Lecture 11: October 4 11.1 Network Reliability
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چکیده
Consider a connected undirected graph G with n vertices and m edges, where each edge has some probability p of failing. What is the probability that G becomes disconnected under random, independent edge failures? This can also be viewed as a counting problem, except that each item now has an associated weight: in particular, we want to compute the sum of the weights of all disconnected subgraphs of G, where the weight of a subgraph with t fewer edges than G is p(1 − p)m−t. We can also generalize the problem to allow a different failure probability for each edge.
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Lecture 11: October 8 11.1 Primal and Dual Problems
j=1 vj`j(x). Lagrange multipliers u ∈ R, v ∈ R. Lemma 11.1 At each feasible x, f(x) = supu≥0,v L(x, u, v), and the supremum is taken iff u ≥ 0 satisfying uihi(x) = 0, i = 1, · · · ,m. Proof: At each feasible x, we have hi(x) ≤ 0 and `(x) = 0, thus L(x, u, v) = f(x) + ∑m i=1 uihi(x) + ∑r j=1 vj`j(x) ≤ f(x). The last inequality becomes equality iff uihi(x) = 0, i = 1, · · · ,m. Proposition 11.2 T...
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تاریخ انتشار 2011