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\hyperbaseurl{http://toc15.dsteurer.org/}
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Search vs decision
The P vs NP question is about decision problems. In this exercise, you will show that P=NP implies that certain search problems have efficient algorithms.
Show that if P=NP then there exists a polynomial-time algorithm that given a 3CNF formula \(\varphi\) outputs a satisfying assignment for \(\varphi\) if such an assignment exists.
Random subset sum
The set \(\{0,1\}\) with addition and multiplication modulo 2 forms a finite field, denoted \(\bbF_2\). The set \(\{0,1\}^n\) with component-wise addition and scalar multiplication forms a vector space over this field, denoted \(\bbF^2\). Just like for vector spaces over the reals, we can use matrices with their algebraic operations to describe linear maps on \(\bbF_2^n\).
Show the following facts:
Let \(x,y\in \bbF_2^n\) with \(x\neq y\). Then, \[\Pr_{r\in \bbF_2^n}\biggl\{ r^T x \neq r^T y \biggr\} =1/2\,.\]
Let \(A\in \bbF_2^{n\times n}\) be an invertible matrix and let \(y\in \bbF_2^n\). Then, \[\Pr_{x\in\bbF_2^n}\biggl\{ A x = y \biggr\} = 2^{-n}\,.\]
Let \(x,y\in \bbF_2^n\) with \(x\neq 0\). Then, \[\Pr_{A \in \bbF_{2}^{n\times n}} \biggl\{ A x = y \biggr\} = 2^{-n}\,.\]
Gap preserving reduction for independent set
In the lecture on 2/9, we sketched a gap preserving reduction from 3Sat to independent set. This exercise asks you to fill in the details for the proof sketch of the lemma.
- Describe the polynomial-time function \(f\) that maps every max3sat instance \(\varphi\) to independent set instance \(G\).
- Describe how every assignment \(x\) to the max3sat instance \(\varphi\) corresponds to an independent set \(S\) in \(G\).
- Describe how every independent set \(S\) in \(G\) corresponds to an assignment \(x\) for \(\varphi\).
Approximation algorithm for MaxQuadEq
In this exercise, you are to fill in details of the approximation algorithm for MaxQuadEq mentioned in the lecture on 2/11.
Show that a random assignment satisfies in expectation at least 1/4 of the equations of a satisfiable system of quadratic equations.
Develop a randomized polynomial-time algorithm that outputs an assignment which satisfies in expectation at least 1/4 of the equations satisfied by an optimum assignment.