imported>Etone |
imported>TCSseminar |
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| | | 截止2018.10.24 13:00,作业3已提交名单如下 |
| == Problem 1==
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| Consider the following optimization problem.
| | |171250623 || 姜勇刚 |
| :'''Instance''': <math>n</math> positive integers <math>x_1<x_2<\cdots <x_n</math>.
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| :Find two ''disjoint'' nonempty subsets <math>A,B\subset\{1,2,\ldots,n\}</math> with <math>\sum_{i\in A}x_i\ge \sum_{i\in B}x_i</math>, such that the ratio <math>\frac{\sum_{i\in A}x_i}{\sum_{i\in B}x_i}</math> is minimized. | | |DZ1733021 || 夏瑞 |
| Give a pseudo-polynomial time algorithm for the problem, and then give an FPTAS for the problem based on the pseudo-polynomial time algorithm.
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| | | |DZ1833028 || 徐闽泽 |
| == Problem 2==
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| In the ''maximum directed cut'' (MAX-DICUT) problem, we are given as input a directed graph <math>G(V,E)</math>. The goal is to partition <math>V</math> into disjoint <math>S</math> and <math>T</math> so that the number of edges in <math>E(S,T)=\{(u,v)\in E\mid u\in S, v\in T\}</math> is maximized. The following is the integer program for MAX-DICUT:
| | |151220130 || 伍昱名 |
| :::<math>
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| \begin{align}
| | |161250027 || 高忆 |
| \text{maximize} &&& \sum_{(u,v)\in E}y_{u,v}\\
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| \text{subject to} && y_{u,v} &\le x_u, & \forall (u,v)&\in E,\\
| | |MP1733012 || 陆璐 |
| && y_{u,v} &\le 1-x_v, & \forall (u,v)&\in E,\\
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| && x_v &\in\{0,1\}, & \forall v&\in V,\\
| | |MF1833009 || 陈越 |
| && y_{u,v} &\in\{0,1\}, & \forall (u,v)&\in E.
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| \end{align}
| | |DZ1833011 || 何雨橙 |
| </math>
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| Let <math>x_v^*,y_{u,v}^*</math> denote the optimal solution to the '''LP-relaxation''' of the above integer program.
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| * Apply the randomized rounding such that for every <math>v\in V</math>, <math>\hat{x}_v=1</math> independently with probability <math>x_v^*</math>. Analyze the approximation ratio (between the expected size of the random cut and OPT).
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| * Apply another randomized rounding such that for every <math>v\in V</math>, <math>\hat{x}_v=1</math> independently with probability <math>1/4+x_v^*/2</math>. Analyze the approximation ratio for this algorithm.
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| == Problem 3==
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| Recall the MAX-SAT problem and its integer program:
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| :::<math>
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| \begin{align}
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| \text{maximize} &&& \sum_{j=1}^my_j\\
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| \text{subject to} &&& \sum_{i\in S_j^+}x_i+\sum_{i\in S_j^-}(1-x_i)\ge y_j, && 1\le j\le m,\\
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| &&& x_i\in\{0,1\}, && 1\le i\le n,\\
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| &&& y_j\in\{0,1\}, && 1\le j\le m.
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| \end{align}
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| </math>
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| Recall that <math>S_j^+,S_j^-\subseteq\{1,2,\ldots,n\}</math> are the respective sets of variables appearing positively and negatively in clause <math>j</math>.
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| Let <math>x_i^*,y_j^*</math> denote the optimal solution to the '''LP-relaxation''' of the above integer program. In our class we learnt that if <math>\hat{x}_i</math> is round to 1 independently with probability <math>x_i^*</math>, we have approximation ratio <math>1-1/\mathrm{e}</math>.
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| We consider a generalized rounding scheme such that every <math>\hat{x}_i</math> is round to 1 independently with probability <math>f(x_i^*)</math> for some function <math>f:[0,1]\to[0,1]</math> to be specified.
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| * Suppose <math>f(x)</math> is an arbitrary function satisfying that <math>1-4^{-x}\le f(x)\le 4^{x-1}</math> for any <math>x\in[0,1]</math>. Show that with this rounding scheme, the approximation ratio (between the expected number of satisfied clauses and OPT) is at least <math>3/4</math>.
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| * Derandomize this algorithm through conditional expectation and give a deterministic polynomial time algorithm with approximation ratio <math>3/4</math>.
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| * Is it possible that for some more clever <math>f</math> we can do better than this? Try to justify your argument.
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| ==Problem 4 ==
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| The following is the weighted version of set cover problem:
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| Given <math>m</math> subsets <math>S_1,S_2,\ldots,S_m\subseteq U</math>, where <math>U</math> is a universe of size <math>n=|U|</math>, and each subset <math>S_i</math> is assigned a positive weight <math>w_i>0</math>, the goal is to find a <math>C\subseteq\{1,2,\ldots,m\}</math> such that <math>U=\bigcup_{i\in C}S_i</math> and the total weight <math>\sum_{I\in C}w_i</math> is minimized.
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| * Give an integer programming for the problem and its linear programming relaxation.
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| * Consider the following idea of randomized rounding: independently round each fractional value to <math>\{0,1\}</math> with the probability of the fractional value itself; and repeatedly apply this process to the variables rounded to 0 in previous iterations until <math>U</math> is fully covered. Show that this can return a set cover with <math>O(\log n)</math> approximation ratio with probability at least <math>0.99</math>.
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| == Problem 5==
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| Recall that the instance of '''set cover''' problem is a collection of <math>m</math> subsets <math>S_1,S_2,\ldots,S_m\subseteq U</math>, where <math>U</math> is a universe of size <math>n=|U|</math>. The goal is to find the smallest <math>C\subseteq\{1,2,\ldots,m\}</math> such that <math>U=\bigcup_{i\in C}S_i</math>. The frequency <math>f</math> is defined to be <math>\max_{x\in U}|\{i\mid x\in S_i\}|</math>.
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| * Give the primal integer program for set cover, its LP-relaxation and the dual LP.
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| * Describe the complementary slackness conditions for the problem.
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| * Give a primal-dual algorithm for the problem. Present the algorithm in the language of primal-dual scheme (alternatively raising variables for the LPs). Analyze the approximation ratio in terms of the frequency <math>f</math>.
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