# Consider a collection of n closed intervals with distinct endpoints [Si, ti]; you can assume without loss of generality that the Si's and ti's are all distinct integers

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Consider a collection of n closed intervals with distinct endpoints [Si, ti]; you can assume without loss of generality

that the Si's and ti's are all distinct integers (in particular you can simply assume that the Si's and the ti's comprise the numbers 1, 2., '2n).

a) Give an algorithm for finding the largest possible subset of non-overlapping intervals. For example, in Figure

1, an optimal subset is {c, e, h, k, o, q}.

b) Consider a related problem of finding the smallest possible subset of points such that each interval overlaps at

least one point. For example, the points labeled (I) through in Figure I accomplish this. Is it a coincidence

that both problems happen to have the same objective value (i.e. 6)?

Let k > 2 be fixed and suppose you are given a matrix M whose columns consist of a single consecutive block of

an integer between 1 and k — 1, and where every row of M sums to a multiple of k, as shown below (for the case k=8);

Prove that it is possible to select a subset of columns and subtract k from their positive elements so that the rows

Of the new matrix sum to zero. For example, if we select columns l, 2, 3, 5, 9, and II in the above, the resulting

matrix

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