高级算法 (Fall 2017)/Problem Set 3: Difference between revisions

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Created page with " == Problem 1== Consider the following optimization problem. :'''Instance''': <math>n</math> positive integers <math>x_1<x_2<\cdots <x_n</math>. :Find two ''disjoint'' nonempt..."
 
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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.
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.
* 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>.
* 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>.
* Derandomize this algorithm through conditional expectation and give a deterministic polynomial time algorithm with approximation ratio <math>3/4</math>.
* Is it possible that for some more clever <math>f</math> we can do better than this? Try to justify your argument.
* Is it possible that for some more clever <math>f</math> we can do better than this? Try to justify your argument.
==Problem 4 ==
The following is the weighted version of set cover problem:
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.
* Give an integer programming for the problem and its linear programming relaxation.
* 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>.


== Problem 4==
== Problem 4==

Revision as of 09:09, 22 November 2017

Problem 1

Consider the following optimization problem.

Instance: [math]\displaystyle{ n }[/math] positive integers [math]\displaystyle{ x_1\lt x_2\lt \cdots \lt x_n }[/math].
Find two disjoint nonempty subsets [math]\displaystyle{ A,B\subset\{1,2,\ldots,n\} }[/math] with [math]\displaystyle{ \sum_{i\in A}x_i\ge \sum_{i\in B}x_i }[/math], such that the ratio [math]\displaystyle{ \frac{\sum_{i\in A}x_i}{\sum_{i\in B}x_i} }[/math] is minimized.

Give a pseudo-polynomial time algorithm for the problem, and then give an FPTAS for the problem based on the pseudo-polynomial time algorithm.

Problem 2

In the maximum directed cut (MAX-DICUT) problem, we are given as input a directed graph [math]\displaystyle{ G(V,E) }[/math]. The goal is to partition [math]\displaystyle{ V }[/math] into disjoint [math]\displaystyle{ S }[/math] and [math]\displaystyle{ T }[/math] so that the number of edges in [math]\displaystyle{ 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:

[math]\displaystyle{ \begin{align} \text{maximize} &&& \sum_{(u,v)\in E}y_{u,v}\\ \text{subject to} && y_{u,v} &\le x_u, & \forall (u,v)&\in E,\\ && y_{u,v} &\le 1-x_v, & \forall (u,v)&\in E,\\ && x_v &\in\{0,1\}, & \forall v&\in V,\\ && y_{u,v} &\in\{0,1\}, & \forall (u,v)&\in E. \end{align} }[/math]

Let [math]\displaystyle{ x_v^*,y_{u,v}^* }[/math] denote the optimal solution to the LP-relaxation of the above integer program.

  • Apply the randomized rounding such that for every [math]\displaystyle{ v\in V }[/math], [math]\displaystyle{ \hat{x}_v=1 }[/math] independently with probability [math]\displaystyle{ x_v^* }[/math]. Analyze the approximation ratio (between the expected size of the random cut and OPT).
  • Apply another randomized rounding such that for every [math]\displaystyle{ v\in V }[/math], [math]\displaystyle{ \hat{x}_v=1 }[/math] independently with probability [math]\displaystyle{ 1/4+x_v^*/2 }[/math]. Analyze the approximation ratio for this algorithm.

Problem 3

Recall the MAX-SAT problem and its integer program:

[math]\displaystyle{ \begin{align} \text{maximize} &&& \sum_{j=1}^my_j\\ \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,\\ &&& x_i\in\{0,1\}, && 1\le i\le n,\\ &&& y_j\in\{0,1\}, && 1\le j\le m. \end{align} }[/math]

Recall that [math]\displaystyle{ S_j^+,S_j^-\subseteq\{1,2,\ldots,n\} }[/math] are the respective sets of variables appearing positively and negatively in clause [math]\displaystyle{ j }[/math].

Let [math]\displaystyle{ 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]\displaystyle{ \hat{x}_i }[/math] is round to 1 independently with probability [math]\displaystyle{ x_i^* }[/math], we have approximation ratio [math]\displaystyle{ 1-1/\mathrm{e} }[/math].

We consider a generalized rounding scheme such that every [math]\displaystyle{ \hat{x}_i }[/math] is round to 1 independently with probability [math]\displaystyle{ f(x_i^*) }[/math] for some function [math]\displaystyle{ f:[0,1]\to[0,1] }[/math] to be specified.

  • Suppose [math]\displaystyle{ f(x) }[/math] is an arbitrary function satisfying that [math]\displaystyle{ 1-4^{-x}\le f(x)\le 4^{x-1} }[/math] for any [math]\displaystyle{ 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]\displaystyle{ 3/4 }[/math].
  • Derandomize this algorithm through conditional expectation and give a deterministic polynomial time algorithm with approximation ratio [math]\displaystyle{ 3/4 }[/math].
  • Is it possible that for some more clever [math]\displaystyle{ f }[/math] we can do better than this? Try to justify your argument.

Problem 4

The following is the weighted version of set cover problem:

Given [math]\displaystyle{ m }[/math] subsets [math]\displaystyle{ S_1,S_2,\ldots,S_m\subseteq U }[/math], where [math]\displaystyle{ U }[/math] is a universe of size [math]\displaystyle{ n=|U| }[/math], and each subset [math]\displaystyle{ S_i }[/math] is assigned a positive weight [math]\displaystyle{ w_i\gt 0 }[/math], the goal is to find a [math]\displaystyle{ C\subseteq\{1,2,\ldots,m\} }[/math] such that [math]\displaystyle{ U=\bigcup_{i\in C}S_i }[/math] and the total weight [math]\displaystyle{ \sum_{I\in C}w_i }[/math] is minimized.

  • Give an integer programming for the problem and its linear programming relaxation.
  • Consider the following idea of randomized rounding: independently round each fractional value to [math]\displaystyle{ \{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]\displaystyle{ U }[/math] is fully covered. Show that this can return a set cover with [math]\displaystyle{ O(\log n) }[/math] approximation ratio with probability at least [math]\displaystyle{ 0.99 }[/math].

Problem 4

Recall that the instance of set cover problem is a collection of [math]\displaystyle{ m }[/math] subsets [math]\displaystyle{ S_1,S_2,\ldots,S_m\subseteq U }[/math], where [math]\displaystyle{ U }[/math] is a universe of size [math]\displaystyle{ n=|U| }[/math]. The goal is to find the smallest [math]\displaystyle{ C\subseteq\{1,2,\ldots,m\} }[/math] such that [math]\displaystyle{ U=\bigcup_{i\in C}S_i }[/math]. The frequency [math]\displaystyle{ f }[/math] is defined to be [math]\displaystyle{ \max_{x\in U}|\{i\mid x\in S_i\}| }[/math].

  • Give the primal integer program for set cover, its LP-relaxation and the dual LP.
  • Describe the complementary slackness conditions for the problem.
  • 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]\displaystyle{ f }[/math].