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</style>
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<div class="section" id="Maximum-Weight-Stable-Set">
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<h1><span class="section-number">2. </span>Maximum Weight Stable Set<a class="headerlink" href="#Maximum-Weight-Stable-Set" title="Permalink to this headline">¶</a></h1>
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<div class="section" id="Problem-definition">
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<h2><span class="section-number">2.1. </span>Problem definition<a class="headerlink" href="#Problem-definition" title="Permalink to this headline">¶</a></h2>
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<p>Given a simple undirected graph <span class="math notranslate nohighlight">\(G=(V,E)\)</span> and weights <span class="math notranslate nohighlight">\(w \in \mathbb{R}^V\)</span>, the problem is to find a stable set <span class="math notranslate nohighlight">\(S \subseteq V\)</span> that maximizes $ <span class="math">\sum</span>_{v <span class="math">\in `V} w_v$. We recall that a subset :math:`S \subseteq V</span> is a <em>stable set</em> if no two vertices of <span class="math notranslate nohighlight">\(S\)</span> are adjacent. This is one of Karp’s 21 NP-complete problems.</p>
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</div>
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<div class="section" id="Random-instance-generator">
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<h2><span class="section-number">2.2. </span>Random instance generator<a class="headerlink" href="#Random-instance-generator" title="Permalink to this headline">¶</a></h2>
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<p>The class <code class="docutils literal notranslate"><span class="pre">MaxWeightStableSetGenerator</span></code> can be used to generate random instances of this problem, with user-specified probability distributions. When the constructor parameter <code class="docutils literal notranslate"><span class="pre">fix_graph=True</span></code> is provided, one random Erdős-Rényi graph <span class="math notranslate nohighlight">\(G_{n,p}\)</span> is generated during the constructor, where <span class="math notranslate nohighlight">\(n\)</span> and <span class="math notranslate nohighlight">\(p\)</span> are sampled from user-provided probability distributions <code class="docutils literal notranslate"><span class="pre">n</span></code> and <code class="docutils literal notranslate"><span class="pre">p</span></code>. To generate each instance, the generator independently samples each <span class="math notranslate nohighlight">\(w_v\)</span> from the user-provided
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probability distribution <code class="docutils literal notranslate"><span class="pre">w</span></code>. When <code class="docutils literal notranslate"><span class="pre">fix_graph=False</span></code>, a new random graph is generated for each instance, while the remaining parameters are sampled in the same way.</p>
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</div>
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<div class="section" id="Challenge-A">
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<h2><span class="section-number">2.3. </span>Challenge A<a class="headerlink" href="#Challenge-A" title="Permalink to this headline">¶</a></h2>
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<ul class="simple">
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<li><p>Fixed random Erdős-Rényi graph <span class="math notranslate nohighlight">\(G_{n,p}\)</span> with <span class="math notranslate nohighlight">\(n=200\)</span> and <span class="math notranslate nohighlight">\(p=5\%\)</span></p></li>
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<li><p>Random vertex weights <span class="math notranslate nohighlight">\(w_v \sim U(100, 150)\)</span></p></li>
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<li><p>500 training instances, 50 test instances</p></li>
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</ul>
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<div class="nbinput nblast docutils container">
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<div class="prompt highlight-none notranslate"><div class="highlight"><pre><span></span>[ ]:
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</pre></div>
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</div>
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<div class="input_area highlight-ipython3 notranslate"><div class="highlight"><pre>
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<span></span><span class="n">MaxWeightStableSetGenerator</span><span class="p">(</span>
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<span class="n">w</span><span class="o">=</span><span class="n">uniform</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mf">100.</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">50.</span><span class="p">),</span>
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<span class="n">n</span><span class="o">=</span><span class="n">randint</span><span class="p">(</span><span class="n">low</span><span class="o">=</span><span class="mi">200</span><span class="p">,</span> <span class="n">high</span><span class="o">=</span><span class="mi">201</span><span class="p">),</span>
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<span class="n">p</span><span class="o">=</span><span class="n">uniform</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mf">0.05</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">0.0</span><span class="p">),</span>
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<span class="n">fix_graph</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
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<span class="p">)</span>
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</pre></div>
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<a class='left-prev' id="prev-link" href="../preliminaries/" title="previous page"><span class="section-number">1. </span>Preliminaries</a>
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<a class='right-next' id="next-link" href="../knapsack/" title="next page"><span class="section-number">3. </span>Multidimensional Knapsack</a>
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