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[Algorithms II] Week 6-2 Linear Programming
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目录
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<div id="toc"><ul><li><a class="toc-href" href="#1-brewer-s-problem" problem'="" s="" title="1. Brewer-">1. Brewer-'s Problem</a></li><li><a class="toc-href" href="#2-simplex-algorithm" title="2. Simplex Algorithm">2. Simplex Algorithm</a></li><li><a class="toc-href" href="#3-simplex-implementations" title="3. Simplex Implementations">3. Simplex Implementations</a><ul><li><a class="toc-href" href="#choosing-pivot-variable-find-entering-column" title="choosing pivot variable (find entering column)">choosing pivot variable (find entering column)</a></li><li><a class="toc-href" href="#choosing-pivot-constraint-find-entering-row" title="choosing pivot constraint (find entering row)">choosing pivot constraint (find entering row)</a></li><li><a class="toc-href" href="#do-the-pivot-column-q-row-p" title="do the pivot (column q, row p)">do the pivot (column q, row p)</a></li><li><a class="toc-href" href="#degeneracy" title="degeneracy">degeneracy</a></li></ul></li><li><a class="toc-href" href="#4-linear-programming-reductions_1" title="4. Linear Programming Reductions">4. Linear Programming Reductions</a><ul><li><a class="toc-href" href="#reduction-to-std-form-equalities" title="reduction to std form (equalities)">reduction to std form (equalities)</a></li><li><a class="toc-href" href="#modeling-of-lp" title="modeling of LP">modeling of LP</a></li><li><a class="toc-href" href="#maxflow-by-lp" title="maxflow by LP">maxflow by LP</a></li><li><a class="toc-href" href="#max-cardinality-bipartite-matching-by-lp" title="max cardinality bipartite matching by LP">max cardinality bipartite matching by LP</a></li><li><a class="toc-href" href="#and-many-others" title="and many others...">and many others...</a></li></ul></li></ul></div>
</div>
</div>
<p>simplex algo: top 10 algo of the 20th century (ever?). </p>
<p>what is linear programming: <br/>
<em>a general problem-solving model</em> that works for: <br/>
shortest-path, maxflow, MST, matching, assignment, ... </p>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image.png"/> </p>
<h1 id="1-brewer-s-problem">1. Brewer-'s Problem</h1>
<p>toy example: choose products to maximize profit. <br/>
... <br/>
<em>feasible region</em>: a convex polygon. <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image002.png"/> </p>
<p>⇒ optimum solution appears at an extreme point. </p>
<p>standard form of LP </p>
<ul>
<li>n non-neg variables (j=1..n) </li>
<li>m linear euqations (i=1..m) </li>
<li>input: a_ij, c_j, b_i </li>
<li>output: x_j </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image003.png"/> <br/>
to convert <em>inequality</em> to <em>equality (as in the standard form above)</em>: add slack var! <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image004.png"/> </p>
<blockquote>
<p>def. convex set <br/>
for any a and b in set ⇒ 1/2(a+b) is also in set. </p>
</blockquote>
<p>extreme point: </p>
<blockquote>
<p>def. <strong>extreme point</strong> <br/>
is a point in set that cannot be written as 1/2(a+b) with a b distinct. </p>
</blockquote>
<p>extreme point property: <br/>
if there exists an potimal solution, then there exists one that is an extreme point. </p>
<ul>
<li>nb of extreme point is finite </li>
<li>but this nb can be exponential </li>
</ul>
<p>greedy property: <br/>
<em>extreme point is optimal iff no better adj extreme points.</em> <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image005.png"/> </p>
<h1 id="2-simplex-algorithm">2. Simplex Algorithm</h1>
<p>algo. simplex </p>
<blockquote>
<ul>
<li>start at some point </li>
<li><strong>pivot</strong> from one extreme point to an adj one (never decrease the obj fcn) </li>
<li>repeat until optimal </li>
</ul>
</blockquote>
<p>We're using the "basis" and "pivoting" to solve LP. </p>
<blockquote>
<p>def. <strong>basis</strong> (基变量) is a subset (size=m) of the n variables. </p>
</blockquote>
<p>vars in basis are always non-zero... </p>
<p>basic feasible solution: </p>
<ul>
<li>set n-m non-basis vars to 0 </li>
<li>solve for remaining m vars (with m constraints) </li>
<li>if unique and feasible (matrix invertable) </li>
</ul>
<p>algo: </p>
<ul>
<li>initial basic-feasible-solution: start slack vars as basis. </li>
<li>choose a non-basic var as <strong>pivot</strong>, add it into basis, take some basis var out </li>
</ul>
<p>ex. pick B as pivot var using constraint 2 (2nd equation): <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image006.png"/> </p>
<ul>
<li>why picking var B? → its obj coeff is positive </li>
<li>why pivot on 2nd constraint (5A+15B+Sc=480)? → <ul>
<li>RHS > 0 (preserves feasibility) </li>
<li>minimum ratio rule: min(480/15, 160/4, 1190/20) </li>
</ul>
</li>
</ul>
<p>stop when no obj-coeff is positive </p>
<h1 id="3-simplex-implementations">3. Simplex Implementations</h1>
<p>encode standard LP formulation into java 2d array: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image007.png"/> </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">Simplex</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">double</span><span class="err">[][]</span><span class="w"> </span><span class="n">a</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="n">m</span><span class="p">,</span><span class="n">n</span><span class="p">;</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">Simplex</span><span class="p">(</span><span class="k">double</span><span class="err">[][]</span><span class="w"> </span><span class="n">A</span><span class="p">,</span><span class="w"> </span><span class="k">double</span><span class="err">[]</span><span class="w"> </span><span class="n">b</span><span class="p">,</span><span class="w"> </span><span class="k">double</span><span class="err">[]</span><span class="w"> </span><span class="n">c</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">m</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">b</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">n</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">c</span><span class="p">.</span><span class="n">length</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">a</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">double</span><span class="o">[</span><span class="n">m+1</span><span class="o">][</span><span class="n">n+m+1</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="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">i</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o"><</span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">i</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="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">j</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o"><</span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</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">a</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">A</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</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="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">j</span><span class="o">=</span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o"><</span><span class="n">m</span><span class="o">+</span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">j-n</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </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="k">for</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">j</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o"><</span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">j</span><span class="o">][</span><span class="n">n+m</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">b</span><span class="o">[</span><span class="n">j</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="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">j</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o"><</span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o">++</span><span class="p">)</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">m</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="n">c</span><span class="o">[</span><span class="n">j</span><span class="o">]</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="err">}</span><span class="w"></span></span>
</pre></div>
<p>simplex algo: just transform initial 2d array into final solution. </p>
<h3 id="choosing-pivot-variable-find-entering-column">choosing pivot variable (find entering column)</h3>
<p><strong>Bland's rule.</strong> find the first column whose obj-coeff is positive. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">private</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">bland</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">for</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">q</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">q</span><span class="o"><</span><span class="n">m</span><span class="o">+</span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">q</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">if</span><span class="p">(</span><span class="n">a</span><span class="o">[</span><span class="n">m</span><span class="o">][</span><span class="n">q</span><span class="o">]></span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="k">return</span><span class="w"> </span><span class="n">q</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">return</span><span class="w"> </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="err">}</span><span class="w"></span></span>
</pre></div>
<h3 id="choosing-pivot-constraint-find-entering-row">choosing pivot constraint (find entering row)</h3>
<p>minimum ratio rule (if a tie, choose first row). </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="n">private</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">minRatioRule</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">q</span><span class="p">)</span><span class="err">{</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="o">=</span><span class="w"> </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="k">for</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">i</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o"><</span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o">++</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="w"> </span><span class="p">(</span><span class="n">a</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">q</span><span class="o">]<=</span><span class="mi">0</span><span class="p">)</span><span class="w"> </span><span class="k">continue</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="o">==-</span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="n">p</span><span class="o">=</span><span class="n">i</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">m+n</span><span class="o">]/</span><span class="n">a</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">q</span><span class="o">]</span><span class="w"> </span><span class="o"><</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">p</span><span class="o">][</span><span class="n">m+n</span><span class="o">]/</span><span class="n">a</span><span class="o">[</span><span class="n">p</span><span class="o">][</span><span class="n">q</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">p</span><span class="o">=</span><span class="n">i</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="k">return</span><span class="w"> </span><span class="n">p</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="err">}</span><span class="w"></span></span>
</pre></div>
<h3 id="do-the-pivot-column-q-row-p">do the pivot (column q, row p)</h3>
<p>like Guassian elimination: </p>
<p>make var q disappear on each row (except for row p); <br/>
on row p: make var q's coeff become 1. </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="k">public</span><span class="w"> </span><span class="n">void</span><span class="w"> </span><span class="k">pivot</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">p</span><span class="p">,</span><span class="w"> </span><span class="nc">int</span><span class="w"> </span><span class="n">q</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">for</span><span class="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">i</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o"><</span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">i</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="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">j</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o"><</span><span class="n">m</span><span class="o">+</span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</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">if</span><span class="w"> </span><span class="p">(</span><span class="n">i</span><span class="o">!=</span><span class="n">p</span><span class="w"> </span><span class="o">&&</span><span class="w"> </span><span class="n">j</span><span class="o">!=</span><span class="n">q</span><span class="p">)</span><span class="w"> </span></span>
<span class="code-line"><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">-=</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">p</span><span class="o">][</span><span class="n">j</span><span class="o">]*</span><span class="n">a</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">q</span><span class="o">]/</span><span class="n">a</span><span class="o">[</span><span class="n">p</span><span class="o">][</span><span class="n">q</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="p">(</span><span class="nc">int</span><span class="w"> </span><span class="n">i</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">i</span><span class="o"><</span><span class="n">m</span><span class="p">;</span><span class="w"> </span><span class="n">i</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">if</span><span class="p">(</span><span class="n">i</span><span class="o">!=</span><span class="n">p</span><span class="p">)</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">i</span><span class="o">][</span><span class="n">q</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">0</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="n">j</span><span class="o">=</span><span class="mi">0</span><span class="p">;</span><span class="w"> </span><span class="n">j</span><span class="o"><</span><span class="n">m</span><span class="o">+</span><span class="n">n</span><span class="p">;</span><span class="w"> </span><span class="n">j</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">if</span><span class="p">(</span><span class="n">j</span><span class="o">!=</span><span class="n">q</span><span class="p">)</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">p</span><span class="o">][</span><span class="n">j</span><span class="o">]</span><span class="w"> </span><span class="o">/=</span><span class="w"> </span><span class="n">a</span><span class="o">[</span><span class="n">p</span><span class="o">][</span><span class="n">q</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">a</span><span class="o">[</span><span class="n">p</span><span class="o">][</span><span class="n">q</span><span class="o">]</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="mi">1</span><span class="p">;</span><span class="w"> </span></span>
<span class="code-line"><span class="err">}</span><span class="w"></span></span>
</pre></div>
<p>so the simplex algo is: </p>
<div class="highlight"><pre><span class="code-line"><span></span><span class="err">public void solve(){ </span></span>
<span class="code-line"><span class="err"> while(true){ </span></span>
<span class="code-line"><span class="err"> int q = bland(); </span></span>
<span class="code-line"><span class="err"> if(q==-1) break; // optimal if -1 </span></span>
<span class="code-line"><span class="err"> int p = minRatioRule(q); </span></span>
<span class="code-line"><span class="err"> if(p==-1) break; // unbounded if -1 </span></span>
<span class="code-line"><span class="err"> pivot(p,q); </span></span>
<span class="code-line"><span class="err"> } </span></span>
<span class="code-line"><span class="err">}</span></span>
</pre></div>
<p>final solution is just in the array: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image008.png"/> </p>
<p><strong>remarkable property</strong> <br/>
<em>in typical applications</em>, simplex terminates after at most 2(m+n) pivots. — whereas nb of extreme points is exp in n !! <br/>
ie. LINEAR time in practice!! </p>
<p>other pivot rules: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image009.png"/> </p>
<h3 id="degeneracy">degeneracy</h3>
<p>when choosing new basis, still stay in the same extreme point... <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image010.png"/> <br/>
→ might cause cycling <br/>
→ bland's rule guarantees finite number of pivots </p>
<p>further improvement: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image012.png"/> </p>
<p>Best practice. <em>Don't implement it yourself......</em> </p>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image013.png"/> <br/>
(AMPL是个好东西...) </p>
<p>算法的力量: <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image014.png"/> </p>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image015.png"/> </p>
<h1 id="4-linear-programming-reductions_1">4. Linear Programming Reductions</h1>
<h3 id="reduction-to-std-form-equalities">reduction to std form (equalities)</h3>
<ul>
<li>Minimization problem: max -1*obj </li>
<li>ineq constraints: add slack var </li>
<li>unbounded var X: replace with X=X0-X1, X0>=0, X1>=0 </li>
</ul>
<h3 id="modeling-of-lp">modeling of LP</h3>
<ul>
<li>identify variables </li>
<li>define constraints </li>
<li>define objective fcn </li>
<li>convert to std form </li>
</ul>
<h3 id="maxflow-by-lp">maxflow by LP</h3>
<ul>
<li>variables: x_uv = flow on edge uv </li>
<li>constraints: capacity, flow conservation </li>
<li>obj: net flow to t </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image016.png"/> </p>
<p>can use LP to solve mincost maxflow easily... </p>
<h3 id="max-cardinality-bipartite-matching-by-lp">max cardinality bipartite matching by LP</h3>
<p>input: bipartite graph <br/>
goal: max cardinatlity matching (set of vertex-disjoint edges) <br/>
<img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image017.png"/> <br/>
can be reduced to maxflow (见algolab...) </p>
<ul>
<li>var: x_ij = indicator of person i assigned to job j (0<=x_ij<=1) </li>
<li>constraints: vertex-disjoint </li>
<li>obj: sum of all x_ij </li>
</ul>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image018.png"/> </p>
<p>non-trival: cause this is an INTEGER LP... </p>
<p>Th (Von Neumann) (and Poincare?..) <br/>
if all RHS=1 ⇒ <em>all extreme points of the polyhedron have integer coord</em>. </p>
<h3 id="and-many-others">and many others...</h3>
<p><img alt="" class="img-responsive" src="../images/algoII_week6_2_LP/pasted_image019.png"/> </p>
<p><strong>the profound question: Is there a universal problem-solving model ?</strong> <br/>
→ P/NP... </p>
<blockquote>
<p><strong>"For the time being, the closest thing that we have to universal problem-solving model is LP "</strong> </p>
</blockquote>
</div>
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Part 12 of series «Algorithms Princeton MOOC II»：
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<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week1_1.html' style="text-align:left">[Algorithms II] Week 1-1 Undirected Graphs</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week1_2.html' style="text-align:left">[Algorithms II] Week 1-2 Directed Graphs</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week2_1.html' style="text-align:left">[Algorithms II] Week 2-1 Minimum Spanning Trees</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week2_2.html' style="text-align:left">[Algorithms II] Week 2-2 Shortest Paths</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week3_1.html' style="text-align:left">[Algorithms II] Week 3-1 Maximum Flow</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week3_2.html' style="text-align:left">[Algorithms II] Week 3-2 Radix Sorts</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week4_1.html' style="text-align:left">[Algorithms II] Week 4-1 Tries</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week4_2.html' style="text-align:left">[Algorithms II] Week 4-2 Substring Search</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week5_1.html' style="text-align:left">[Algorithms II] Week 5-1 Regular Expressions</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week5_2.html' style="text-align:left">[Algorithms II] Week 5-2 Data Compression</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week6_1_reductions.html' style="text-align:left">[Algorithms II] Week 6-1 Reductions</a></li>
<li class="list-group-item"><a class="btn btn-primary" href='https://x-wei.github.io/algoII_week6_2_LP.html' style="text-align:left">[Algorithms II] Week 6-2 Linear Programming</a></li>
<li class="list-group-item"><a class="btn btn-default" href='https://x-wei.github.io/algoII_week6_3_intractability.html' style="text-align:left">[Algorithms II] Week 6-3 Intractability</a></li>
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<a href="https://x-wei.github.io/algoII_week6_1_reductions.html">[Algorithms II] Week 6-1 Reductions</a>
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<div id="toc"><ul><li><a class="toc-href" href="#1-brewer-s-problem" problem'="" s="" title="1. Brewer-">1. Brewer-'s Problem</a></li><li><a class="toc-href" href="#2-simplex-algorithm" title="2. Simplex Algorithm">2. Simplex Algorithm</a></li><li><a class="toc-href" href="#3-simplex-implementations" title="3. Simplex Implementations">3. Simplex Implementations</a><ul><li><a class="toc-href" href="#choosing-pivot-variable-find-entering-column" title="choosing pivot variable (find entering column)">choosing pivot variable (find entering column)</a></li><li><a class="toc-href" href="#choosing-pivot-constraint-find-entering-row" title="choosing pivot constraint (find entering row)">choosing pivot constraint (find entering row)</a></li><li><a class="toc-href" href="#do-the-pivot-column-q-row-p" title="do the pivot (column q, row p)">do the pivot (column q, row p)</a></li><li><a class="toc-href" href="#degeneracy" title="degeneracy">degeneracy</a></li></ul></li><li><a class="toc-href" href="#4-linear-programming-reductions_1" title="4. Linear Programming Reductions">4. Linear Programming Reductions</a><ul><li><a class="toc-href" href="#reduction-to-std-form-equalities" title="reduction to std form (equalities)">reduction to std form (equalities)</a></li><li><a class="toc-href" href="#modeling-of-lp" title="modeling of LP">modeling of LP</a></li><li><a class="toc-href" href="#maxflow-by-lp" title="maxflow by LP">maxflow by LP</a></li><li><a class="toc-href" href="#max-cardinality-bipartite-matching-by-lp" title="max cardinality bipartite matching by LP">max cardinality bipartite matching by LP</a></li><li><a class="toc-href" href="#and-many-others" title="and many others...">and many others...</a></li></ul></li></ul></div>
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if($('#affix-toc').hasClass("affix"))
return $('#sidebar').height();
return $('#sidebar').height() - $('#affix-toc').height();
},
bottom: function (){
return $("#fcfooter").offset().top -
$("#article-content").offset().top -
$("#article-content").height() + 20;
}
}
});
$('#affix-toc').width($('#sidebar').width());
});
$(window).resize(function () {
$('#affix-toc').width($('#sidebar').width());
});
</script>
</body>
</html>