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POAd distribution problem: an optimal solution?
 

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  1. POAd distribution problem: an optimal solution?
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    <p>I'm asked to find a 2 approximate solution to this problem:</p> <p>You’re consulting for an e-commerce site that receives a large number of visitors each day. For each visitor i, where i € {1, 2 ..... n}, the site has assigned a value v[i], representing the expected revenue that can be obtained from this customer.</p> <p>Each visitor i is shown one of m possible ads A1, A2 ..... Am as they enter the site. The site wants a selection of one ad for each customer so that each ad is seen, overall, by a set of customers of reasonably large total weight. </p> <p>Thus, given a selection of one ad for each customer, we will define the <em>spread</em> of this selection to be the minimum, over j = 1, 2 ..... m, of the total weight of all customers who were shown ad Aj.</p> <p>Example: Suppose there are six customers with values 3, 4, 12, 2, 4, 6, and there are m = 3 ads. Then, in this instance, one could achieve a spread of 9 by showing ad A1 to customers 1, 2, 4, ad A2 to customer 3, and ad A3 to customers 5 and 6.</p> <p>The ultimate goal is to find a selection of an ad for each customer that <em>maximizes</em> the spread. </p> <p>Unfortunately, this optimization problem is NP-hard (you don’t have to prove this). </p> <p>So instead give a polynomial-time algorithm that approximates the maximum spread within a factor of 2.</p> <p>The solution I found is the following:</p> <pre><code>Order visitors values in descending order Add the next visitor value (i.e. assign the visitor) to the Ad with the current lowest total value Repeat </code></pre> <p>This solution actually seems to always find the optimal solution, or I simply can't find a counterexample. Can you find it? Is this a non-polinomial solution and I just can't see it?</p>
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    1. COSome information is missing. Just assign one ad to the minimum value customer and split the rest among the others. By spread did you mean max-min? The usage of spread makes more sense in that case... And 'reasonably large total weight' is not precise enough.
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    2. COThat is exactly what os written in my textbook. Yes By spread it means max-min.
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    3. COI suggest you edit your question to reflect that.
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