Archive for April, 2010

Homework #2

April 19, 2010

Due: Monday, May 3, at the beginning of class. Turn in physically or by email to Thach.

Remember to take a look at the grading guidelines.

In solving the problem sets, you are allowed to collaborate with fellow students taking the class, but remember that each submission can have at most one author. If you do collaborate in any way, you must acknowledge, for each problem, the people you worked with on that problem.

The problems have been carefully chosen for their pedagogical value, and hence might be similar to those given in past offerings of this course at UW, or similar to other courses at other schools.  Using any pre-existing solutions from these sources, for from the web, constitutes a violation of the academic integrity you are expected to exemplify, and is strictly prohibited.

Most of the problems only require one or two key ideas for their solution.  It will help you a lot to spell out these main ideas so that you can get most of the credit for a problem even if you err on the finer details.

A final piece of advice:  Start working on the problem sets early!  Don’t wait until the day (or few days) before they’re due.

Problems

Each problem is worth 15 points unless otherwise noted.

  1. Let A be the language of properly nested parentheses. For example (()) and (()()(())) are in A, but )( is not. Show that A is in L.
  2. Let CONNECTED denote the language consisting of directed graphs which are strongly connected: there is a path from a to b, for every pair of vertices a,b. Show that CONNECTED is NL-complete.
  3. An undirected graph is bipartite if the vertices can be partitioned into two sets such that all edges go from a node in one set to a node in the other. A graph is bipartite if and only if it does not have any cycles of odd length. Let BIPARTITE denote the language consisting of bipartite graphs. Show that BIPARTITE \in NL.
  4. Show that SAT \notin TISP(n^c, n^d) for all constants c,d such that c(c+d)<2.
  5. If S = \{ S_1,S_2,\dotsc,S_m \} is a collection of subsets of a finite set U, the VC dimension of S is the size of the largest set X \subset U such that for every X' \subset X, there is a set S_i in S for which S_i \cap X = X'. A boolean circuit \mathcal{C} represents collection S if S_i consists of exactly those elements x \in U for which C(i,x) = 1. Define the language VC-DIMENSION to consist of those pairs <C,k> such that C is a circuit representing a family of VC dimension at least k. Show that VC-DIMENSION is in \Sigma_3^p, and is complete for this class.
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Homework #1

April 5, 2010

Due: Monday, April 19, at the beginning of class. Turn in physically or by email to Thach.

Remember to take a look at the grading guidelines.

In solving the problem sets, you are allowed to collaborate with fellow students taking the class, but remember that each submission can have at most one author. If you do collaborate in any way, you must acknowledge, for each problem, the people you worked with on that problem.

The problems have been carefully chosen for their pedagogical value, and hence might be similar to those given in past offerings of this course at UW, or similar to other courses at other schools.  Using any pre-existing solutions from these sources, for from the web, constitutes a violation of the academic integrity you are expected to exemplify, and is strictly prohibited.

Most of the problems only require one or two key ideas for their solution.  It will help you a lot to spell out these main ideas so that you can get most of the credit for a problem even if you err on the finer details.

A final piece of advice:  Start working on the problem sets early!  Don’t wait until the day (or few days) before they’re due.

Problems

Each problem is worth 15 points unless otherwise noted.

  1. Exercise 2.8 from the text: Let HALT be the Halting Language defined in Theorem 1.11 of the text: HALT = \{ a,x | M_a(x) \text{ halts in a finite number of steps} \}. Show that HALT is NP-hard. Is it NP-complete?
  2. Exercise 2.10 from the text: Suppose L_1, L_2 are in NP. Then is L_1 \cup L_2 in NP? What about L_1 \cap L_2?
  3. Exercise 2.15 from the text: A clique of an undirected graph is a set of vertices in which every two vertices have an edge between them. A vertex cover is a subset such that every edge of the graph is incident to one of the vertices from the set. Let CLIQUE be language \{G,k | \text{ the graph G has a clique of size k } \} , and let VERTEX COVER be the language \{ G,k | \text{the graph G has a vertex cover of size k} \} . Prove that both of these problems are NP-complete.
  4. Exercise 3.9 from the text: Suppose we pick a random language C by choosing every string to be in C independently with probability 1/2. Prove that with high probability P^C is not the same as NP^C .
  5. A digital signature scheme is a triple of randomized polynomial time algorithms: Generate, Sign, and Check, with the following properties. On input  1^n , Generate outputs a pair of strings (public, secret), such that for any message m, Check(m,Sign(m,secret),public) always outputs 1, yet for any polynomial time algorithm A, the probability that Check(m,A(m,public),public) outputs 1 is at most O(n^{-100}) . Show that such a scheme cannot exist if P=NP.