hw1 kaiwenjin

hw1/kaiwenjin/9.6.jl

@@ -0,0 +1,44 @@
+using Test
+function pp_dp(w::Vector{Int}, v::Vector{Int}, W::Int)
+    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && W >= 0
+    @assert length(w) == length(v) 
+    n = length(w)
+    dp = zeros(Int, n+1, W+1) 
+    for i in 1:n
+        for j in 0:W
+            if j < w[i]
+                dp[i+1, j+1] = dp[i, j+1]
+            else
+                dp[i+1, j+1] = max(dp[i, j+1], dp[i, j - w[i] + 1] + v[i])
+            end
+        end
+    end
+    return dp[n+1, W+1]
+end
+
+function pp(w::Vector{Int}, v::Vector{Int}, W::Int)
+    @assert length(w) == length(v)
+    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && W >= 0
+    n = length(w)
+    res2 = -1
+    for k = 0:2^n-1
+        chosen = []
+        for i in 1:n
+            push!(chosen, 1&k)
+            k >>= 1
+        end
+        val = sum(chosen .* v)
+        if val > res2 && sum(chosen .* w) <= W
+            res2 = val
+        end
+    end
+    return res2
+end
+@testset "package_problem" begin
+    w = [2,3,4,7]
+    v = [1,3,5,9]
+    W = 10
+    res1 = pp_dp(w,v,W)
+    res2 = pp(w,v,W)
+    @test res1 == res2
+end

hw1/kaiwenjin/9.7.jl

@@ -0,0 +1,59 @@
+using Test,Random
+function pp_dp(w::Vector{Int}, v::Vector{Int}, W::Int)
+    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && W >= 0
+    @assert length(w) == length(v) 
+    n = length(w)
+    dp = zeros(Int, n+1, W+1) 
+    for i in 1:n
+        for j in 0:W
+            if j < w[i]
+                dp[i+1, j+1] = dp[i, j+1]
+            else
+                dp[i+1, j+1] = max(dp[i, j+1], dp[i, j - w[i] + 1] + v[i])
+            end
+        end
+    end
+    return dp[n+1, W+1]
+end
+
+function pp(w::Vector{Int}, v::Vector{Int}, W::Int)
+    @assert length(w) == length(v)
+    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && W >= 0
+    n = length(w)
+    res2 = -1
+    for k = 0:2^n-1
+        chosen = []
+        for i in 1:n
+            push!(chosen, 1&k)
+            k >>= 1
+        end
+        val = sum(chosen .* v)
+        if val > res2 && sum(chosen .* w) <= W
+            res2 = val
+        end
+    end
+    return res2
+end
+
+function pp_fptas(w::Vector{Int}, v::Vector{Int}, W::Int, ε::Float64)
+    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && W >= 0 && 0 < ε < 1
+    @assert length(w) == length(v)
+    n = length(w)
+    v_max = maximum(v)
+    c = ε * v_max / n
+    v_scaled = floor.(Int, v ./ c)
+    return pp_dp(w, v_scaled, W) * c
+end
+
+@testset "pp_fptas" begin
+    Random.seed!(6)
+    N = 20
+    M = 100
+    w = rand(collect(1:M), N)
+    v = rand(collect(1:M), N)
+    W = sum((M÷2)*(N÷2))
+    ε = 0.1
+    res0 = pp(w, v, W)
+    res1 = pp_fptas(w, v, W, ε)
+    @test abs(res1 - res0) <= ε * res0
+end

hw1/kaiwenjin/hw1.md

@@ -0,0 +1,32 @@
+### 9.6
+The algorithm and test refer to [algorithm 9.6](9.6.jl)
+
+$n$ objects, $v_{max}$ values and each entry needs $O(n)$ comparisons. So the time complexity is $T = O(n^2v_{max})$.
+
+Why "P≠NP": this $T$ depends on $v_{max}$, which may be much larger than $n$.
+
+### 9.7
+The algorithm and test refer to [algorithm 9.7](9.7.jl)
+
+Proof: consider the scaling factor $c = \frac{n}{\epsilon v_{max}}$
+$$T = O(n^2v_{max}') = O(\frac{n^3}{\epsilon})$$
+
+And the error is not more than (assuming that $v_{max}\leq V$)
+$$1*1/c*n = \epsilon v_{max}\leq \epsilon V$$
+
+QED.
+
+### 9.16
+Original form:
+$$\max(c^Tx)\\
+Ax \leq b\\x \geq 0$$
+
+Standard form $L$:
+$$\max(c^Tx)\\
+[A,I]\left[\begin{array}{c}x\\s\end{array}\right] = b\\x \geq 0, s \geq 0$$
+
+Without loss of generality, assume that $b \geq 0$. Then we get an LP with a trivial feasible solution whose optimal solution is feasible for $L$:
+$$\min e^T s\\
+[A,I]\left[\begin{array}{c}x\\s\end{array}\right] = b\\x \geq 0, s \geq 0$$
+
+A feasible solution is $x = 0, s = b$. Its optimal solution must exist and is a feasible solution for $L$.