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>You all should have received a message from me today on Slack or email assigning you a partner and a problem</li>
  <li class="fragment">This is a significantly different assignment, where you will create a <strong>video presentation</strong> about a certain NP complete problem</li>
  <li class="fragment">The video itself is due next week on <strong>Thursday, May 7th</strong></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 class="fragment"><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>

<hr />

<ul>
  <li class="fragment"><strong>Theorem</strong> (Cook, Levin): <code class="language-plaintext highlighter-rouge">$\text{SAT}$</code> is NP complete</li>
</ul>

<h2 id="our-first-reduction">Our First Reduction</h2>
<ul>
  <li>Last time we showed that <code class="language-plaintext highlighter-rouge">$\text{CLIQUE} \le_p \text{IND-SET}$</code> which are the languages:</li>
</ul>

<hr />

<p><code class="language-plaintext highlighter-rouge">$$\text{CLIQUE} = \{\langle G, k\rangle \mid \text{$G$ contains a clique of}\\\text{size at least $k$}\}$$</code></p>

<hr />

<p><code class="language-plaintext highlighter-rouge">$$\text{IND-SET} = \{\langle G, k\rangle \mid \text{$G$ contains an independent}\\\text{set of size at least $k$}\}$$</code></p>

<h2 id="our-first-reduction-1">Our First Reduction</h2>
<ul>
  <li><strong>Main Idea:</strong> Modify the graph <code class="language-plaintext highlighter-rouge">$G=(V,E)$</code> by “inverting” its edges to create <code class="language-plaintext highlighter-rouge">$\widehat{G} = (\widehat{V}, \widehat{E})$</code>
    <ul>
      <li>If <code class="language-plaintext highlighter-rouge">$\{x,y\}\in E$</code>, then <code class="language-plaintext highlighter-rouge">$\{x,y\}\not\in\widehat{E}$</code></li>
      <li>If <code class="language-plaintext highlighter-rouge">$\{x,y\}\not\in E$</code>, then <code class="language-plaintext highlighter-rouge">$\{x,y\}\in\widehat{E}$</code></li>
      <li class="fragment">More concisely, <code class="language-plaintext highlighter-rouge">$\widehat{G} = (V, \overline{E})$</code></li>
    </ul>
  </li>
</ul>

<h2 id="our-first-reduction-2">Our First Reduction</h2>
<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>M_f = On input &lt;G,k&gt where G = (V,E);
  1. Construct G' = (V, E') by inverting the edges of G
  2. Output &lt;G',k&gt
</code></pre></div></div>

<ul>
  <li><code class="language-plaintext highlighter-rouge">$f$</code> runs in poly-time because it only needs to iterative over <code class="language-plaintext highlighter-rouge">$V$</code> and <code class="language-plaintext highlighter-rouge">$E$</code> once to construct <code class="language-plaintext highlighter-rouge">$G'$</code></li>
  <li>If <code class="language-plaintext highlighter-rouge">$G$</code> contains a <code class="language-plaintext highlighter-rouge">$k$</code>-clique, then the inverted graph must contain a <code class="language-plaintext highlighter-rouge">$k$</code>-independent set</li>
  <li>Similarly, if <code class="language-plaintext highlighter-rouge">$G'$</code> contains a <code class="language-plaintext highlighter-rouge">$k$</code>-independent set, then the original graph must contain a <code class="language-plaintext highlighter-rouge">$k$</code>-clique</li>
</ul>

<h2 id="textind-set-is-in-textnp"><code class="language-plaintext highlighter-rouge">$\text{IND-SET}$</code> is in <code class="language-plaintext highlighter-rouge">$\text{NP}$</code></h2>
<ul>
  <li>We also discussed that <code class="language-plaintext highlighter-rouge">$\text{IND-SET}\in\text{NP}$</code> because we can use the following verifier:</li>
</ul>

<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>V = "On input &lt;G,k,C&gt; where G=(V,E) is an undirected graph
  1. If C is not a description of a subset of V, then REJECT
  2. If |C| < k, then REJECT
  3. For each vertex u in C do:
     a. For each vertex v in C do:
        i. If u is connected to v by an edge in E, then REJECT
  4. If we get here, then C is an independent set of size >= k,
     so ACCEPT"
</code></pre></div></div>

<h2 id="other-np-complete-problems">Other NP Complete Problems</h2>
<p><code class="language-plaintext highlighter-rouge">$$\text{3SAT} = \{ \langle\Phi\rangle\mid\Phi\text{ is a 3-CNF satisfiable formula}\}$$</code></p>

<ul>
  <li class="fragment">Boolean formulas in <strong>conjunctive normal form</strong> (<strong>CNF</strong>) are like: <code class="language-plaintext highlighter-rouge">$\Phi = (x_1\lor x_2\lor \overline{x_3})\land(x_4\lor \overline{x_2}\lor \overline{x_1})$</code>
    <ul>
      <li class="fragment">Each triplet of disjunction is called a <strong>clause</strong></li>
    </ul>
  </li>
</ul>

<hr />

<ul>
  <li class="fragment"><strong>Theorem</strong>: <code class="language-plaintext highlighter-rouge">$\text{SAT}\le_p\text{3SAT}$</code>
    <ul>
      <li class="fragment">…so it is NP hard</li>
      <li class="fragment">…and since it is also in NP, it is NP complete</li>
    </ul>
  </li>
</ul>

<h2 id="other-np-complete-problems-1">Other NP Complete Problems</h2>
<ul>
  <li><strong>Theorem</strong>: <code class="language-plaintext highlighter-rouge">$\text{3SAT} \le_p \text{CLIQUE}$</code>
    <ul>
      <li class="fragment">…so <code class="language-plaintext highlighter-rouge">$\text{CLIQUE}$</code> is NP hard</li>
      <li class="fragment">…and since it is also in NP, it is NP complete</li>
    </ul>
  </li>
</ul>

<hr />

<ul>
  <li class="fragment"><strong>Corollary</strong>: <code class="language-plaintext highlighter-rouge">$\text{IND-SET}$</code> is NP complete
    <ul>
      <li class="fragment">…it is in NP because it has a verifier</li>
      <li class="fragment">…it is NP hard since <code class="language-plaintext highlighter-rouge">$\text{CLIQUE}\le_p\text{IND-SET}$</code> and <code class="language-plaintext highlighter-rouge">$\text{CLIQUE}$</code> is NP complete</li>
    </ul>
  </li>
</ul>

<h1 id="more-reduction-examples">More Reduction Examples</h1>

<h2 id="vertex-cover">Vertex Cover</h2>
<ul>
  <li>A <strong>vertex cover</strong> of an undirected graph <code class="language-plaintext highlighter-rouge">$G = (V,E)$</code> is a set <code class="language-plaintext highlighter-rouge">$C\subseteq V$</code> such that for every edge <code class="language-plaintext highlighter-rouge">$\{x,y\}\in E$</code>, either <code class="language-plaintext highlighter-rouge">$x\in C$</code> or <code class="language-plaintext highlighter-rouge">$y \in C$</code>.</li>
  <li class="fragment">The <strong>Vertex Cover Problem</strong> is the language:</li>
</ul>

<p class="fragment"><code class="language-plaintext highlighter-rouge">$$\text{VERTEX-COVER} = \{\langle G, k\rangle \mid \text{$G$ contains}\\\text{a vertex cover of size at least $k$}\}$$</code></p>

<hr />

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

<h2 id="vertex-cover-1">Vertex Cover</h2>
<ul>
  <li><strong>3-2-1-GO:</strong> What do we need to show?
    <ul>
      <li class="fragment">That it is in <code class="language-plaintext highlighter-rouge">$\text{NP}$</code> and that it is NP hard</li>
    </ul>
  </li>
  <li class="fragment">How do we show that it is in <code class="language-plaintext highlighter-rouge">$\text{NP}$</code>?
    <ul>
      <li class="fragment">Write a verifier for it</li>
      <li class="fragment">Given a set <code class="language-plaintext highlighter-rouge">$C\subseteq V$</code> as a certificate, we can check if it is indeed a vertex cover</li>
    </ul>
  </li>
</ul>

<div class="language-plaintext fragment highlighter-rouge"><div class="highlight"><pre class="highlight"><code>V = On input &lt;G,k,C&gt
  1. Check if C is a subset of the vertices of G; if not, REJECT
  2. Check if |C| < k, if so REJECT
  3. For each edge {x,y} of G:
     a. Check if x or y are in C; if neither are, then REJECT
  4. If all edges are covered by C, then ACCEPT
</code></pre></div></div>

<h2 id="vertex-cover-2">Vertex Cover</h2>
<ul>
  <li><strong>3-2-1-GO:</strong> How do we show that it is NP hard?
    <ul>
      <li class="fragment">Reduce an NP complete problem to it</li>
    </ul>
  </li>
</ul>

<hr />

<ul>
  <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>, or <code class="language-plaintext highlighter-rouge">$\text{IND-SET}$</code></li>
      <li class="fragment">Although <code class="language-plaintext highlighter-rouge">$\text{CLIQUE}$</code> and <code class="language-plaintext highlighter-rouge">$\text{IND-SET}$</code> look similar, it is surprisingly hard to reduce one of them directly</li>
      <li class="fragment">Surprisingly, <code class="language-plaintext highlighter-rouge">$\text{3SAT}$</code> is a versatile language that can be reduced to a large variety of problems</li>
    </ul>
  </li>
  <li class="fragment"><strong>Claim:</strong> <code class="language-plaintext highlighter-rouge">$\text{3SAT} \le_p \text{VERTEX-COVER}$</code></li>
</ul>

<h2 id="vertex-cover-3">Vertex Cover</h2>
<ul>
  <li>To complete the reduction, we need to convert a formula <code class="language-plaintext highlighter-rouge">$\langle\Phi\rangle$</code> into a <code class="language-plaintext highlighter-rouge">$\langle G, k\rangle$</code> such that
    <ul>
      <li><code class="language-plaintext highlighter-rouge">$\Phi$</code> is satisfiable <code class="language-plaintext highlighter-rouge">$\iff$</code> <code class="language-plaintext highlighter-rouge">$G$</code> has a vertex cover of size <code class="language-plaintext highlighter-rouge">$k$</code></li>
    </ul>
  </li>
</ul>

<hr />

<ul>
  <li class="fragment">Consider <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>
    <ul>
      <li class="fragment">This formula has three <strong>variables</strong> and two <strong>clauses</strong></li>
      <li class="fragment">For our reduction to work, we will need to preserve this information somehow in the graph</li>
      <li class="fragment">Ideally, any satisfying assignment to <code class="language-plaintext highlighter-rouge">$\Phi$</code> should immediately yield a vertex cover</li>
      <li class="fragment">Similarly, any vertex cover should immediately yield a satisfying assignment</li>
    </ul>
  </li>
</ul>

<h2 id="vertex-cover-4">Vertex Cover</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>