A Short Course on Complexity Theory

A Short Course on Complexity Theory

Lecture 1: Introduction

You have seen algorithms for a variety of problems. This begs the question of whether there is an algorithm for any given problem?. Furthermore is there an efficient algorithm for all problems?. Even if you have an algorithm for the problem, can it be made more efficient?. This course will be addressing these questions. For answering these questions, we first need to define:

  1. What is a problem?
  2. What is computer/programing language, where you can execute/write the algorithms?
  3. What do you mean by efficient?
We will start with very simplified definitions for these questions and try to find the answers for this setting.

Decision Problems

Typically the output of an algorithm could be a graph, a number etc. But we will consider only problems with an YES or NO answer. Furthermore, we will define the problem to be just the set of all YES-instances. You are familiar with this definition from your automata theory class. For example take the REACHABILITY problem where the input is \( (G(V,E), s,t) \) where \( s,t \in V \), and the problem is whether there is a path from \( s \) to \( t \) in the graph \( G \). So we will define the problem itself as the set of instances \( (G,s,t) \) where a path exists.

$$ \mbox{REACHABILITY} = \{ (G,s,t) : \mbox{ there is a path in $G$ from } s \mbox{ to } t \} $$

The number of instances of graphs having a particular number of vertices \( n \) is finite. But the set of all instances is the union of such instances with \( n=1 \) to \( \infty \). Hence each instance has a size \( n \) which is the number of vertices in the graph. Size of an instance is a fundamental concept, and we will be defining it for all the instances. Later we will see, that size plays an important role in defining what we mean by an efficient algorithm.

It was easy to define REACHABILITY because the original problem had an YES or NO answer. But can we write problems for which the outputs are numbers, as decision problems? Lets consider the the problem of finding the chromatic number of a graph. Chromatic number of a graph \( G \) is the minimum number of distinct colors required to color a graph such that for every edge, end points are of different colors. Note that for a graph on \( n \) vertices, the chromatic number is \( \leq n \).

The descision version of the chromatic number problem takes the graph \( G \) and a number \( k \) between \( 1 \) and \( n \) as input. The output is YES, if the chromatic number is \( \leq k \). So

$$ \text{CHROMATIC-NUMBER} = \{ (G,k) : \text{ chromatic number of graph } G \text{ is } \leq k \} $$

Suppose you have an algorithm that takes \( T(n) \) steps for deciding the CHROMATIC-NUMBER problem, you can find the chromatic number also. For this we will create \( n \) instances of CHROMATIC-NUMBER from the input graphs, setting \( k=1 \) to \( n \). Note that these instances will transition from NO-instances to YES-instance and remain YES-instances. The chromatic number is the least \( k \) for which the instance is an YES-instance. Hence you have algorithm which takes \( n\times T(n) \) steps for finding the exact chromatic number.

!bu-problem Can you find a algorithm, which takes \( (\log n) T(n) \) steps? !eu-problem

Turing Machines

The simplfied model for computer/programming language will be the Turning Machine (TM). The discovery of the TM has a long and interesting history. See Logicomix, for a nice read in comic book format.

The Turing Machine has many tapes (which is similar to the hard disk in a modern computer) and a limited set of states (which is similar to the RAM or the cache). There is a read-only input tape, where the input is written and a write-only output tape where the output is expected when machine is finished. There is also multiple work tapes, which can be used for storing intermediate results in the computation. The work tapes are assumed to be of infinite length. This is needed because as the size of the input instances becomes larger, the intermediate memory also needs to be large. The set of states of the TM is fixed and does not change with input size.

The input is assumed to be a string from a finite alphabet \( \Sigma \). A tape cell can store an additional blank symbol indicating that it is unused.

The algorithm is implemented in a TM by means of a state function. The state function takes as arguments:

  1. the symbol under the heads of the TM in each of the tapes,
  2. well as the current state
and returns
  1. the next state
  2. the symbols to be written under each of the work tape heads
  3. the directions to move each of the heads one cell to the left, right or not move.
Now we define a TM using mathematical notation. \( \Sigma \) is the alphabet in which input and work tape is written (for eg. binary ie. \( \Sigma = \{0,1\} \)). The blank symbol for unwritten tape cell is \( \_ \). Let \( \Omega \) be the set of states of the TM. There are 2 special states of the TM called \( \omega_{\text{start}} \) and \( \omega_{\text{stop}} \) called the start and the stop states. The state transition function is denoted by $$ \delta : \Omega \times \Sigma \cup \{ \_ \} \rightarrow \Omega \times \Sigma^k \times \{ < , > , - \}^{k+2} $$

Now lets see an example of a TM for solving the following problem: $$ PALINDROMES = \{ x \in \{0,1\}^n : x \text{ is a palindrome}\} $$

First lets try to write a pseudo code for a TM solving the problem:

  1. Copy the input in to the second tape
  2. Move the head in the input tape to end and second tape to begining
  3. Compare each bit by moving the input head from end to beggining and second head from beggining to end
  4. If any bit is different reject
  5. If all bits match accept
We will use a turing machine simulator for implementing this. See the example : http://turingmachinesimulator.com/shared/oihvkhvacu

Note that it is extremely tedious to write algorithms as a TM. Neverthless, all the algorithms can be written as TM.

!bu-problem Design a TM which decides the language : $$ DIV5 = \{ x \in \{0,1\}^n : x \text{ is binary number divisible by } 5\} $$

Write the program first as a pseudo code and then in the format in https://turingmachinesimulator.com/. !eu-problem

We will use a word "decidable" to indicate that a descision problem is solvable. A TM is said to decide a descision problem if there is a TM, that accepts all the YES-instances and rejects all other strings. Note that this TM will never go into an infinite loop.

Robustness of TM

We will now see that doing some modifications to the TM definitions, does not change what it can compute. We will not be going into details, but only sketch the proof.

Convert \( k \) tape TMs to single tape TMs.

We will merge the \( k \) tapes in to a single tape which has an alphabet that encodes all the \( k \) symbols. But we will also need to encode whether the head in the \( k \) tape TM was placed over a symbol. So if the orignal alphabet was \( \{0,1\} \) then the new alphabet will be \( \{0,1, \dot 0, \dot 1 \}^k \).

Each step of the \( k \) tape TM will be simulated by the one tape TM, by doing a full pass over its tape, reading all the alphabets with \( \{\dot 0, \dot 1\} \). Then it will erase and the write the dots according to the appropriate tape movement (given by the transition function of the old TM).

Convert alphabet from any \( \Sigma \) to binary.

Suppose we have a TM with alphabet size \( |\Sigma| = t \). We will encode the \( t \) symbols into binary strings of length \( \log t \). For each step of the old TM, the new TM will go over \( \log t \) positions in its binary tape to read the symbol, and update \( \log t \) positions for writing the new symbol according to the transition funciton of the old TM.

It is important to note that, while doing these transformations, the alphabet size \( |\Sigma| \) and the state size \( |\Omega| \) remain constant, independent of the input size.

Efficient Algorithms

Recall that for every instance of a problem, we will have a size defined (denoted by \( n \)). The number of steps taken by the TM to decide an instance will grow with \( n \). Hence the running time of a problem will be defined in terms of the size \( n \). Futhermore we will be considering worst case running time. That is the running time for intances of size \( n \) will be defined as the the largest running time, among all intances of size \( n \).

For example, consider the Dijkstra's algorithm for shortest path. There are instances of graph of size \( n \) like a \( n \) length path on which the algorithm takes only \( O(n) \) steps. However if the graph is a complete graph, it takes \( O(n^2) \) steps. But it never takes more than \( O(n^2) \) steps on instances of size \( n \). Hence running time of Dijkstra's algorithm will be considered as \( O(n^2) \).

!bu-problem Let \( f(n) \) and \( g(n) \) be any two of the following functions. Determine whether \( f(n) = O(g(n)) \), \( f(n) = \Omega(g(n)) \) and \( f(n) = \Theta(g(n)) \).

  1. \( n! \)
  2. \( n^{\log n} \)
  3. \( n^2 \) when \( n \) is odd and \( 2^n \) otherwise
  4. \( n^3 \)
  5. \( n^{n} \)
  6. \( 2^n \)

Polynomial time vs Exponential Time

Now lets consider the CHROMATIC-NUMBER problem. There is very simple brute force algorithm for it. Given an instance \( (G,k) \), for every assignment of \( k \) colors to the vertices, check for every edge, that the end points have different colors. If we find a color assignment which passes all checks, the TM can say YES. If for all assignments some check fails, then the TM says NO. This is a valid algorithm. Note that for a NO-instance, the TM has to loop over all the color assignments which is \( k^n \) in number, and do all the checks for each edge. Hence the running time is \( O(k^n\times n^2) \).

Note that the running time for the above algorithm grows exponentialy in \( n \). Even if \( n=100 \), the running time is a very huge number that even a fast computer cannot hope to solve it. So now the question is where there is another algorithm for CHROMATIC-NUMBER, which is faster.

Our definition of efficient algorithms will be polynomial time algorithms, that is algorithms for which the running time is of the form \( O(n^t) \) for some fixed integer \( t \).


Chapter 3. Michael Sipser, Introduction to Theory of Computation

Chapter 1, Chapter 2. Christos Papadimitriou, Computational Complexity

Misc: Logicomix

Lecture 2: Paradox's and Diagonalization

Russell's Paradox and Diagonalization

There is a village where the barbers shave only those people who do not shave by themselves. Now does a barber shave himself?

If YES, then he cannot shave himself. If NO, then he can shave himself.

See Godel, Escher, Bach, for more paradoxes and their history.

Now we will solve a problem using the Russell's paradox. Let \( \mathbb N \) be the set of natural numbers and \( \mathcal{P}(N) \) is the set of all subsets of \( \mathbb N \) (or the power set of \( \mathbb N \)). Show that there cannot be a one to one mapping between \( \mathbb N \) and \( \mathcal{P}(N) \) that has range equal to \( \mathcal{P}(N) \) (that is for every subset of \( \mathbb N \), there is some integer that maps to it).

Suppose there is a one to one mapping \( f \) that has range equal to \( \mathcal{P}(N) \). Consider the set $$ S = \{ x : x \notin f(x) \} $$ Since \( f \) that has range equal to \( \mathcal{P}(N) \), there is some integer \( y \) that maps to \( S \) (that is \( f(y) = S \)). Now does \( y \in S \)?

!bu-problem Let STRINGS be the set of all infinite length binary sequences. That is $$ \text{STRINGS} = \{ x:\mathbb{N} \rightarrow \{0,1\} \} $$ That is any infinite length binary sequence is simply a function that maps every integer to the bit at the corresponding position. Show that there cannot be a one to one mapping between \( \mathbb N \) and STRINGS. !eu-problem

Universal Turing Machines

Computer as we know them are general purpose machines. That is, you can give them any program in a particular format (a programming langauge, or assembly), and they can execute it. However the Turing Machines that we saw in the previous lecture, seems to be tailor made for a particular problem.

The first step to making Turing Machines general purpose is to encode a TM as a string, which could be fed as input to a general purpose TM, which we will call Universal Turing Machines. As we saw previously, a TM is a tuple

$$ M = (\Omega, \Sigma, \mathcal{T}, \delta, \omega_{\text{start}}, \omega_{\text{accept}}, \omega_{\text{reject}}) $$

where \( \Omega \) is a set of states, \( \Sigma \) is the alphabet for the language, \( \mathcal{T} \) is the tape alphabet, \( \delta \) is a transition function and \( \omega_{\text{start}}, \omega_{\text{accept}}, \omega_{\text{reject}} \) are the special start, reject, accept states respectively. Note that the states, the alphabets and the special states, are all finite and can be encoded as a finite alphabet. \( \delta \) function also is finite and could be represented as a set of rules (similar to how the turning machine simulater is doing). We will denote the string representing a TM \( M \) by \( \langle M \rangle \).

A Universal Turing Machine takes a string \( (\langle M \rangle, x) \), and simulates the running of the TM \( M \) on the input \( x \). It is just another TM having its own transition function, alphabets and states. However it is designed such that it can accept a TM encoding \( \langle M \rangle \) as in part of the input, and the transition function is defined such that it simulates the running of \( M \) on a string \( x \).

Configuration. An important concept for doing simulation is the configuration of a TM. Lets stick to 1-tape TMs for now. The "state" of the algorithm (not the TM state) is really consist of the the TM's state, the contents of its tape as well as the position of the tape heads. If a TM is at state \( \omega_i \), has the string 101101111 in its tape and the tape head is at the 5th position, then its current configuration is 1011 \( \omega_i \) 01111. Note that this is a string over the alphabet \( \mathcal{T} \cup \Omega \). The computation of TM is a graph in the space of all TM configurations. If the TM halts, it is a path which ends in a configuration having an accept or reject state. If it loops, there will be a cycle in this graph.

A universal TM, goes over a configuration string, then goes over the encoding of the delta function, finds out which rule in the delta function to use, and can update the configuration, to reflect the next configuration of the TM begin simulated.

Halting Problem

Recall that we started with the question whether all problems can be solved by a TM. Consider the following descision problem: $$ H = \{ (\langle M \rangle,x) : M \text{ accepts } x \} $$ That is, given a TM encoding \( \langle M \rangle \) and a string \( x \), check if \( M \) accepts \( x \). Can you think of a TM which decides \( H \). One way is to simulate \( M \) on \( x \) like the universal TM. But its possible that \( M \) loops on \( x \) and the UTM will also loop.

Suppose there a TM \( M_H \) which decides \( H \). We will define a new TM \( D \) which takes an encoding of a TM \( \langle M \rangle \) as input , which does the following

  1. Run \( M_H \) on \( (\langle M \rangle, \langle M \rangle) \).
  2. Output the opposite of what \( M_H \) outputs.
So \( D \) accepts \( \langle M \rangle \) if \( M \) rejects the input \( \langle M \rangle \) and it rejects of if \( M \) accepts the input \( \langle M \rangle \). Now the question is what is the output of of \( D \) when run on \( \langle D \rangle \)?

You will see that there is a problem, very similar to the Russell's paradox here. Hence there cannot be a TM \( D \). But we can construct \( D \) if there is a TM \( M_H \). So neither can there be a TM \( M_H \) which decides \( H \).

\( H \) is popularly known as the halting problem which was shown by Turing to be not decidable (or undecidable).

!bu-problem Let \( \text{EMPTY} = \{ \langle M \rangle : M \text{ rejects all inputs } \} \). Show that this problem is undecidable. !eu-problem

Lecture 3: Non Determinism, NP and Search Problems

Non Deterministic Turing Machines

Last lecture we saw that TM's can be encoded as strings and simulated by a universal TM using the configurations. We also saw that simulation of a TM, is essentialy tracing a path in the configuration space. But in the TMs that we defined, the out degree of any node (a configuration) in the graph (in the configuration space) is one.

Similar to the Nondeterministic finite automata, we can also define TMs with delta rules that results in multiple next states (called Nondeterministic Turing Machines or NTMs). Then the delta rules will be of the form \( \delta:\Omega\times \mathcal{T} \rightarrow \mathcal{P}(\Omega\times \mathcal{T}\times \{\langle, \rangle, -\}) \). (\( \mathcal{P} \) denotes the power set). So the configuration graph of a NTM can have out degree greater than \( 1 \). However the it still has to be a finite number since the size of \( \mathcal{P}(\Omega\times \mathcal{T}\times \{\langle, \rangle, -\}) \) is finite. Following the different paths, an NTM could accept, reject or keep looping. So we need to define what is meant by deciding a language by an NTM.

An NTM is said to decide a language (a descision problem) \( L \) iff, for all strings in the language, there exists one path in the configuration space that results in accept state. For strings not in the language, all paths in the configuration space should result in reject state.

We know that NFA (Nondeterministic Finite automata) can always be converted to a deterministic one. However for Pushdown Automaton this coverstion is not possible always. We will see that for TMs, this conversion can always be done. That is the set of languages that can be decided by TMs does not change by allowing nondeterminism.

Recall that UTM simulated a deterministic TM, by tracing the path in the configuration space. But for NTMs, the graph in the configuration space is a tree. A simple idea is for a UTM to do a graph traversal. DFS might be a bad idea, because some of the paths go into infinite loops. Hence it can to BFS. The first time, it finds that the NTM has reached the accept state, the UTM can also accept. If it never finds an accept state, the simulating TM rejects.

Nondeterministic Polynomial Time : NP

As we disscussed earlier, an NTM can take different paths in the configuration space. The length of the path is essentialy the number of steps. Now we will define the worst case running time for a NTM.

For a language \( L \), on inputs of size \( n \), the worst case running time of an NTM is the length of the longest path in the configuration space on any of the inputs of size \( n \).

NP or Nondeterministic Polynomial time is the class of descision problems for which, there is an NTM which decides it in worst case polynomial time.

NP : Verifier Definition

We will define descision problems that are verifiable. A descision problem is said to be verifable if there exists a deterministic TM \( M \) that takes two inputs \( (x,y) \) where \( x \) is an instance of the descision problem and \( y \) is called a certificate which has length of atmost \( p(n) \) where \( p \) is a polynomial. For YES-instances \( x \), there should exist a certificate \( y \) such the \( M \) accepts the input \( (x,y) \). For instances not in the language, for every \( y \) of length \( p(n) \), \( M \) should reject on \( (x,y) \). Also the running time of \( M \) must be polynomial time in size of \( x \).

!bu-problem Show that NP is the same as the set of verifable languages. !eu-problem

An example of a verifiable language is the Clique problem.

$$ \text{CLIQUE} = \{ (G,k): G \text{ has a clique of size } k \} $$

A certificate for YES-instance \( (G,k) \), is just the list of vertices in a $k$-clique. The polynomial time verifier just checks if there is an edge between all pairs of vertices. If \( (G,k) \) is not in the language, then for any set of \( k \) vertices, that you can give to the verifier, it will reject, since there will not be an edge between some pair in the list.

!bu-problem Show that the CHROMATIC-NUMBER problem defined in Lecture 1 is verifiable. !eu-problem

Descision vs Search Problems

Lecture 4: Reductions, Cook-Levin Theorem and NP-Completeness

The notes are mostly from Section 7.4, 7.5 in the Sipser book.

Polynomial Time Reductions

Last lecture, we showed that for some problems, the search problem can be solved in polytime, if the corresponding desciion problem can be solved in polytime. In this lecture, we will show that many desicion problems can be solved by solving one particular desciion problem called $3$-SAT.

For this we need first define a reduction. The following problems were defined previously

  1. $3$-SAT = \( \{ \phi : \phi \text{ is a 3CNF formula that is satisfiable} \} \).
  2. CLIQUE = \( \{ (G,k) : G \text{ has a clique of size } k \}. \)
We will say that \( f \) is a (poly time) reduction from $3$-SAT to CLIQUE, if it maps $3$-CNF formulas \( \phi \) to a tuple \( (G,k) \) where \( G \) is a graph and \( k \) is a number such that
  1. \( f \) can be computed by a polynomial time TM.
  2. \( \phi \) is satisfiable if and only if \( G \) has a \( k \) clique.
If we have such an \( f \), any polynomial time algorithm for CLIQUE can be used to design a polynomial time algorithm for $3$-SAT.

Verify that: If we have such an \( f \), give a polynomial time algorithm for $3$-SAT, assuming there is a polynomial time algorithm for CLIQUE.

The algorithm for computing \( f \) is as follows:

  1. For every clause in \( \phi \), put \( 3 \) new verticies correponding to each literal in the clause.
  2. Put all edges in the graph except:
    1. between the 3 veritices correponding to the same clause.
    2. \( x_i \) an \( \bar x_i \) for all \( i \).
  3. Set \( k \) to be equal to the number of clauses.
Verify the following:
  1. \( f \) is polynomial time.
  2. Show that if \( \phi \) is satisfiable \( G \) has a \( k \) CLIQUE.
  3. Show that if \( G \) has a \( k \) CLIQUE then \( \phi \) is satisfiable.
Such a reduction says that CLIQUE is a harder problem than $3$-SAT, because an algo for CLIQUE gives an algo for $3$-SAT and we dont know if the reverse is True. Hence it is denote as $$ \text{CLIQUE} \geq_p 3\text{-SAT}.$$

Cook-Levin Theorem

The Cook-Levin Theorem tells that the reverse reduction also exists. In fact, it states that any language in NP can be reduced to SAT

Cook-Levin Theorem. For any language \( L \in NP \), SAT \( \geq_p \) L.

For doing this, for any language \( L \), that has a nondeterministic polynomial time TM, we need to come up with a polynomial time reduction, which satisfies the conditions, given in the previous section.

See Theorem 7.37 for the proof of Cook-Levin Theorem.

!bu-problem Show that $3$-SAT \( \geq_p \) SAT. !eu-problem

NP Compeleness / Hardness

Vertex Cover 3SAT Reduction.

!bu-problem Solve Problem 7.21 in Sipser book !eu-problem

!bu-problem Solve Problem 7.23 in Sipser book !eu-problem

!bu-problem Solve Problem 7.27 in Sipser book !eu-problem

Lecture 5: More Time Complexity

NP-completeness of Vertex Cover and Subset Sum

See Section 7.5 in Sipser 2nd Edition

EXP and Time Heirachy Theorem

EXPTIME is the set of languages for which there is a poly time determinitic TM that runs in \( 2^{n^k} \) for some integer \( k \).

Clearly \( \text{P} \subseteq \text{NP} \subseteq \text{EXPTIME} \).

As we discussed, the question if P=NP or P \( \subsetneq \) NP is an open problem. But what about P \( \subsetneq \) EXP?

This is easy to prove using diagonalization. Let DTIME(\( f(n) \)), be the set of languages that can be decided in time \( f(n) \) by a determinitic TM. Note that P \( \subseteq \) DTIME(\( 2^n \)). We will design a language that is different from all languages in DTIME(\( 2^n \)) by can be decided in EXPTIME.

We define the language by giving a TM \( D \). \( D \) takes encodings of TMs as input \( \langle M \rangle \). D simulates M on \( \langle M \rangle \) for \( 2^n \) steps. If M halts in \( 2^n \) steps, it gives the opposite answer. Otherwise it rejects. Due to the overheads involved in simulating a TM, the TM \( D \) will take more time than \( 2^n \), however halts in less than \( 2^{n^2} \) steps. Hence the language decided by \( D \) is in EXPTIME.

But can the language decided by \( D \) be in DTIME(\( 2^n \)). That is their another TM \( D' \) that decided this language in \( 2^n \) steps. If so what is the output of \( D \) on the input \( \langle D' \rangle \).

You can see that this results in a contradiction and hence the language decided by \( D \) is not in P. So P \( \subsetneq \) EXPTIME.

The question of whether NP = EXPTIME is again open.

coNP and Map of Complexity classes

The complement of a language \( L \) is $$ \bar L = \{ x : x \not in L \}.$$ Let coNP \( = \{ L : \bar L \in \text{NP} \} \).

Another way of defining coNP is: it is the set of languages \( L \) for which there is a determinitic TM \( M \) that takes 2 inputs \( x,y \) such that

  1. If \( x \in L \) then for every \( y \), \( M(x,y) \) accepts.
p If \( x \notin L \) then there exist a \( y \), for which \( M(x,y) \) rejects.

!bu-problem Show that these two definitions gives the same set of languages. !eu-problem

coNP is a set of languages for which there is a certificate for non membership. Similar to NP-complete, we can define: $$\text{coNP-complete} = \{ L \in \text{coNP} : \forall L' \in \text{coNP}, L' \leq_p L \}$$

!bu-problem Show that UNSAT = \( \{ \phi : \phi \text{ is a boolean formula that is unsatisfiable} \} \), is coNP-complete (a formula is unstatisfiable when for all assignments to the variable, the formula evaluates to False). !eu-problem

Note that P = coP.

Map of complexity classes

Lecture 6: Space Complexity

Space Complexity classes and some Relationships

Let SPACE$(f(n))$ be the set of languages that can be decided in space \( f(n) \), then $$\text{TIME}(f(n)) \subseteq \text{SPACE}(f(n)) \subseteq \text{TIME}(2^{cf(n)}).$$

Let coSPACE$(f(n))$ be the complements of languages in SPACE$(f(n))$, then $$\text{coSPACE}(f(n)) = \text{SPACE}(f(n)).$$

Let L be the set of languages that can be decided in space \( c\log n \) for some constant \( c \), and PSPACE the set of languages that can be deciding in polynomial space. Then $$ \text{L} \subsetneq \text{PSPACE}.$$

!bu-problem Show that

  1. \( \text{L} = \text{coL} \subseteq \text{P} \subset \text{PSPACE} \subseteq \text{EXPTIME}. \)
  2. \( \text{L} \subsetneq \text{PSPACE}. \)

Non Determinism and Space : NL-complete, Savith's Theorem

Just like we defined NP, we can define NL as the set of languages that is decided by a non deterministic log-space TM. Whether L = NL is again a well known open problem like P = NP. We can do reductions in NL. As we will see later \( NL \subseteq P \). Therefore any two languages in NL are polynomial time reducible (in fact a polytime machine solve the problem itself). Hence for defining NL-complete langauges, we use log-space reductions that runs in \( O(\log n) \) space (denoted by \( \leq_l \)). That is $$\text{NL-complete} = \{ R : R \text{ is a language such that for all other languages R' } \in L, R' \leq_l R \}.$$

!bu-problem Show that

  1. \( \text{REACHABILITY} \in \text{NL-complete}. \)
  2. \( \text{NL} \subseteq \text{P}. \)

Actually all languages is NL can be decided by a deterministic \( O(\log^2 n) \) space TM. This theorem is known as Savitch's Theorem.

Savitch's theorem says that \( \text{NPSPACE}(f(n)) \subseteq \text{SPACE}(f^2(n)) \). See proof in proof of Theorem 8.5 in Sipser book.

Overview of next part of course

Streaming algorithms for Well paranthesised expressions. Lowerbound using communication complexity.


Chapter 8, Introduction to Theory of Computation by Micheal Sipser, Edition 2.

Chapter 4, Computational Complexity: A Modern Approach Sanjeev Arora and Boaz Barak http://theory.cs.princeton.edu/complexity/book.pdf

Lecture 7: NL = coNL, Randomization

Recapping Nondeterminism

When do we say a Nondeterministic TM accepts a languages?

Why is it not obiviously true that \( NP = coNP \).?

Verifier definition of NP, NL.

Immerman-Szelepcsenyi Theorem : NL = coNL

In time complexity, we didnt have a proof of NP = coNP. This remains an open problem. However in space complexity, we know that NL = coNL. This theorem is known as the Immerman-Szelepcsenyi Theorem. See Section 8.6 in Sipser book.

!bu-problem Given an NL algorithm (use verifier definition) :

  1. which on input a graph \( G(V,E) \), vertices \( s,t \in V \) and a number \( i \), checks if \( s \) is reachable from \( t \) in \( \leq i \) steps.
  2. which on input a graph \( G(V,E) \), verticex \( s \in V \) and a numbers \( i, c \), checks if the number of vertices reachable from \( s \) in \( i \) steps is \( \geq c \).
  3. which on input a graph \( G(V,E) \), verticex \( s \in V \) and a numbers \( i, c \), checks if the number of vertices not reachable from \( s \) in \( i \) steps is \( =c \).

Randomized TMs and Complexity Classes

!bu-problem Show that

  1. RP \( \subseteq \) BPP
  2. coRP \( \subseteq \) BPP
  3. BPP \( \subseteq \) PSPACE
  4. RP \( \subseteq \) NP
  5. BPP \( = \) coBPP

!bu-problem Bonus Problem A forest is a graph which is a disjoint union of trees (ie tree on different set of vertices). Show that REACHABILITY in a forest can be solved using \( O(\log n) \) space. !eu-problem


Chapter 8, Introduction to Theory of Computation by Micheal Sipser, Edition 2.

Chapter 4, Computational Complexity: A Modern Approach Sanjeev Arora and Boaz Barak http://theory.cs.princeton.edu/complexity/book.pdf

Lecture 8: Randomized Algos

BPP and Amplification

Last lecture we saw that \( \text{RP}_{1/100} = \text{RP}_{99/100} \). This lecture we said that \( \text{BPP}_{1/2 + \epsilon} = \text{BPP}_{1 - \epsilon} \) for any small constant \( \epsilon \).

!bu-problem Let \( \text{BPP}_\alpha \) be the class of languguages L that have a PTM M such that

  1. If \( x\in L \) then \( \Pr[M(x) \text{accepts}] \geq \alpha \).
  2. If \( x\notin L \) then \( \Pr[M(x) \text{rejects}] \geq \alpha \).
Show that:
  1. \( \text{BPP}_{1/2} \) is the set of all languages.
  2. \( \text{BPP}_{2/3} = \text{BPP}_{20/27} \) (without using Chernoff's Bounds).

See Chapter 10.2 in Sipser

Approx MAX-CUT

See Chapter 10.1 in Sipser for approximation algorithms. See Section 2.3.4 in https://people.seas.harvard.edu/~salil/pseudorandomness/power.pdf

!bu-problem The Approximate-$7/8$-$3$SAT problem is given an input a $3$-SAT boolean formula, find an assignment that satisfies atleast \( 7/8 \) fraction of the clauses (ie if there are \( m \) clauses, find an assignment that satisfies \( m\times 7/8 \) clauses).

Give a randomized algorithm which solves Approximate-7/8-3SAT (answer just need to be correct in expectation as we discussed for the MAXCUT problem in class.). !eu-problem


See Section 2.4 in https://people.seas.harvard.edu/~salil/pseudorandomness/power.pdf

!bu-problem Let M be the adjacency matrix of a undirected $d$-regular graph \( G \) (all vertices have exactly \( d \) neighbours). Show that:

  1. \( d \) is an eigenvalue of \( M \).
  2. If \( c \) is the number of connected components in \( G \), then there are \( c \) linearly independent eigenvectors of \( M \) with eigen value \( d \).
  3. If \( G \) is a bipartite graph then \( -d \) is an eigenvalue of \( M \).

!bu-problem Bonus Problem Let M be the adjacency matrix of a undirected $d$-regular graph \( G \) (all vertices have exactly \( d \) neighbours). Show that:

  1. If there are \( c \) linearly independent eigenvectors of \( M \) with eigen value \( d \) then \( G \) has \( c \) connected components.
  2. If \( -d \) is an eigenvalue of \( M \) then \( G \) is a bipartite graph,.


Chapter 10, Intro to Theory of Comp., Edition 2, Sipser

Chapter 2, Pseudorandomness survey, Salil Vadhan https://people.seas.harvard.edu/~salil/pseudorandomness/power.pdf https://people.seas.harvard.edu/~salil/pseudorandomness/

Chapter 7, Computational Complexity: A Modern Approach Sanjeev Arora and Boaz Barak http://theory.cs.princeton.edu/complexity/book.pdf