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10 Decision Problems
A decision problem is a general question to be answered, usually
possessing several parameters, or free variables, whose values are
left unspecified. An instance of a problem is obtained by
specifying particular values for all the problem parameters.
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Example.
Hamiltonian Circuit
instance A Graph G = (V,E).
question Does G contain a Hamiltonian Circuit?
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"Does the following graph have a Hamiltonian Circuit?" is a
question. It refers to a specific instance.
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is a cycle that contains every vertex.
A solution to a problem is an algorithm that answers the question
that results from each instance. A decision problem is decidable
if a solution exist and it is undecidable otherwise.
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In general we are concerned with the questions of whether certain
decision problems are decidable and if so, whether they can be
decided in polynomial time.
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A "problem" is not the same as a "question"
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According to Church's thesis, every computational device can be
simulated by some Turing machine. Thus, a decision problem is
decidable if and only if there is a Turing machine that solves the
problem. To show that a decision problem is undecidable, we take
the point of view that it suffices to show that no Turing machine
solves the problem. Conversely, to show that a Turing machine
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exists to solve a decision problem, it suffices to present an
informal algorithm. If a decision problem is decidable, we want to
know whether it is feasible. That is we want to know whether
there is a polynomial time-bounded turing machine (This notion
will be formally defined during this course.) that decides the
problem.
Recalling that input to a Turing machine must be presented as a
word over a finite alphabet, the first technical issue we must
address is one of reasonable encodings. In order for a Turing
machine to solve a decision problem about graphs, for example,
graphs must be encoded as words over some finite alphabet.
Furthermore, the encoding must be reasonable in the sense that the
length of the word that represents the graph must be no more than
a polynomial in the length of whatever is considered to be a natural
presentation of a graph.
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Reasonable encoding should be concise and not" padded" with
unnecessary information or symbols. Numbers should be
represented in
binary, or any other concise
representation, and should not be represented in unary.
If we restrict ourselves to encoding schemes with these
properties, then the particular choice of representation will not
affect whether a given problem is feasible. For example, one might
present a graph G = (V,E) either by a Vertex List and an Edge List or
with an Adjacency matrix. For our purposes, either is acceptable.
Once such a decision is made, it is straightforward to encode the
presentation as a word over a finite alphabet. To wit, suppose that·
V = {1, ... , k} and E consists of pairs of the form (e l , e z). Then,
1.
Denote each integer by its 2-adic representation. Let
c(i) be the 2-adic representation of the number i.
2.
For each I, I = 1, ... k, let [c(i)] represent the vertex I.
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In general, if Xl , ..., xn represent the objects Xl , .... ><0, let
(XI' ... , xn) represent the object (Xl , .... ><0). Thus, in
particular, an edge (ell e z) is represented by the string
([c(e l )], [c(eJ]).
In this manner every graph is representable as a string over the
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finite alphabet { 0, 1, [, ], (, )}. Other data structures are encoded
in a similar manner.
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