Algorithm Design

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Chapter 8 NP and ComputatiOnal Intractability
We now further constrain the instance so that the only way to solve it will
be to group together a subset of the jobs whose durations add up precisely to
W. To do this, we define an (n + 1) st job; it has a release time of W, a deadline
of W + 1, and a duration of 1.
Now consider any feasible solution to this instance of the scheduling
problem. The (n + 1) st job must be run in the interval [W, W + 1]. This leaves
S available time units between the common release time and the common
deadline; and there are S time units worth of jobs to run. Thus the machine
must not have any idle time, when no jobs are running. In particular, if jobs
ik are the ones that run before time W, then the corresponding numbers
wig in the Subset Sum instance add up to exactly W.
Conversely, if there are numbers wil ..... wig that add up to exactly W,
then we can schedule these before job n + I and the remainder after job n + !;
this is a feasible solution to the schednling instance. TM
Caveat: Subset Sum with Polynomially Bounded Numbers
There is a very common source of pitfalls involving the Subset Sum Problem,
and while it is closely connected to the issues we have been discussing already,
we feel it is worth discussing explicitly¯ The pitfall is the following..
Consider the special case of Subset Sum, with n input numbers, in which W
is bounded by a polynomial ~nction o[ n. Assuming ~P ~ ~N~P, this special
case is not NP-comptete.
It is not NP-complete for the simple reason that it can be solved in time O(nW),
by our dynamic programming algorithm from Section 6.4; when W is bounded
by a polynomial function of n, this is a polynomial-time algorithm.
All this is very clear; so you may ask: Why dwell on its. The reason is that
there is a genre of problem that is often wrongly claimed to be NP-complete
(even in published papers) via reduction from this special case of Stibset Sum.
Here is a basic example of such a problem, which we will call Component
Grouping.
Given a graph G that is not connected, and a number k, does there exist a
subset o[ its connected components whose union has size exactly k?
Incorrect Claim. Component Grouping is NP-complete.
Incorrect Proof. Component Grouping is in ~q~Y, and we’ll skip the proof
of this. We now attempt to show that Subset Sum