===========================================================================
CSC 363            Homework Exercise 5 -- Marking Scheme          Fall 2008
===========================================================================


NOTE TO STUDENTS:

You will find below the marking scheme used for your homework, including
the meaning of marking codes and number of marks associated with each one.
This file also contains my instructions to the marker (so you can get an
idea of how the homework was marked) and the marker's comments about each
question.

Please take the time to read this carefully before you ask questions about
the grading of your homework.


NOTE TO MARKER:

Be picky!  On any homework, it is the responsibility of students to show
that they understand how to solve each problem and to write up their
answers carefully.  At the same time, keep in mind that exercises are worth
only 1.5% of the final grade, so your marking should be somewhat coarse.
And remember that marking is not just about evaluating the students's
performances, but also about giving them feedback so that they can learn
from their mistakes.  This is especially important for students who made
numerous or more serious mistakes, as they are likely to need more feedback
in order to understand why their answers were incorrect.

For each question, I list solution elements with an associated code for
writing on student papers (the letter(s) between underscores _) and a
number of marks.  There are also general errors (with associated codes)
given below, with a maximum number of marks to take off for each type of
general error (as a percentage of the value of the question).  You will
likely encounter other common errors, or maybe decide to break down the
marking scheme further.  Simply make note of these changes/additions to
the marking scheme, and introduce new code letters (or short words) to
allow you to quickly give accurate feedback to the students (both in terms
of what they did wrong and how many marks it cost them).


GENERAL ERRORS (marked negatively, in addition to any other errors):

    _N_otation [up to 20%]:  incorrect/ambiguous notation

    _V_agueness [up to 20%]:  incorrect/unjustified/vague claim


1.  [8 marks]

    (a) [3 marks]

        _S_tructure [1 mark]:  clear attempt to describe a polytime decider
            for A n B, based on the polytime deciders for A and B, and to
            justify the correctness of the new decider briefly

        _D_ecider [1 mark]:  correct polytime decider for A n B

        _C_orrectness [1 mark]:  good argument of correctness

    (b) [5 marks]

        _S_tructure [1 mark]:  clear attempt to describe a polytime
            verifier for A*, based on the polytime verifier for A, and to
            justify the correctness of the new verifier briefly

        _V_erifier [2 mark]:  correct polytime verifier for A*

        _C_orrectness [2 marks]:  good argument of correctness


2.  [7 marks]

    (a) [2 marks]

        _C_over [1 mark]:  correct vertex cover

        _I_ndependent set [1 mark]:  correct observation that complement of
            vertex cover is independent set

    (b) [5 marks]

        _S_tructure [1 mark]:  clear attempt to start with an arbitrary
            input <G,k> for VC, to describe the construction of a specific
            input <G',k'> for IS, to argue that the construction takes
            polytime and that <G,k> (- VC iff <G',k'> (- IS

        _R_eduction [2 marks]:  correct construction of <G',k'> from <G,k>
            (Note that this construction f(G,k) = <G',k'> has to satisfy
            the property that <G,k> in VC iff <G',k'> in IS, but the
            function f does *not* have to find the independent set or
            vertex cover within the graph G, something that cannot be done
            in polynomial time.)

        _C_orrectness [2 marks]:  good argument of correctness