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Gybe’s, the Anthea. Daffodil is the daughter of the owner of the yacht that is named after Mr. Luff’s daughter. Mr. Windward’s daughter is named Lalage. Who is Jonquil’s father? 3. Prove that for y ∈ {0, 1}n, 0 ≤ Ff (y) ≤ 1. 4 For a 3-CNF formula φ = C1 C2 · · · Cm over Boolean variables x1 , x2 , . . , xn , let x be the vector (x1 , x2, . . , xn) in {0, 1}n. For each variable xj , 1 ≤ j ≤ n, Exercises 29 define a corresponding real variable yj , and let y be the vector (y1 , y2 , . . , yn ) in Rn .

1. Now, we prove the “if” part. Suppose that (E, I) is not a matroid. Then we can find a subset F of E such that F has two maximal independent subsets I and I with |I| > |I |. Define, for any e ∈ E, ⎧ ⎨1+ , c(e) = 1, ⎩ 0, if e ∈ I , if e ∈ I \ I , if e ∈ E \ (I ∪ I ), where is a positive number less than 1/|I | (so that c(I) > c(I )). A produces the solution set I , which is not optimal. The following are some examples of matroids. 8 Let E be a finite set of vectors and I the family of linearly independent subsets of E.

Without loss of generality, assume that c(e1 ) ≥ c(e2 ) ≥ · · · ≥ c(en ). (2) Set I ← ∅. (3) For i ← 1 to n do if I ∪ {ei } ∈ I then I ← I ∪ {ei }. (4) Output IG ← I. A. We will see that c(IG )/c(I ∗ ) has a simple upper bound that is independent of the cost function c. For any F ⊆ E, a set I ⊆ F is called a maximal independent subset of F if no independent subset of F contains I as a proper subset. For any set I ⊆ E, let |I| denote the number of elements in I. Define u(F ) = min{|I| | I is a maximal independent subset of F }, v(F ) = max{|I| | I is an independent subset of F }.

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