CSE596, Fall 2015
Problem Set 9
Due Fri. Dec. 4
Reading. Next Monday will finish NP-completeness and then move on from ALR ch. 28, section 4
to section 5, which is the last section of the ALR notes to be covered. Completeness (under logspace
reductions) for NL and P will be introduced, then completeness for PSPACE which makes sense
under either ≤pm or ≤log
m . This will be in tandem with the coverage in the text’s section 7.5 of
log-space reductions and the same results. That ≤log
m is transitive—i.e., that logpsace-computable
functions are closed under composition—is not a trivial result, and it will set up ideas that go into
NL ⊆ DSPACE[log2 n] (which implies the rest of Savitch’s Theorem by padding-and-translation) and
the PSPACE-completeness of the QBF language. The following week will cover the earlier portions
of Homer-Selman, chapter 7, but skipping the first part of section 7.3 and all of section 7.4 on the
polynomial hierarchy.
(A) Discussion Problem: Harking back to the discussion problem on Problem Set 2, show that the
following problem is NP-hard. More precisely, show that the following problem is complete for co-NP
under polynomial-time many-one reductions (3 pts. checkoff):
Instance: A regular expression R with no Kleene stars and an integer n > 0.
Question: Does R match all strings in {0, 1}n ?
(1) Prove that the problem (c) on question (1d) of Prelim II is NP-hard. More precisely, prove
that the complement of the language {(N, n) : {0, 1}n ⊆ L(N )} is NP-complete under ≤pm . Use a
reduction from 3SAT by transforming a 3CNF formula φ(x1 , . . . , xn ) into an NFA Nφ such that Nφ
fails to accept some y ∈ {0, 1}n iff y satisfies all clauses. Explain your construction of Nφ , why it is
computable in polynomial time given φ in 3CNF, and why your reduction is correct. (18 pts.)
(2) Show that the following decision problem is NP-complete:
Dominating Set
Instance: An undirected graph G = (V, E) and an integer k, 1 ≤ k ≤ |V |.
Question: Does there exist a subset U ⊆ V of size at most k such that every other vertex
in V is adjacent to one in U (that is, (∀v ∈ V \ U )(∃u ∈ U ) : (u, v) ∈ E)?
Needless to say, it is forbidden to look this up on the Internet.
For some helpful examples, if G is a “triangle,” any one node forms a dominating set that is not a
vertex cover. If G has 6 vertices and edges in the form of the letter ‘H’ then the only dominating set
of size 2 is obtained by choosing the two middle vertices, and these do not form an independent set.
(24 pts.)
(3) Tied s-t Path. You are given a directed acyclic graph G = (V, E) in which each node has one
“left” out-arc and one “right” out-arc, with a distinguished source node s and sink node t. You are
also given a list of “ties” (u, v) which say that if you take the left [right] edge out of u, then you must
also take the left [right] edge out of v. Is there a path from s to t subject to the ties? Show that
this decision problem is NP-complete. (30 pts., for 75 regular-credit points on the set, including the
checkoff credit for A)
[18 pts. extra credit for doing this with the extra condition that no node is tied more than
once, i.e., the ties are disjoint pairs of nodes—you may wish to lean on special properties of the 3CNF
formulas that translate circuits in my Cook-Levin theorem proof.]