NP完全性与近似算法: NP完全问题的证明、困难问题的近似算法设计、复杂性理论等。

Problem 1 Given a graph $G=(V, E)$ let $\mathcal{I}=\{S \subseteq V \mid$ there is a matching $M$ in $G$ that covers $S\}$. Prove that $(V, \mathcal{I})$ is a matroid.

The theory of thProblem 1 Given a graph $G=(V, E)$ let $\mathcal{I}=\{S \subseteq V \mid$ there is a matching $M$ in $G$ that covers $S\}$. Prove that $(V, \mathcal{I})$ is a matroid.e harmonic oscillator gives the average kinetic energy as $\bar{V}=\frac{1}{2} E$, i.e., $\frac{1}{2} m \omega^2 A^2=\frac{1}{4} \hbar \omega$, where $\mathrm{w}=\sqrt{g / l}$ and $A$ is the root-meansquare amplitude of the zero-point oscillation. Hence
Thus the zero-point oscillation of a macroscopic pendulum is negligible.
(b) If we regard the width and height of the rigid obstacle as the width and height of a gravity potential barrier, the tunneling probability is
\begin{aligned} T & \approx \exp \left[-\frac{2 w}{\hbar} \sqrt{2 m\left(m g H-\frac{1}{2} m v^2\right)}\right] \\ & =\exp \left(-\frac{2 m w}{\hbar} \sqrt{2 g H-v^2}\right), \end{aligned}
where
$$\frac{2 m w}{\hbar} \sqrt{2 g H-v^2} \approx 0.9 \times 10^{30} \text {. }$$
Hence
$$T \sim e^{-0.9 \times 10^{30}} \approx 0$$
That is, the tunneling probability for the marble is essentially zero.
(c) The de Broglie wavelength of the tennis ball is
$$\lambda=\mathrm{h} / \mathrm{p}=h / m v=1.3 \times 10^{-30} \mathrm{~cm},$$

Problems and Solutions on Electromagnetism
and the diffraction angles in the horizontal and the vertical directions are respectively
$$\theta_1 \approx \lambda / D=1.3 \times 10^{-32} \mathrm{rad}, \quad \theta_2 \approx \lambda / L=9 \times 10^{-33} \mathrm{rad} .$$
Thus there is no diffraction in any direction.

Problem 2 Let $G=(V, E)$ be a graph and let $\mathcal{I}=\{J \subseteq E:|J \cap E(U)| \leq 2|U|-3, U \subseteq$ $V,|U|>1\}$. Show that $(E, \mathcal{I})$ is a matroid.

In order to show that $(E, \mathcal{I})$ is a matroid, we need to verify the following three axioms of a matroid:

1. **The empty set is independent:** It’s clear that the empty set satisfies the condition for all subsets $U$ of $V$, hence $\emptyset \in \mathcal{I}$.

2. **Any subset of an independent set is independent:** Let $I$ be an independent set and $J$ be any subset of $I$. For all $U \subseteq V$, we have $|J \cap E(U)| \leq |I \cap E(U)| \leq 2|U| – 3$. Hence, $J \in \mathcal{I}$.

3. **If $A$ and $B$ are in $\mathcal{I}$ and $|A|<|B|$, then there exists an element in $B$ but not in $A$ that can be added to $A$ without violating the property of $\mathcal{I}$:** Let’s consider two independent sets $A$ and $B$ in $\mathcal{I}$ with $|A|<|B|$. We need to find an edge $e \in B \setminus A$ such that $A \cup \{e\} \in \mathcal{I}$. Let’s denote by $U$ the vertices incident to $e$.

– If $|U|=1$, then $|A \cup \{e\} \cap E(U)| \leq 2|U|-3 = -1$, a contradiction, which implies that there is no edge with a single vertex in the graph.

– If $|U|>1$, then we have $|A \cup \{e\} \cap E(U)| \leq |A \cap E(U)| + 1 \leq 2|U| – 2 \leq 2|U| – 3$ because the addition of the edge $e$ could at most add one to the number of edges in $A$ that are incident to vertices in $U$. This implies $A \cup \{e\} \in \mathcal{I}$.

Therefore, $(E, \mathcal{I})$ satisfies all the properties of a matroid.

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