Np Completeness Continued | CS 139: Theory of Computation

<h2 id="np-completeness">NP Completeness</h2>
<h1 id="continued">Continued</h1>

<hr />

<h2 id="assignment-9">Assignment 9</h2>
<ul>
  <li><strong>Due:</strong> Thursday, May 7th, at 11:59 PM</li>
  <li class="fragment">Suggested Ways to Record
    <ul>
      <li class="fragment">Integrated into Slack is a video calling feature using <strong>Microsoft Teams</strong>
        <ul>
          <li>Has a built-in <strong>recording</strong> option</li>
          <li>Could call your partner, show slides by sharing screen, and present the material</li>
        </ul>
      </li>
      <li class="fragment"><strong>Blackboard Collaborate</strong>
        <ul>
          <li>I can enable presenting and video recording for students on-demand</li>
        </ul>
      </li>
    </ul>
  </li>
  <li class="fragment">Send me a link to your video when you are done</li>
</ul>

<h2 id="final-exam">Final Exam</h2>
<ul>
  <li><strong>Final Exam</strong> is on Wednesday, May 13th
    <ul>
      <li><strong>Open</strong> notes, <strong>open</strong> text, 110 minute exam</li>
      <li>Covers Sipser chapters 1, 3, 4, 5, and 7</li>
    </ul>
  </li>
  <li class="fragment">Exam procedures:
    <ul>
      <li class="fragment">Administered via a Gradescope feature that allows timed assessments <strong>without</strong> printing or scanning</li>
      <li class="fragment">You may take the exam at any time from
        <ul>
          <li>12:00 AM CDT to 11:59 PM CDT</li>
        </ul>
      </li>
      <li class="fragment">Once you start, you will have 110 minutes to complete the exam</li>
      <li class="fragment">You may <strong>not</strong> use Google or receive help from another other source other than your notes and the text</li>
    </ul>
  </li>
</ul>

<h2 id="final-exam-1">Final Exam</h2>
<ul>
  <li>Old Practice problems from Sipser:
    <ul>
      <li><strong>Chapter 1</strong>: 2, 6, 7, 11, 12, 14, 18, 20, 22, 29, 30, 31, 38, 40, 43, 46</li>
      <li><strong>Chapter 3</strong>: 7, 8, 15, 16</li>
      <li><strong>Chapter 4</strong>: 2, 3, 4, 5, 10, 12, 13, 21</li>
      <li><strong>Chapter 5</strong>: 4, 5, 6, 7</li>
    </ul>
  </li>
  <li>New Practice problems from Sipser:
    <ul>
      <li><strong>Chapter 5</strong>: 9, 10, 14, 20, 22</li>
      <li><strong>Chapter 7</strong>: 1, 5, 6, 7, 9, 10, 16, 18, 20</li>
    </ul>
  </li>
</ul>

<h1 id="np-completeness-1">NP Completeness</h1>

<h2 id="np-completeness-2">NP Completeness</h2>
<ul>
  <li>A language <code class="language-plaintext highlighter-rouge">$B$</code> is called <strong>NP hard</strong> if <code class="language-plaintext highlighter-rouge">$A\le_p B$</code> holds for each language <code class="language-plaintext highlighter-rouge">$A\in\text{NP}$</code></li>
  <li><code class="language-plaintext highlighter-rouge">$B$</code> is called <strong>NP complete</strong> if the following are true:
    <ol>
      <li><code class="language-plaintext highlighter-rouge">$B\in\text{NP}$</code></li>
      <li><code class="language-plaintext highlighter-rouge">$B$</code> is NP hard</li>
    </ol>
  </li>
</ul>

<hr />

<ul>
  <li><code class="language-plaintext highlighter-rouge">$A \le_p B$</code> means there is a poly-time computable function <code class="language-plaintext highlighter-rouge">$f$</code>, where for every <code class="language-plaintext highlighter-rouge">$w\in\Sigma^\ast$</code>:
    <ul>
      <li><code class="language-plaintext highlighter-rouge">$w\in A$</code> if and only if <code class="language-plaintext highlighter-rouge">$f(w) \in B$</code></li>
    </ul>
  </li>
</ul>

<h2 id="np-complete-problems">NP Complete Problems</h2>
<ul>
  <li>We’ve now seen that the following problems are NP complete:
    <ul>
      <li><code class="language-plaintext highlighter-rouge">$\text{SAT}$</code></li>
      <li><code class="language-plaintext highlighter-rouge">$\text{CLIQUE}$</code></li>
      <li><code class="language-plaintext highlighter-rouge">$\text{IND-SET}$</code></li>
      <li><code class="language-plaintext highlighter-rouge">$\text{VERTEX-COVER}$</code></li>
    </ul>
  </li>
</ul>

<h2 id="hamiltonian-path">Hamiltonian Path</h2>
<ul>
  <li>The <strong>Directed Hamiltonian Path</strong> problem is:</li>
</ul>

<p><code class="language-plaintext highlighter-rouge">$$\text{HAMPATH} = \{\langle G, s, t\rangle \mid \text{$G$ has a simple path from $s$ to $t$}\\\text{that visits every vertex exactly once}\}$$</code></p>

<hr />

<ul>
  <li class="fragment"><strong>Claim:</strong> <code class="language-plaintext highlighter-rouge">$\text{HAMPATH}$</code> is NP complete</li>
</ul>

<h2 id="hamiltonian-path-1">Hamiltonian Path</h2>
<ul>
  <li><strong>3-2-1-GO:</strong> What do we need to show?
    <ul>
      <li>That it is in NP and that it is NP hard</li>
    </ul>
  </li>
  <li class="fragment">How do we show that it is in NP?
    <ul>
      <li class="fragment">Write a verifier for it</li>
    </ul>
  </li>
  <li class="fragment">What could we use as the certificate?
    <ul>
      <li class="fragment">Given a path <code class="language-plaintext highlighter-rouge">$C = (v_1, v_2, \ldots, v_n)$</code> as a certificate, we can check if it is indeed a Hamiltonian path</li>
    </ul>
  </li>
</ul>

<div class="language-plaintext fragment highlighter-rouge"><div class="highlight"><pre class="highlight"><code>V = On input &lt;G,s,t,C&gt
  1. Check if C = (v0, v1, ..., vn) is a sequence of vertices of G;
     if not, then REJECT
  2. Check if |C| = |V|; if not then REJECT
  3. Check if v0 = s and vn = t and that all the other vertices
     are unique
  4. For i = 0...n-1:
     a. Check if vi ---> v(i+1) is an edge in G; if not, then REJECT
  4. If we get here, then C is a valid Hamiltonian path, so ACCEPT
</code></pre></div></div>

<h2 id="hamiltonian-path-2">Hamiltonian Path</h2>
<ul>
  <li>How can we show that <code class="language-plaintext highlighter-rouge">$\text{HAMPATH}$</code> is NP hard?
    <ul>
      <li class="fragment">Reduce an NP complete problem to it</li>
    </ul>
  </li>
  <li class="fragment"><strong>3-2-1-GO:</strong> What problem should we try reducing to it?
    <ul>
      <li><code class="language-plaintext highlighter-rouge">$\text{SAT}$</code>, <code class="language-plaintext highlighter-rouge">$\text{3SAT}$</code>, <code class="language-plaintext highlighter-rouge">$\text{CLIQUE}$</code>, <code class="language-plaintext highlighter-rouge">$\text{IND-SET}$</code>, or <code class="language-plaintext highlighter-rouge">$\text{VERTEX-COVER}$</code></li>
      <li class="fragment">Surprisingly,<code class="language-plaintext highlighter-rouge">$\text{3SAT}$</code> is also the easiest here</li>
    </ul>
  </li>
  <li class="fragment"><strong>Claim:</strong> <code class="language-plaintext highlighter-rouge">$\text{3SAT} \le_p \text{HAMPATH}$</code></li>
</ul>

<h2 id="hamiltonian-path-3">Hamiltonian Path</h2>
<p><code class="language-plaintext highlighter-rouge">$\Phi = (x_1\lor x_2\lor \overline{x}_3)\land (\overline{x}_1\lor \overline{x}_1\lor x_3)$</code></p>

<hr />

<ul>
  <li class="fragment"><strong>Main Idea:</strong>
    <ul>
      <li class="fragment">For each variable <code class="language-plaintext highlighter-rouge">$x_i$</code>, create a <strong>gadget</strong> in the graph</li>
      <li class="fragment">For each clause such as <code class="language-plaintext highlighter-rouge">$(x_i \lor \overline{x}_j \lor x_n)$</code>, create a different <strong>gadget</strong> in the graph</li>
    </ul>
  </li>
</ul>

<h2 id="undirected-hamiltonian-path">Undirected Hamiltonian Path</h2>
<ul>
  <li>The <strong>Undirected Hamiltonian Path</strong> problem is:</li>
</ul>

<p><code class="language-plaintext highlighter-rouge">$$\text{UHAMPATH} = \{\langle G, s, t\rangle \mid \text{$G$ has a simple path from $s$ to $t$}\\\text{that visits every vertex exactly once}\}$$</code></p>

<ul>
  <li class="fragment">The only difference here is that <code class="language-plaintext highlighter-rouge">$G$</code> is an <strong>undirected</strong> graph instead of a directed one</li>
</ul>

<hr />

<ul>
  <li class="fragment"><strong>Claim:</strong> <code class="language-plaintext highlighter-rouge">$\text{UHAMPATH}$</code> is NP complete</li>
</ul>

<h2 id="undirected-hamiltonian-path-1">Undirected Hamiltonian Path</h2>
<ul>
  <li>Obviously <code class="language-plaintext highlighter-rouge">$\text{UHAMPATH}$</code> is in NP—verifier would be nearly identical to <code class="language-plaintext highlighter-rouge">$\text{HAMPATH}$</code>’s</li>
  <li class="fragment">Can we show <code class="language-plaintext highlighter-rouge">$\text{HAMPATH}\le_p\text{UHAMPATH}$</code>?</li>
</ul>

<h2 id="double-sat">Double SAT</h2>
<ul>
  <li>Show that the following language is NP complete
<code class="language-plaintext highlighter-rouge">$$\text{DOUBLE-SAT} = \{\langle \Phi\rangle \mid \text{$\Phi$ has at least two}\\\text{unique satisfying assignments }\}$$</code></li>
</ul>

<h2 id="double-sat-1">Double SAT</h2>

<p><code class="language-plaintext highlighter-rouge">$$\text{DOUBLE-SAT} = \{\langle \Phi\rangle \mid \text{$\Phi$ has at least two}\\\text{unique satisfying assignments }\}$$</code></p>

<hr />

<ul>
  <li><strong>Exercise:</strong> Show that it is in NP
    <ul>
      <li class="fragment"><strong>Idea:</strong> Have a machine nondeterministically select two assignments, make sure they’re different, and check if they both satisfy the formula</li>
    </ul>
  </li>
  <li class="fragment"><strong>Exercise:</strong> Show that it is NP hard
    <ul>
      <li class="fragment">Show that <code class="language-plaintext highlighter-rouge">$\text{3SAT} \le_p \text{DOUBLE-SAT}$</code></li>
    </ul>
  </li>
</ul>