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CS 6520: Computational Complexity
Problem Set 2
Due April 3, 2008
Problem 1
Prove that BPL ⊆ P.
Problem 2
Give a directed graph that has exponential cover time (in contrast with the polynomial bound for undirected graphs as shown in class). Conclude that the RL-algorithm for USTCONN based on a random walk fails when the graph is an arbitrary directed graph and thus fails to decide STCONN.
Problem 3
The complexity class ZPP is the class of languages decidable in expected polynomial time. That is, a language L ∈ ZPP is there is a probabilistic algorithm A and constants c, d such that for every x ∈ { 0 , 1 }∗, both of the following conditions hold:
- A(x) = χL(x) always.
- The expected running time of A on x is at most c|x|d.
Prove that ZPP = RP ∩ coRP.
Problem 4
Prove that if NP ⊆ BPP, then RP = NP.
Problem 5
A branching program on variables x 1 ,... , xn is a directed acyclic graph where all nodes are labeled with a variable xi, except for two output nodes, one of which is labeled with 0 and the other labeled with 1. Both of the output nodes have outdegree 0. Every node other than the two output nodes has
outdegree 2, of which one outgoing edge is labeled with 0 and the other labeled with 1. One node with indegree 0 is designated as the start node. A branching program on variables x 1 ,... , xn defines a Boolean function f : { 0 , 1 }n^ → { 0 , 1 }, where for each α = (α 1 ,... , αn) ∈ { 0 , 1 }n, the value f (α) is defined as follows: Begin at the start node, and follow the path determined by taking the outgoing edge from each node that is labeled by the value assigned by α to the indicated variable,^1 until an output node is reached. The label of the output node reached is the value of f (α).
- Prove that the class of Boolean functions computable by a polynomial- sized branching program is precisely L/Poly.
- A branching program is read-once if every variable appears at most once in every path from the start node to an output node. Prove that the problem of deciding whether two read-once branching programs compute the same Boolean function, is in coRP. (Hint: You may find Polynomial Identity Testing useful. )
Problem 6
Prove that
- Every language that has an interactive proof with a deterministic ver- ifer, is in NP. More generally, every language that has an interactive proof with zero soundness error is in NP.
- The class IP is not changed if the prover is allowed to be probabilistic.
- IP ⊆ PSPACE.
- Allowing a probability of error in the completeness condition does not change the class IP. That is, if a language has an interactive proof with completeness c and soundness error s, for any c and s satisfying 0 ≤ s < c ≤ 1, then it has an interactive proof with perfect completeness.
(^1) That is, if the current node in the path is labeled with xi, then take the edge labeled with αi to reach the next node.
Problem 9 (Extra Credit)
Prove that every deterministic algorithm for Polynomial Identity Test but with only oracle access to the tested polynomial f : Fn^ → F, where F is a finite field, the degree of f equals n which is the number of variables, and |F| > n, requires at least 2n^ queries to the f -oracle. Does this imply that P 6 = RP? Justify your answer.
Problem 10 (Extra Credit)
In class the elegant RL-algorithm for USTCONN based on a random walk was analyzed using linear algebra. The following are a few results that were stated but not proven in class. Let M be the transition matrix for a random walk on a d-regular undi- rected multi-graph^2 G = (V, E) where V = { 1 ,... , n}. Prove the following statements.
- Every eigenvalue of M has absolute value at most 1.
- G is connected if and only if 1 is an eigenvalue of multiplicity 1.
- Prove that the second largest eigenvalue of M (not in absolute value) equals
max x⊥~ 1 ,‖x‖=
〈M x, x〉 = 1 −
d
· min x⊥~ 1 ,‖x‖=
{i,j}∈E
(xi − xj )^2 ,
where both max and min are taken over unit vectors orthogonal to the all-1 vector.
- Using the above, prove that if G is connected, then every eigenvalue of M other than 1 is at most
1 − O
dn^3
- G is bipartite if and only if −1 is an eigenvalue of M.
- Using all above, prove that if G is connected and non-bipartite,
λ 2 (G) ≤ 1 − O
dn^3
(^2) A multi-graph is a graph where parallel edges and self loops are allowed.
- Establish a tighter bound
1 − O
dDn
for Part (4) and (6), where D is the diameter of G.
