The Class Np | CS 139: Theory of Computation

<h1 id="the-class-np">The Class NP</h1>

<hr />

<h2 id="administrivia">Administrivia</h2>
<ul>
  <li>Assignment 8 is due Tuesday at 11:59 PM</li>
</ul>

<h1 id="brief-review">Brief Review</h1>

<h2 id="big-o-notation">Big-O Notation</h2>
<ul>
  <li><code class="language-plaintext highlighter-rouge">$f(n) = O(g(n))$</code> for two functions <code class="language-plaintext highlighter-rouge">$f,g:\mathbb{N}\rightarrow\mathbb{R}^+$</code> if:
    <ul>
      <li><strong>THERE EXIST</strong> positive integers <code class="language-plaintext highlighter-rouge">$c$</code> and <code class="language-plaintext highlighter-rouge">$n_0$</code> such that:
        <ul>
          <li><strong>FOR EVERY</strong> integer <code class="language-plaintext highlighter-rouge">$n \ge n_0$</code> we have:
            <ul>
              <li><code class="language-plaintext highlighter-rouge">$f(n) \le c\cdot g(n)$</code></li>
            </ul>
          </li>
        </ul>
      </li>
    </ul>
  </li>
  <li class="fragment">Intuitively it means that <strong>eventually</strong> <code class="language-plaintext highlighter-rouge">$g(n)$</code> will grow <strong>at least as fast</strong> as <code class="language-plaintext highlighter-rouge">$f(n)$</code></li>
</ul>

<h2 id="time-complexity">Time Complexity</h2>
<ul>
  <li>Let <code class="language-plaintext highlighter-rouge">$M$</code> be a Turing machine that always halts</li>
  <li>The (<strong>worst-case</strong>) <strong>running time</strong> of <code class="language-plaintext highlighter-rouge">$M$</code> is the function <code class="language-plaintext highlighter-rouge">$f:\mathbb{N}\rightarrow\mathbb{N}$</code> where <code class="language-plaintext highlighter-rouge">$f(n)$</code> is the <em>maximum</em> number of steps <code class="language-plaintext highlighter-rouge">$M$</code> takes on an input of length <code class="language-plaintext highlighter-rouge">$n$</code></li>
</ul>

<h2 id="time-complexity-1">Time Complexity</h2>
<ul>
  <li>Let <code class="language-plaintext highlighter-rouge">$t:\mathbb{N}\rightarrow\mathbb{R}^+$</code> be a function</li>
  <li>Then <code class="language-plaintext highlighter-rouge">$\text{TIME}(t(n))$</code> is the collection of languages that are decidable by a single-tape deterministic Turing machine in <code class="language-plaintext highlighter-rouge">$O(t(n)$</code> time</li>
</ul>

<ul>
  <li><code class="language-plaintext highlighter-rouge">$\text{P}$</code> is the set of the languages decidable in <strong>polynomial time</strong> and are considered “easy” to solve</li>
</ul>

<p><code class="language-plaintext highlighter-rouge">$$\text{P} = \bigcup_{k\in\mathbb{N}} \text{TIME}(n^k)$$</code></p>

<ul>
  <li class="fragment"><code class="language-plaintext highlighter-rouge">$\text{EXPTIME}$</code> is the set of languages decidable in <strong>exponential time</strong> and are considered “intractable”</li>
</ul>

<p class="fragment"><code class="language-plaintext highlighter-rouge">$$\text{EXPTIME} = \bigcup_{k\in\mathbb{N}} \text{TIME}(2^{n^k})$$</code></p>

<h2 id="exercise">Exercise</h2>
<ul>
  <li>An undirected graph <code class="language-plaintext highlighter-rouge">$G = (V,E)$</code> <strong>contains a triangle</strong> if there is a fully connected subgraph of size 3</li>
  <li>Show that the following language is in <code class="language-plaintext highlighter-rouge">$\text{P}$</code>:</li>
</ul>

<p><code class="language-plaintext highlighter-rouge">$$\text{TRIANGLE} = \{\langle G\rangle \mid G\text{ contains a triangle}\}$$</code></p>

<h2 id="exercise-1">Exercise</h2>

<div class="language-plaintext highlighter-rouge"><div class="highlight"><pre class="highlight"><code>M = "On input &lt;G&gt; where G = (V, E) is an undirected graph:
  1. For each u in V:
     A. For each v in V:
        i. For each w in V
           a. Check if {u,v}, {u,w}, and {v,w} are in E
           b. If so, check if u != v and v != w, if so then ACCEPT
  2. If all triples are not fully connected, then REJECT"
</code></pre></div></div>

<h1 id="the-class-np-1">The Class NP</h1>

<h2 id="super-mario-bros">Super Mario Bros</h2>
<p><strong>3-2-1-GO:</strong> How would you prove the following claim?</p>

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<ul>
  <li><strong>Claim:</strong> Super Mario Bros for NES can be completed in less than 5 minutes—from start to finish.</li>
</ul>

<hr />

<ul>
  <li class="fragment">…by doing a speedrun in under 5 minutes, recording it, and sending it to me to watch it</li>
</ul>

<h2 id="finding-versus-checking">Finding Versus Checking</h2>
<ul>
  <li><strong>Finding</strong> a route to beat Mario in five minutes is <strong>hard</strong> and can take hundreds of hours of practice planning</li>
  <li class="fragment"><strong>Checking</strong> if a speedrun is under five minutes is <strong>easy</strong>—at most it requires watching a five minute video</li>
</ul>

<h2 id="another-example-satisfiability">Another Example: Satisfiability</h2>

<ul>
  <li>Recall that a <strong>Boolean formula</strong> is consists of one or more variables using AND, OR, and NOT operators such as:</li>
</ul>

<ul>
  <li class="fragment"><strong>3-2-1-GO:</strong> Is there an assignment of Boolean values to <code class="language-plaintext highlighter-rouge">$x$</code> and <code class="language-plaintext highlighter-rouge">$y$</code> that causes <code class="language-plaintext highlighter-rouge">$\Phi$</code> to be true?
    <ul>
      <li class="fragment">Yes!  Here it is: <code class="language-plaintext highlighter-rouge">$x = \text{True}$</code> and <code class="language-plaintext highlighter-rouge">$y = \text{False}$</code></li>
    </ul>
  </li>
  <li class="fragment">To convince a skeptic that <code class="language-plaintext highlighter-rouge">$\Phi$</code> is satisfiable, we need only show them the assignment and let them <strong>verify</strong> it works</li>
</ul>

<h2 id="boolean-satisfiability-problem">Boolean Satisfiability Problem</h2>
<p>The <strong>satisfiability problem</strong> is the language:</p>

<p><code class="language-plaintext highlighter-rouge">$$\text{SAT} = \{ \langle\Phi\rangle\mid\Phi\text{ is a satisfiable Boolean formula}\}$$</code></p>

<hr />

<ul>
  <li class="fragment"><strong>Interesting Facts</strong>:
    <ul>
      <li class="fragment">Given a formula <code class="language-plaintext highlighter-rouge">$\Phi$</code>, we don’t know of a polynomial time algorithm that decides whether <code class="language-plaintext highlighter-rouge">$\Phi\in\text{SAT}$</code></li>
      <li class="fragment">However, for every <code class="language-plaintext highlighter-rouge">$\Phi\in\text{SAT}$</code>, there is an assignment that can be <strong>verified</strong> in polynomial time</li>
    </ul>
  </li>
</ul>

<h2 id="language-verifiers">Language Verifiers</h2>
<ul>
  <li>Let <code class="language-plaintext highlighter-rouge">$A$</code> be a language</li>
  <li>A <strong>verifier</strong> for <code class="language-plaintext highlighter-rouge">$A$</code> is a Turing machine <code class="language-plaintext highlighter-rouge">$V_A$</code> that satisfies:
    <ul>
      <li><code class="language-plaintext highlighter-rouge">$w\in A$</code> if and only if:
        <ul>
          <li><strong>THERE EXISTS</strong> a string <code class="language-plaintext highlighter-rouge">$c$</code> such that:
            <ul>
              <li><code class="language-plaintext highlighter-rouge">$V_A$</code> accepts the pair <code class="language-plaintext highlighter-rouge">$\langle w, c\rangle$</code></li>
            </ul>
          </li>
        </ul>
      </li>
    </ul>
  </li>
</ul>

<hr />

<ul>
  <li class="fragment">The string <code class="language-plaintext highlighter-rouge">$c$</code> is called a <strong>certificate</strong> or <strong>proof</strong> or <strong>witeness</strong> that testifies to <code class="language-plaintext highlighter-rouge">$w\in A$</code></li>
</ul>

<h2 id="the-class-np-2">The Class NP</h2>
<ul>
  <li><code class="language-plaintext highlighter-rouge">$\text{NP}$</code> is the class of languages with <strong>polynomial time verifiers</strong></li>
</ul>

<hr />

<h2 id="sat-is-in-np">SAT is in NP</h2>
<ul>
  <li>Remember how we could verify that <code class="language-plaintext highlighter-rouge">$\Phi\in\text{SAT}$</code> if we were given a satisfying assignment as proof?
    <ul>
      <li class="fragment">We can formalize this as an algorithm!</li>
    </ul>
  </li>
</ul>

<hr />

<div class="language-plaintext fragment highlighter-rouge"><div class="highlight"><pre class="highlight"><code>V_SAT = "On input &lt;F,c&gt; where F is a Boolean formula
  1. If c is not a description of an assignment to the
     variables of F, then REJECT
  2. Evaluate the formula F by substituting the assignment
     encoded in the certificate c
  3. If F evaluates to true, then ACCEPT; otherwise REJECT"
</code></pre></div></div>

<hr />

<ul>
  <li class="fragment">Since all of these steps run in polynomial time, this verifier shows that <code class="language-plaintext highlighter-rouge">$\text{SAT}\in\text{NP}$</code></li>
</ul>

<h2 id="the-class-np-3">The Class NP</h2>
<ul>
  <li><code class="language-plaintext highlighter-rouge">$\text{NP}$</code> stands for <strong>nondeterministic polynomial time</strong></li>
  <li class="fragment">A language <code class="language-plaintext highlighter-rouge">$A$</code> is in <code class="language-plaintext highlighter-rouge">$\text{NP}$</code> if and only if it can be decided by a nondeterministic polynomial time Turing machine</li>
</ul>

<p class="fragment"><code class="language-plaintext highlighter-rouge">$$\text{NP} = \bigcup_{k\in\mathbb{N}}\text{NTIME}(n^k)$$</code></p>

<ul class="fragment">
  <li><code class="language-plaintext highlighter-rouge">$\text{NTIME}(t(n))$</code> are the languages decidable by a nondeterministic Turing machine in <code class="language-plaintext highlighter-rouge">$O(t(n)$</code> time</li>
</ul>

<h2 id="nondeterministic-tm-for-sat">Nondeterministic TM for SAT</h2>
<ul>
  <li>We could have also proved that <code class="language-plaintext highlighter-rouge">$\text{SAT}\in\text{NP}$</code> by giving a polynomial time nondeterministic decider for it</li>
</ul>

<div class="language-plaintext fragment highlighter-rouge"><div class="highlight"><pre class="highlight"><code>N = "On input &lt;F&gt; where F is a Boolean formula
  1. For each variable x_i in F:
     a. Nondeterministically select an assignment of True or
        False for the variable x
  2. Evaluate the formula F by substituting the assignment
     we generated in step 1
  3. If F evaluates to true, then ACCEPT; otherwise REJECT""
</code></pre></div></div>

<ul>
  <li class="fragment">Notice that <code class="language-plaintext highlighter-rouge">$N$</code> nondeterministically verifies <strong>all possible assignments</strong> at the same time</li>
</ul>

<h2 id="exercise-traveling-salesman">Exercise: Traveling Salesman</h2>
<ul>
  <li>Suppose there are <code class="language-plaintext highlighter-rouge">$n$</code> cities and each city has an airport that can fly to all other cities</li>
  <li class="fragment">The <strong>cost</strong> of traveling from city <code class="language-plaintext highlighter-rouge">$i$</code> to city <code class="language-plaintext highlighter-rouge">$j$</code> is the positive number <code class="language-plaintext highlighter-rouge">$C(i,j)$</code> where <code class="language-plaintext highlighter-rouge">$C$</code> is a <code class="language-plaintext highlighter-rouge">$n\times n$</code> matrix</li>
  <li class="fragment">A <strong>route</strong> <code class="language-plaintext highlighter-rouge">$r = (c_1, c_2, \ldots c_m)$</code> is a finite sequence of the cities that visits each city at least once</li>
  <li class="fragment">The <strong>total cost</strong> of the route <code class="language-plaintext highlighter-rouge">$r$</code> is the sum <code class="language-plaintext highlighter-rouge">$c_r = \sum_{i=1}^{m-1}c_{i, i+1}$</code></li>
</ul>

<p class="fragment"><code class="language-plaintext highlighter-rouge">$$\text{TSP} = \{\langle C,k\rangle \mid\text{ there is a route $r$ with $c_r \le k$}\}$$</code></p>

<h2 id="exercise-traveling-salesman-1">Exercise: Traveling Salesman</h2>
<p><code class="language-plaintext highlighter-rouge">$$\text{TSP} = \{\langle C,k\rangle \mid\text{ there is a route $r$ with $c_r \le k$}\}$$</code></p>

<hr />

<ul>
  <li>Show that <code class="language-plaintext highlighter-rouge">$\text{TSP}\in\text{NP}$</code> using both a verifier and nondeterministic Turing machine</li>
  <li class="fragment">What would the verifier look like?</li>
  <li class="fragment">What would the nondeterministic TM look like?</li>
</ul>

<h1 id="p-versus-np">P Versus NP</h1>

<h2 id="p-versus-np-1">P Versus NP</h2>
<ul>
  <li>The biggest open problem in computer science right now is the following question:Is <code class="language-plaintext highlighter-rouge">$\text{P} = \text{NP}$</code>?</li>
  <li class="fragment">Clearly it is true that <code class="language-plaintext highlighter-rouge">$P \subseteq NP$</code> since every language <em>solvable</em> in poly time can be <em>checked</em> in poly time (just compute the answer and compare)</li>
</ul>

<hr />

<ul>
  <li class="fragment">Philosophically, the P versus NP question is asking:
    <ul>
      <li>Is <strong>finding</strong> a solution harder than <strong>verifying</strong> a solution”?</li>
    </ul>
  </li>
  <li class="fragment">Most computer scientists think that <code class="language-plaintext highlighter-rouge">$\text{P} \ne \text{NP}$</code></li>
</ul>

<h2 id="p-versus-np-2">P Versus NP</h2>
<ul>
  <li>How can we solve the <code class="language-plaintext highlighter-rouge">$\text{P}$</code> versus <code class="language-plaintext highlighter-rouge">$\text{NP}$</code> problem?</li>
  <li class="fragment">It sounds so easy!
    <ol>
      <li class="fragment">To show that <code class="language-plaintext highlighter-rouge">$\text{P} \ne \text{NP}$</code>, all we need is <strong>one language</strong> that is in <code class="language-plaintext highlighter-rouge">$\text{NP}$</code> but not in <code class="language-plaintext highlighter-rouge">$\text{P}$</code></li>
      <li class="fragment">To show that <code class="language-plaintext highlighter-rouge">$\text{P} = \text{NP}$</code>, you only need to come up with general strategy for doing Mario speedruns</li>
    </ol>
  </li>
</ul>

<hr />

<ul>
  <li class="fragment"><strong>Theorem:</strong> <code class="language-plaintext highlighter-rouge">$\text{P} = \text{NP}$</code> if and only if general Mario levels can be speedrun efficiently</li>
  <li class="fragment"><strong>Theorem:</strong> <code class="language-plaintext highlighter-rouge">$\text{P} = \text{NP}$</code> if and only if <code class="language-plaintext highlighter-rouge">$\text{SAT}\in\text{P}$</code></li>
</ul>