get the $p$-isomorphism between two $N P$-complete problems. This technique is, however, still not simple enough because, to prove that $n N P$-complete problems are all $p$-isomorphic to each other, we need to construct $n(n-1) \leq_{i n v, l i}^{P}{ }^{-}$ reductions. In this section, we present a more general technique that further simplifies the task.

## 数学代考|计算复杂性理论代写computatiknal complexity theory代考|Density of NP-Complete Sets

We established, in the last section, a proof technique with which we were able to show that many natural N P-complete languages are $p$-isomorphic to SAT. This proof technique is, however, not strong enough to settle the Berman-Hartmanis conjecture, as we do not know whether every N P complete set is paddable. Indeed, we observe that if all N P-complete sets are p-isomorphic to each other, then P \neq N P. To see this, we observe that a finite set cannot be $p$-isomorphic to an infinite set. Thus, if the BermanHartmanis conjecture is true, then every nonempty finite set is in P but is not N P-complete and, hence, is a witness for P \neq N P. From this simple observation, we see that it is unlikely that we can prove the BermanHartmanis conjecture without a major breakthrough. Nevertheless, one might still gain some insight into the structure of the N P-complete problems through the study of this conjecture. In this section, we investigate the density of N P-complete problems and provide more support for the conjecture.

The density or the census function of a language A is a function $C_{A}: N \rightarrow N$ defined by $C_{A}(n)=|{x \in A:|x| \leq n}|$, that is, $C_{A}(n)=\left|A_{\leq n}\right|$. Recall that a language A is sparse if there exists a polynomial q such that for every $n \in N, C_{A}(n) \leq q(n)$.

## 数学代考|计算复杂性理论代写COMPUTATIKNAL COMPLEXITY THEORY代考|POLYNOMIAL-TIME ISOMORPHISM

CLIQUE：给定一个图 G 和一个整数 k>0，确定 G 是否有一个大小至少为 k 的 clique。

## Matlab代写

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