Problem:
0-1 Knapsack Problem

You have N items that you want to put them into a knapsack. Item i has value vi and weight wi.

You want to find a subset of items to put such that:

The total value of the items is as large as possible.

The items have combined weight at most W, that is capacity of the knapsack.

Find the maximum total value of items in the knapsack.

Input

N W
v1 w1
v2 w2
:
vN wN

The first line consists of the integers N and W. In the following lines, the value and weight of the i-th item are given.

Output

Print the maximum total values of the items in a line.

Constraints

1 ≤ N ≤ 100

1 ≤ vi ≤ 1000

1 ≤ wi ≤ 1000

1 ≤ W ≤ 10000

Sample Input 1

4 5
4 2
5 2
2 1
8 3

Sample Output 1

13

Sample Input 2

2 20
5 9
4 10

Sample Output 2

9

Solution in Ruby (judge verdict: Time Limit Exceeded)

@data = '50
5
60,10
100,20
120,30
210,45
10,4
20
8
10,7
4,2
5,9
1,4
7,9
3,7
6,4
3,5
0
'.split("\n")

# @data = readlines

P_CROSS = 0.6
P_MUTATION = 0.01

Item = Struct.new(:value, :weight)

Problem = []

def init(count = 0)
  if @data[0] != '0'
    Problem << {Max_weight: @data[0].to_i, N: @data[1].to_i, Items: Array.new(@data[1].to_i)}
    @data[1].to_i.times do |i|
      Problem[count][:Items][i] = Item.new(*@data[i+2].split(',').map(&:to_i))
    end
#    p Problem[count]
    @data = @data[(@data[1].to_i+2)...@data.length]
    init(count+1)
  end
end

def aoj
  n,w = gets().chomp.split(' ').map(&:to_i)
  @data = []
  @data << w
  @data << n
  n.times do |i|
    @data << gets().chomp.split(' ').join(',')
  end

  Problem << {Max_weight: @data[0], N: @data[1], Items: Array.new(@data[1])}
  @data[1].times do |i|
    Problem[0][:Items][i] = Item.new(*@data[i+2].split(',').map(&:to_i))
  end
end

class GA

  # 0: ??\???????????? 1: ??\?????????
  GENE = Struct.new(:score, :chromosome)

  def initialize(population, items, max)
    @generation = 1
    @population = population # ????????°
    @items = items
    @max = max # ??????????????????????????§??????
    @size = items.length # ????????????
    @genes = Array.new(@population) { GENE.new(0, Array.new(@items.length)) }
    start_time = Time.now
#    p @items

#    srand(start_time.sec)

    @genes.each do |gene|
      gene.chromosome.map! { |locas| locas = rand(2) }
#      puts gene
    end

    evaluate(@genes)

    sort_genes
#   puts @genes

  end

  def evaluate(genes)
    genes.each do |gene|
      value_sum = 0
      weight_sum = 0
      gene.chromosome.each_with_index do |locas, index|
        if locas == 1
          value_sum += @items[index].value
          weight_sum += @items[index].weight
        end
      end
      if weight_sum <= @max && weight_sum != 0
        gene.score = value_sum
      else
        gene.score = 1
      end
    end
#   puts genes
  end

  def sort_genes
    @genes.sort_by! { |gene| -gene.score }
  end


  def start
    prev = 0
    end_count = 0
    loop do
      yield(@generation, @genes[0, @population], @genes[0], avg)
      selection
      mutation

      evaluate(@genes)
      sort_genes
#      puts @genes

      if prev == @genes[0].score && prev != 1
        break if end_count > 100
        end_count += 1
      else
        end_count = 0
      end
      prev = @genes[0].score
      @generation += 1
    end

    value_sum = 0
    weight_sum = 0
    @genes[0].chromosome.each_with_index do |locas, index|
        if locas == 1
          value_sum += @items[index].value
          weight_sum += @items[index].weight
        end
    end
#    puts "value_sum : " + value_sum.to_s
    puts value_sum.to_s
#    puts "weight_sum : " + weight_sum.to_s


    yield(@generation, @genes[0,@population], @genes[0], avg)
  end


  def selection
    # ??¨???????????????
    # ??¨??????????????????????????????????????????
    # http://mikilab.doshisha.ac.jp/dia/research/person/jiro/reports/GAparams/GAparams03.html
    # ??????????????????????????????????????¶?????°?????????"??????????????????
    # ?¬?????????£??§?????????????????????????????°???????????????

    # ??¨????????????????????????????????????????????? ???
    #    @genes.first.chromosome = @parents.first.chromosome

    #    @parents = Marshal.load(Marshal.dump(@genes))
    @parents = Array.new(@population)
    @elite = Marshal.load(Marshal.dump(@genes.first))
    @parents[0] = @elite

#    p @genes

    linear_scaling

    # ????????¬????????????
    #    1.upto((@size)/2) do |i|
    # ??????????????????????????????????????¬?????????????????????
#    p sum
#    p @genes
    1.upto(@population-1) do |i|
      @parents[i] = Marshal.load(Marshal.dump(@genes[roulette(rand(sum))]))
#      crossover(roulette,roulette,@size-(i*2-1),@size-(i*2))
    end

    @genes = []

    # ????????????????????????????????????,?¬?????????£????????????????????? ? ???????????????????????????????????§?????¶ => N ~ 2N ???????
    #    count = 0
    @population.times do |i|
      @population.times do |j|
        if rand < P_CROSS
          crossover(i, j) #, count, count+1)
#          count += 2
#          break if count > @population
          break
        else
          @genes << @parents[i]
          break
        end
      end
    end

  end


  # ?????¢?????±???????????°
  # ????????? fi ??? f'i ??????????????????
  # a = 1 , b = (fN - f1*N) / N -1  =>  f'i = a * fi + b
  def linear_scaling
    f_avg = avg
    f_max = @genes[0].score
    f_min = @genes[-1].score
#    p f_avg, f_min

    return if f_avg == f_min

    a = 0
    b = 0
    c = 2.0
    n = @parents.length

#    p f_min >= (c * f_avg - f_max)/(c - 1.0)

    if f_min >= (c * f_avg - f_max)/(c - 1.0)
      a = (c - 1.0) * f_avg / (f_max - f_avg)
      b = f_avg * (f_max - c*f_avg)/(f_max - f_avg)
    else
      a = f_avg / (f_avg - f_min)
      b = -f_min * f_avg / (f_avg - f_min)
    end
    @genes.each { |gene| gene.score = a * gene.score + b }
  end

  def roulette(prob)
#    puts "sum : %d" % sum
    @genes.each_with_index do |gene, index|
      prob -= gene.score
      return index if prob < 0
    end
  end

  # 2?????????
  def crossover(p1, p2) #, c1, c2)
#    p p1, p2
    parent1 = Marshal.load(Marshal.dump(@parents[p1]))
    parent2 = Marshal.load(Marshal.dump(@parents[p2]))

    start_id = rand(@size)
    cross_length = rand(@size - start_id)

    parent1.chromosome[start_id, cross_length], parent2.chromosome[start_id, cross_length] = parent2.chromosome[start_id, cross_length], parent1.chromosome[start_id, cross_length]
    @genes << parent1
    @genes << parent2
#@genes[c1].chromosome = parent1
#@genes[c2].chromosome = parent2
#    evaluate(@genes[-1])
#    evaluate(@genes[-2])
  end

  def mutation
    @genes.each_with_index do |gene, index|
      if rand < P_MUTATION
        i = rand(@size)
        gene.chromosome[i] = (gene.chromosome[i] - 1).abs
      end
    end
  end

  def sum
#    p @genes
#    p @genes.map{|item| item[:score]}
     @genes.map { |item| item[:score] }.inject(:+)
  end

  def avg
    sum / @genes.length
    #sum = @genes.select{|item| item[:score]>1}
    #sum.map{|item| item[:score]}.inject(:+)/sum.length
    #    p sum.inject{|arr,item| arr + item[:score]}#/sum.length
  end

end


if __FILE__ == $0
  aoj
#  init
#  p Problem
#  p Problem[0][:Items][0].weight

  population = 500

  Problem.each do |problem|
    ga = GA.new(population, problem[:Items], problem[:Max_weight])
    ga.start do |generation, genes, max_score, average|
#      puts 'generation : ' + generation.to_s
#      puts 'elite : ' + genes[0].to_s
#      genes.each do |item|
#        p item
#      end
#      break if gets =~ /x/
    end

  end


end