Tutorial week 12

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Tutorial week 12

• Li Yongkun

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outline

• FFT example
• hw6 answer
• hw7 answer

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FFT --one example

• Compute the FFT of (1,0,1,0,).
• FFT (2,0,2,0)

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Hw6 problem1 --description

• 3-Coloring is a yes/no question, but we can
  phrase it as an optimization problem as follows.
• Suppose we are given a graph G (V, E), and we
  want to color each
• node with one of three colors, even if we arent
  necessarily able to give different
• colors to every pair of adjacent nodes. Rather,
  we say that an edge (u, v) is
• satisfied if the colors assigned to u and v are
  different. Consider a 3-coloring that
• maximizes the number of satisfied edges, and let
  c denote this number. Give a
• polynomial-time algorithm that produces a
  3-coloring that satisfies at least 2/3c
• edges. If you want, your algorithm can be
  randomized in this case, the expected
• number of edges it satisfies should be at least
  2/3c.

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Hw6 problem1 --algorithm

• Algorithm
• Initialize for each vertex v ? V color v
  lt- 0
• //use 0, 1, 2 represent three colors
• For each vertex v ? V
• Randomly pick one color c
• Coloring v color v lt- c

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Hw6 problem1 --efficiency

• We want the expected number of edges it
  satisfies should be at least 2/3c.
• For each node, there are 3 choices. So, for each
  edge, there are 339 color state. Only when two
  nodes have the same color, the color state is not
  satisfied. So there are 3 states are not
  satisfied for each edge. For each edge, the
  probability that it satisfies the condition is
  2/3.
• For each edge e, define r.v. X(e), 1 e is
  satisfied, 0 e is not satisfied
• P(X(e)1)2/3.
• Define r.v. X number of edge that is satisfied,
  X

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Hw6 problem2 --description

• Let G (V, E) be an undirected graph with n
  nodes and m edges. For a
• subset X ? V, we use GX to denote the sub-graph
  induced on Xthat is, the
• graph whose node set is X and whose edge set
  consists of all edges of G for
• which both ends lie in X. We are given a natural
  number k n and are interested in
• finding a set of k nodes that induces a dense
  sub-graph of G well phrase this
• concretely as follows. Give a polynomial-time
  algorithm that produces, for a given
• natural number k n, a set X ? V of k nodes with
  the property that the induced
• sub-graph GX has at least mk(k-1)/n(n-1)
  edges. You may give either
• a deterministic algorithm, or
• a randomized algorithm that has an expected
  running time that is polynomial, and that only
  outputs correct answers.

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Hw6 problem2 --algorithm

• Algorithm
• Initialize X is empty
• While( GX is not satisfied) // less than
  mk(k-1)/n(n-1) edges
• Randomly pick k nodes
• Add these nodes into X

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Hw6 problem2 --analysis

• Step1 Prove For each GX generated by randomly
  picking k nodes. The expectation of number of
  edges is mk(k-1)/(n(n-1)).
• For each edge e, define r. v. Y(e) e is chosen
  or not. 1 yes, 0 no
• e is chosen iff both ends of e lie
  in X.
• Define r. v. Y number of edges of GX

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Hw6 problem2 --analysis

• Step2 Prove For each randomly constructed GX,
  Pr EX mk(k-1)/(n(n-1)) 1/(mn(n-1)).
• Define p(i) GX has i edges
• Define
• So

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Hw6 problem2 --analysis

• Step3 After polynomial times loop, you can get
  the GX which has at least mk(k-1)/(n(n-1))
  edges
• Pr EX mk(k-1)/(n(n-1)) 1/(mn(n-1)).
• O(mn(n-1))
• If the condition becomes GX has at least
  edges
• O (m) is enough

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Hw6 problem3 --description

• Suppose youre designing strategies for selling
  items on a popular auction Web site. Unlike other
  auction sites, this one uses a one-pass auction,
  in which each bid must be immediately (and
  irrevocably) accepted or refused. Specifically,
  the site works as follows.
• First a seller puts up an item for sale.
• Then buyers appear in sequence.
• When buyer i appears, he or she makes a bid bi gt
  0.
• The seller must decide immediately whether to
  accept the bid or not.
• If the seller accepts the bid, the item is sold
  and all future buyers are turned away. If the
  seller rejects the bid, buyer i departs and the
  bid is withdrawn and only then does the seller
  see any future buyers. Suppose an item is offered
  for sale, and there are n buyers, each with a
  distinct bid. Suppose further that the buyers
  appear in a random order, and that the seller
  knows the number n of buyers. Wed like to design
  a strategy whereby the seller has a reasonable
  chance of accepting the highest of the n bids. By
  a strategy, we mean a rule by which the seller
  decides whether to accept each presented bid,
  based only on the value of n and the sequence of
  bids seen so far.

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Hw6 problem3 --description

• For example, the seller could always accept the
  first bid
• presented. This results in the seller accepting
  the highest of the n bids with
• Probability only 1/n, since it requires the
  highest bid to be the first one presented.
• Give a strategy under which the seller accepts
  the highest of the n bids with
• probability at least 1/4, regardless of the value
  of n. (For simplicity, you may
• assume that n is an even number.) Prove that
  your strategy achieves this
• probabilistic guarantee.

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Hw6 problem3 --algorithm

• Algorithm
• Auction (n)
• if n 5
• Skip first 2 bids
• i from 3 to 4
• if bid (i) is larger than all first 2 bids,
  accept i-th bid
• else accept the 5th bid
• else
• Skip first bids
• i from 1 to n-1
• If bid (i) is larger than all first
  bids
• Accept i-th bid
• If still not accept any bid, accept n-th bid

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Hw6 problem3 --analysis

• Step1 compute the probability that the accepted
  bid is highest
• when adopt the strategy of
  skipping first k bids.
• Define r. v. X the order of the highest bid
  (possible outcome1, 2, 3, , n)
• Define event A the accepted bid is the largest
  one.
• We want probability that A
  occurs when skip first k bids is adopted.

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Hw6 problem3 --analysis

• So,
• Define g (x)xln(1/x) . g (x)ln(1/x)-1.
• So when x1/e, g (x)0 gtk n/e.
• That means (k/n)ln(n/k) obtains the largest
  value when kn/e.
• k must be integer, let k
• Verify.
• When n is large enough, right.
• n 5, consider separately

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Hw6 problem3 --analysis

• Verify
• We have proved
• ngt5
• ngt8
• 5ltn8
• n5
• 1 n 4 obviously right

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Hw7 problem1 --description

• Consider the following linear program.
• maximize 5x 3y
• 5x - 2y 0
• x y 7
• x 5
• x 0
• y 0
• Plot the feasible region and identify the optimal
  solution.

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Hw7 problem1 --answer

• Draw the figure and parallel
• move the objective function
• The optimal solution is (5,2)

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Hw7 problem2

• The Canine Products company offers two dog foods,
  Frisky Pup and Husky Hound, that are made from a
  blend of cereal and meat. A package of Frisky Pup
  requires 1 pound of cereal and 1.5 pounds of
  meat, and sells for 7. A package of Husky Hound
  uses 2 pounds of cereal and 1 pound of meat, and
  sells for 6. Raw cereal costs 1 per pound and
  raw meat costs 2 per pound. It also costs 1.40
  to package the Frisky Pup and 0.60 to package
  the Husky Hound. A total of 240,000 pounds of
  cereal and 180,000 pounds of meat are available
  each month. The only production bottleneck is
  that the factory can only package 110,000 bags of
  Frisky Pup per month. Needless to say, management
  would like to maximize pro?t.
• (a) Formulate the problem as a linear program in
  two variables.
• (b) Graph the feasible region, give the
  coordinates of every vertex, and circle the
  vertex maximizing pro?t. What is the maximum
  pro?t?

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Hw7 problem2 --answer

• Formulate the problem as a linear program in two
  variables
• Define x packages of Frisk Pup. y packages of
  Husky Hound
• LP max 1.6x1.4y
• s. t.
• x2y240,000
• 1.5xy180,000
• x110,000
• x0 and y0

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Hw7 problem2 --answer

• The optimal solution is
• (60,000, 90,000)
• Maximum profit 222,000

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Hw7 problem2

• set cover The set cover problem is as follows
  Given a set U and a collection of subsets S
  S1, , Sk, and a cost function c which gives
  each set Si a positive integer c(Si), find a
  minimum cost subcollection of S that covers all
  elements of U. That is, each element of U is
  contained in at least one subset Si that you
  picked.
• The frequency f(u) of an element u in U is the
  number of sets Si it is in. The frequency f max
  f(u).
• Write down an integer programming for this task,
  relax it to a LP, and then find a rounding
  method, and prove that it gives a polynomial
  time algorithm with approximation ratio f. That
  is, the algorithm outputs a value c s. t. c ? c
  ? f c, where c is the optimal solution of the
  set cover problem.

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Hw7 problem2 --answer

• Define r. v. X(i) ?0,1 with each S(i). X(i)1
  iff S(i) is in a (fixed) min set cover.
• The objective
• min ?S(i)?S X(i)C(S(i)).
• Constraints
• ?u?S(i) X(i)1 ?u?U
• X(i) ?0,1 ?i

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Hw7 problem2 --answer

• IP
• min ?S(i)?S X(i)C(S(i)).
• s. t.
• ?u?S(i) X(i)1 ?u?U
• X(i) ?0,1 ?i
• Relax to LP
• min ?S(i)?S X(i)C(S(i)).
• s. t.
• ?u?S(i) X(i)1 ?u?U
• 0 X(i) 1 ?i

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Hw7 problem2 --answer

• rounding procedure if X(i) 1/f, then pick it.
• Prove denote
• S an optimal set cover
• X an solution of the LP
• R(X) the rounding solution from X
• Step1 S R(X)
• For any element u since ?u?S(i) X(i) 1, at
  least one of X(i) (u ? S(i)) is 1\f, which
  will be picked to join the set.
• Step2R(x) fS
• ?iX(i) ?iX(i)1/f X(i) // we throw some
  part away
• ?iX(i)1/f 1/f //
  X(i) 1/f
• (1/f)R(x)

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• Thanks!