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    <p>Seems like an application of linear algebra.</p> <p>You have a set of linear equations which you need to solve. The variables being the lengths of the tasks (or edge weights).</p> <p>For instance if the tasks lengths were t1, t2, t3 for 3 tasks.</p> <p>And you are given</p> <pre><code>t1 + t2 = 2 (task 1 and 2 take 2 hours) t1 + t2 + t3 = 7 (all 3 tasks take 7 hours) t2 + t3 = 6 (tasks 2 and 3 take 6 hours) </code></pre> <p>Solving gives <code>t1 = 1, t2 = 1, t3 = 5</code>.</p> <p>You can use any linear algebra techniques (for eg: <a href="http://en.wikipedia.org/wiki/Gaussian_elimination" rel="nofollow noreferrer">http://en.wikipedia.org/wiki/Gaussian_elimination</a>) to solve these, which will tell you if there is a unique solution, no solution or an infinite number of solutions (no other possibilities are possible).</p> <p>If you find that the linear equations do not have a solution, you can try adding a very small random number to some of the task weights/coefficients of the matrix and try solving it again. (I believe falls under <a href="http://en.wikipedia.org/wiki/Perturbation_theory" rel="nofollow noreferrer">Perturbation Theory</a>). Matrices are notorious for radically changing behavior with small changes in the values, so this will likely give you an approximate answer reasonably quickly. </p> <p>Or maybe you can try introducing some 'slack' task in each walk (i.e add more variables) and try to pick the solution to the new equations where the slack tasks satisfy some linear constraints (like 0 &lt; s_i &lt; 0.0001 and minimize sum of s_i), using <a href="http://en.wikipedia.org/wiki/Linear_programming" rel="nofollow noreferrer">Linear Programming</a> Techniques. </p>
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    1. CODo excuse the delay on commenting this. Yes, you're correct in that straightforward linear algebra is one approach, but that only goes for the first, most naive scenario and leaves the beef, scenarios two and three completely untended. This was actually the first solution I came up with, but a prompt realization of the downfalls of this approach left me in a dead end - one to which I wanted to avoid leading any other answers. Seeing the lack of answers, though, I'll be adding up the ante with a bounty shortly.
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    2. CO@Ezku: Your question in #2 and #3 isn't really clear. For instance, it is not at all clear why a linear algebra method won't work there. What 'downfalls' are you talking about? Perhaps that will help clarify the question further. Perhaps some examples will help too.
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    3. CO@Ezku: Added a paragraph to the answer based on your edit of the question.
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