CSE596, Fall 2015
Problem Set 7
Due Fri. Nov. 6
Reading. For next week, read Homer-Selman section 5.4 (notice that in the ALR notes some
corresponding theorems are stated without proof) and start on 5.5. In the parallel part of the
ALR notes, you may skim/skip the “nondeterministic time hierarchy theorem.”
I have decided to set Prelim II on the Monday before Thanksgiving, Monday Nov. 23,
in class period.
(1) Give a reduction f from the language DT M such that for all e, if e ∈ DT M then Mf (e)
runs in time n + 1, while if e ∈
/ DT M , then Mf (e) is not total. Deduce that for any particular
running-time function t(n) ≥ n + 1, the decision problem
Instance: A deterministic Turing machine M ;
Question: Does M run in time t(n)?
is undecidable—indeed, its language is not c.e. Then show that the language is co-c.e. (12+9
= 21 pts.)
(2) A symmetric n × n 0-1 matrix A represents an n-vertex undirected graph GA = (V, E)
with Aij = 1 ⇐⇒ G has an edge between vertex i and vertex j. We’ll assume that GA has
no self-loops; i.e., Aii = 0 for all i. Represent A in row-major order as a binary string of length
N = n2 . Define L2 to be the language of A such that V can be broken into V1 ∪ V2 such that
all edges in GA go between V1 and V2 .
(a) Show that L2 belongs to P. You don’t have to design a Turing machine M to accept
L2 —you can write pseudocode using data structures (such as sets) and appeal to the
polynomial-time simulation of a RAM/assembly-program by a TM as covered earlier in
the course. As a function of the input length N (not n), what is the asymptotic running
time of your pseudocode? (This is how running time is measured in CSE531. 15+3 = 18
pts.)
(b) A useful theorem stays that if a graph is not in L2 , then it has an odd cycle—i.e., a path
of m edges that starts at some vertex v and comes back to v, such that m is odd. Sketch
how a nondeterministic TM can verify this condition using only O(log N ) space to track a
guessed path, thus showing that the complement of L2 is in NL. (Since NL is closed under
complement—a “shock” theorem whose proof at the end of chapter 5 will be beyond our
scope—this also shows that weirdly L2 is in NL as well. 12 pts.)
(3) Define L3 to be the language of undirected graphs G whose vertex set V can be broken
into V1 ∪ V2 ∪ V3 such that all edges from Vi go to Vj where j 6= i. For example, the 5vertex ”bowtie graph” G with E(G) = { (1, 2), (2, 3), (1, 3), (3, 4), (3, 5), (4, 5) } belongs to L3 ,
but the complete graph on 4 vertices does not. Show that L3 belongs to NP. (Think about
why your approach in the case of L2 fails, but don’t spend any time trying to patch it into a
polynomial-time algorithm here. 12 pts., for 63 on the set.)