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[Algorithms II] Week 6-3 Intractability
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-introduction-to-intractability" title="1. Introduction to Intractability">1. Introduction to Intractability</a></li><li><a class="toc-href" href="#2-search-problems" title="2. Search Problems">2. Search Problems</a></li><li><a class="toc-href" href="#3-p-vs-np" title="3. P vs. NP">3. P vs. NP</a></li><li><a class="toc-href" href="#4-classifying-problems" title="4. Classifying Problems">4. Classifying Problems</a><ul><li><a class="toc-href" href="#sat-poly-reduces-to-ilp" title="SAT poly-reduces to ILP">SAT poly-reduces to ILP</a></li></ul></li><li><a class="toc-href" href="#5-np-completeness_1" title="5. NP-Completeness">5. NP-Completeness</a></li><li><a class="toc-href" href="#6-coping-with-intractability" title="6. Coping with Intractability">6. Coping with Intractability</a><ul><li><a class="toc-href" href="#exploit-intractability" title="exploit intractability">exploit intractability</a></li><li><a class="toc-href" href="#coping-with-intractability" title="Coping with intractability">Coping with intractability</a></li><li><a class="toc-href" href="#halmiton-path" title="Halmiton path">Halmiton path</a></li></ul></li></ul></div>
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<h1 id="1-introduction-to-intractability">1. Introduction to Intractability</h1>
<p>recall model of computation: DFA <br/>
a <em>univeral</em> model of computation: turing machine <br/>
→ no more powerful model of computation. <br/>
Turing machine can compute any function that can be computed by a physically harnessable process of the natural world. </p>
<p>bottom line: turing machine is a simple and universal model of computation. </p>
<p>Q. which algos are <em>useful in practice</em>? </p>
<p>useful in practice = polynomial time for all inputs </p>
<p>def. a pb is <strong>intractable</strong> if it cannot be solved in polynomial time. </p>
<p>2 pbs that <em>can be proved</em> to require exp time: </p>
<ol>
<li>Given a constant-size programme, does it halt in <=K steps ? </li>
<li>Given a N*N chess board position, can the first player force a win ? </li>
</ol>
<p>Bad news: very few pbs can be proved to require exp time... </p>
<h1 id="2-search-problems">2. Search Problems</h1>
<p><strong>Four fundamental problems: </strong> </p>
<ul>
<li><strong>LSLOVE</strong> </li>
</ul>
<p>Given a system of linear equations, find a solution <br/>
var: real numbers <br/>
→ guassian elimination </p>
<ul>
<li><strong>LP</strong> </li>
</ul>
<p>Given a system of linear inequaties, find a solution. (not necessarily find the opt) <br/>
var: real numbers </p>
<ul>
<li><strong>ILP</strong> </li>
</ul>
<p>Given a system of linear inequaties, find a <strong>0-1</strong> solution. <br/>
var: 0 or 1 </p>
<ul>
<li><strong>SAT</strong> </li>
</ul>
<p>Given a system of <em>boolean equations</em>, find a binary solution. </p>
<p>Which ones of the 4 foundamental pbs have poly-time solutions? </p>
<ul>
<li>LSLOVE: Gaussian elimination works in O(n3) </li>
<li>LP: Ellipsoid works in poly-time (simplex also poly-time <em>in practice</em>..) </li>
<li>ILP, SAT: No poly-time algorithm known (or believed to exist) ! </li>
</ul>
<p>All 4 pbs are examples of search problems. </p>
<p><strong>Search pb</strong>: given an instance <code>I</code>, find a solution <code>S</code> / report there's no solution. <br/>
<em>requirement</em>: able to efficiently (poly-time) <em>check</em> that <code>S</code> is a solution. (that's the case for the above 4 fundamental pbs) </p>
<p>another example: <br/>
<strong>FACTOR</strong>: given a n-bit integer, find a nontrival factor. <br/>
(given a solution, simply need to long-divide to check...) </p>
<h1 id="3-p-vs-np">3. P vs. NP</h1>
<p>def. <strong>NP</strong> is the class of all search pbs. (ie. solution be checked efficiently) <br/>
NB: classical definition limits to yes-no pbs... <br/>
Significance: NP pbs are what scientists and engineers <em>aspire to compute feasibly</em>. <br/>
examples: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image.png"/> </p>
<p>def. <strong>P</strong> is the class of search pbs that <em>are solvable</em> in poly-time. </p>
<p>(What scientists and engineers <em>do compute feasibly</em>.) </p>
<p>examples: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image001.png"/> </p>
<p><strong>Nondeterminism</strong> <br/>
Nondeterminism machine can <em>guess</em> the solution (donot exist in natural world..). → NFA tries to simulate such a machine... <br/>
Ex. <code>int[] a = new int[N];</code> <br/>
・ Java: initializes entries to 0 . <br/>
・ Nondeterministic machine: <em>initializes entries to the solution!</em> <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image002.png"/> </p>
<p>NP: <em>Search problems solvable in poly time on a nondeterministic Turing machine</em>. </p>
<p>Extended Church-Turing thesis: <br/>
P: Search pbs solvable in poly time <em>in natural world</em>. </p>
<p>do we have non-determinism in natural world? ---> natural computers ? <br/>
ex. STEINER tree: set of segments connecting given N points. <br/>
use soap → doesn't really work... <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image003.png"/> </p>
<p>another example for P/NP: automating creativity <br/>
<em>being creative VS appreciating creativity</em> <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image004.png"/> </p>
<p>The central question: does P=NP? <br/>
(can you alway avoid brute-force searching and do better?) <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image005.png"/> <br/>
Millennium prize by Clay instute. <br/>
(among all ways of earning 1M dollars, this might be the most complicated way... @_@...) </p>
<h1 id="4-classifying-problems">4. Classifying Problems</h1>
<p>classify pbs like classifying elements into perodic table. </p>
<p>key pb: satisfiablity <br/>
SAT. given a sys of boolean eq, find a solution. </p>
<p>exhaustive search: try 2^n possible solutions. <br/>
conjecture: no poly-time algo for SAT (ie. intractable) </p>
<p><strong>Assumption</strong>: assume the intractability for SAT. <br/>
Tool: reduction <br/>
def. pb X reduces to pb Y: we can solve pb X with the algo for pb Y. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image006.png"/> </p>
<blockquote>
<p>if SAT poly-reduces to pb Y ⇒ pb Y in (probably) intractable. </p>
</blockquote>
<h3 id="sat-poly-reduces-to-ilp">SAT poly-reduces to ILP</h3>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image007.png"/> <br/>
(all SAT pb can be reduced to 3SAT) </p>
<p>⇒ can be converted to an ILP pb: <br/>
for each eq, introduce a var Ci: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image009.png"/> </p>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image008.png"/> </p>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image010.png"/> </p>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image011.png"/> </p>
<h1 id="5-np-completeness_1">5. NP-Completeness</h1>
<blockquote>
<p>def. an NP pb is <strong>NP-complete</strong> if all pbs in NP poly-reduces to it. </p>
</blockquote>
<p>prop. <em>SAT id NP-complete.</em> <br/>
any pb in NP poly-reduces to SAT (reverse direction as last lecture) <br/>
pf sketch: convert non-dertiministic turing machine notation to SAT notation... </p>
<p>cor. poly time algo for SAT iff P=NP... </p>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image012.png"/> </p>
<p>⇒ there pbs are equivalent ! </p>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image013.png"/> </p>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image014.png"/> <br/>
summary: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image015.png"/> </p>
<p>==... <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image016.png"/> </p>
<h1 id="6-coping-with-intractability">6. Coping with Intractability</h1>
<h3 id="exploit-intractability">exploit intractability</h3>
<p>cryptography ecopoits the hardness of FACTOR pb <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image017.png"/> </p>
<p>Can factor an n-bit integer in n 3 steps on a "<em>quantum computer</em>.” </p>
<h3 id="coping-with-intractability">Coping with intractability</h3>
<p>relax one of desired features... </p>
<ul>
<li>special cases </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image018.png"/> </p>
<ul>
<li>Develop a heuristic, and hope it produces a good solution. </li>
</ul>
<p>no guarantee <br/>
ex. TSP </p>
<ul>
<li>Approximation algorithm. Find solution of provably good quality. </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image019.png"/> </p>
<h3 id="halmiton-path">Halmiton path</h3>
<p>remark: Euler path (each edge once) easy, Halmiton path (each vertex once) NPC... <br/>
dfs solution for Halmiton path: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span><span class="w"> </span><span class="k">class</span><span class="w"> </span><span class="n">Halmiton</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="k">boolean</span><span class="err">[]</span><span class="w"> </span><span class="n">marked</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="nf">count</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="n">nb</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">Halmiton</span><span class="w"> </span><span class="n">paths</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">public</span><span class="w"> </span><span class="n">Halmiton</span><span class="p">(</span><span class="n">Graph</span><span class="w"> </span><span class="n">G</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">marked</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="k">new</span><span class="w"> </span><span class="k">boolean</span><span class="o">[</span><span class="n">G.V()</span><span class="o">]</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">for</span><span class="w"> </span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">v</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">v</span><span class="o"><</span><span class="n">G</span><span class="p">.</span><span class="n">V</span><span class="p">();</span><span class="w"> </span><span class="n">v</span><span class="o">++</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">G</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="err">}</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">private</span><span class="w"> </span><span class="n">void</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">Graph</span><span class="w"> </span><span class="n">G</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">v</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="k">depth</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="k">depth</span><span class="o">==</span><span class="n">G</span><span class="p">.</span><span class="n">V</span><span class="p">())</span><span class="w"> </span><span class="nf">count</span><span class="o">++</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">marked</span><span class="o">[</span><span class="n">v</span><span class="o">]=</span><span class="k">true</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">for</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="nl">w</span><span class="p">:</span><span class="w"> </span><span class="n">G</span><span class="p">.</span><span class="n">adj</span><span class="p">(</span><span class="n">v</span><span class="p">))</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">if</span><span class="p">(</span><span class="n">marked</span><span class="o">[</span><span class="n">w</span><span class="o">]==</span><span class="k">false</span><span class="p">)</span><span class="w"> </span><span class="n">dfs</span><span class="p">(</span><span class="n">G</span><span class="p">,</span><span class="w"> </span><span class="n">w</span><span class="p">,</span><span class="w"> </span><span class="k">depth</span><span class="o">+</span><span class="mi">1</span><span class="p">);</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">marked</span><span class="o">[</span><span class="n">v</span><span class="o">]=</span><span class="n">flase</span><span class="p">;</span><span class="w"> </span><span class="o">//</span><span class="w"> </span><span class="n">backtrack</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="err">}</span><span class="w"> </span></span>
<span class="code-line"><span class="err">}</span><span class="w"></span></span>
</pre></div>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_3_intractability/pasted_image020.png"/> </p>
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