From 094d35ce4415854060119ce20ea5f3baad5548d5 Mon Sep 17 00:00:00 2001
From: Yui <2946723935@qq.com>
Date: Mon, 12 May 2025 18:07:07 +0800
Subject: [PATCH 1/2] hw1 kaiwenjin

---
 hw1/kaiwenjin/9.6.jl | 44 ++++++++++++++++++++++++++++++++
 hw1/kaiwenjin/9.7.jl | 61 ++++++++++++++++++++++++++++++++++++++++++++
 hw1/kaiwenjin/hw1.md | 26 +++++++++++++++++++
 3 files changed, 131 insertions(+)
 create mode 100644 hw1/kaiwenjin/9.6.jl
 create mode 100644 hw1/kaiwenjin/9.7.jl
 create mode 100644 hw1/kaiwenjin/hw1.md

diff --git a/hw1/kaiwenjin/9.6.jl b/hw1/kaiwenjin/9.6.jl
new file mode 100644
index 0000000..c8f1f2b
--- /dev/null
+++ b/hw1/kaiwenjin/9.6.jl
@@ -0,0 +1,44 @@
+using Test
+function pp_dp(w::Vector{Int}, v::Vector{Int}, V::Int)
+    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && V >= 0
+    @assert length(w) == length(v) 
+    n = length(w)
+    dp = zeros(Int, n+1, V+1) 
+    for i in 1:n
+        for j in 0:V
+            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, V+1]
+end
+
+function pp(w::Vector{Int}, v::Vector{Int}, V::Int)
+    @assert length(w) == length(v)
+    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && V >= 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) <= V
+            res2 = val
+        end
+    end
+    return res2
+end
+@testset "package_problem" begin
+    w = [2,3,4,7]
+    v = [1,3,5,9]
+    V = 10
+    res1 = pp_dp(w,v,V)
+    res2 = pp(w,v,V)
+    @test res1 == res2
+end
diff --git a/hw1/kaiwenjin/9.7.jl b/hw1/kaiwenjin/9.7.jl
new file mode 100644
index 0000000..a56096d
--- /dev/null
+++ b/hw1/kaiwenjin/9.7.jl
@@ -0,0 +1,61 @@
+using Test,Random
+function pp_dp(w::Vector{Int}, v::Vector{Int}, V::Int)
+    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && V >= 0
+    @assert length(w) == length(v) 
+    n = length(w)
+    dp = zeros(Int, n+1, V+1) 
+    for i in 1:n
+        for j in 0:V
+            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, V+1]
+end
+
+function pp(w::Vector{Int}, v::Vector{Int}, V::Int)
+    @assert length(w) == length(v)
+    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && V >= 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) <= V
+            res2 = val
+        end
+    end
+    return res2
+end
+
+function pp_fptas(w::Vector{Int}, v::Vector{Int}, V::Int, ε::Float64)
+    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && V >= 0 && 0 < ε < 1
+    @assert length(w) == length(v)
+    n = length(w)
+    v_max = maximum(v)
+    c = ε * v_max / n
+    w_scaled = floor.(Int, w ./ c)
+    v_scaled = floor.(Int, v ./ c)
+    V_scaled = floor(Int, V / c)
+    return pp_dp(w_scaled, v_scaled, V_scaled) * 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)
+    V = sum((M÷2)*(N÷2))
+    ε = 0.1
+    res0 = pp(w, v, V)
+    res1 = pp_fptas(w, v, V, ε)
+    @test abs(res1 - res0) <= ε * res0
+end
\ No newline at end of file
diff --git a/hw1/kaiwenjin/hw1.md b/hw1/kaiwenjin/hw1.md
new file mode 100644
index 0000000..273db3e
--- /dev/null
+++ b/hw1/kaiwenjin/hw1.md
@@ -0,0 +1,26 @@
+### 9.6
+The algorithm and test refers 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 not "P=NP": this $T$ depends on $v_{max}$, which may be much larger than $n$.
+
+### 9.7
+The algorithm and test refers 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 (assume 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:
+$$\max(c^Tx)\\
+[A,I]\left[\begin{array}{c}x\\s\end{array}\right] = b\\x \geq 0, s \geq 0$$
\ No newline at end of file

From 91236171d6acd0a818f37bd35d303d245832eedc Mon Sep 17 00:00:00 2001
From: Yui <2946723935@qq.com>
Date: Tue, 13 May 2025 15:21:44 +0800
Subject: [PATCH 2/2] hw1 update

---
 hw1/kaiwenjin/9.6.jl | 22 +++++++++++-----------
 hw1/kaiwenjin/9.7.jl | 30 ++++++++++++++----------------
 hw1/kaiwenjin/hw1.md | 20 +++++++++++++-------
 3 files changed, 38 insertions(+), 34 deletions(-)

diff --git a/hw1/kaiwenjin/9.6.jl b/hw1/kaiwenjin/9.6.jl
index c8f1f2b..8a77498 100644
--- a/hw1/kaiwenjin/9.6.jl
+++ b/hw1/kaiwenjin/9.6.jl
@@ -1,11 +1,11 @@
 using Test
-function pp_dp(w::Vector{Int}, v::Vector{Int}, V::Int)
-    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && V >= 0
+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, V+1) 
+    dp = zeros(Int, n+1, W+1) 
     for i in 1:n
-        for j in 0:V
+        for j in 0:W
             if j < w[i]
                 dp[i+1, j+1] = dp[i, j+1]
             else
@@ -13,12 +13,12 @@ function pp_dp(w::Vector{Int}, v::Vector{Int}, V::Int)
             end
         end
     end
-    return dp[n+1, V+1]
+    return dp[n+1, W+1]
 end
 
-function pp(w::Vector{Int}, v::Vector{Int}, V::Int)
+function pp(w::Vector{Int}, v::Vector{Int}, W::Int)
     @assert length(w) == length(v)
-    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && V >= 0
+    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && W >= 0
     n = length(w)
     res2 = -1
     for k = 0:2^n-1
@@ -28,7 +28,7 @@ function pp(w::Vector{Int}, v::Vector{Int}, V::Int)
             k >>= 1
         end
         val = sum(chosen .* v)
-        if val > res2 && sum(chosen .* w) <= V
+        if val > res2 && sum(chosen .* w) <= W
             res2 = val
         end
     end
@@ -37,8 +37,8 @@ end
 @testset "package_problem" begin
     w = [2,3,4,7]
     v = [1,3,5,9]
-    V = 10
-    res1 = pp_dp(w,v,V)
-    res2 = pp(w,v,V)
+    W = 10
+    res1 = pp_dp(w,v,W)
+    res2 = pp(w,v,W)
     @test res1 == res2
 end
diff --git a/hw1/kaiwenjin/9.7.jl b/hw1/kaiwenjin/9.7.jl
index a56096d..fc73c53 100644
--- a/hw1/kaiwenjin/9.7.jl
+++ b/hw1/kaiwenjin/9.7.jl
@@ -1,11 +1,11 @@
 using Test,Random
-function pp_dp(w::Vector{Int}, v::Vector{Int}, V::Int)
-    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && V >= 0
+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, V+1) 
+    dp = zeros(Int, n+1, W+1) 
     for i in 1:n
-        for j in 0:V
+        for j in 0:W
             if j < w[i]
                 dp[i+1, j+1] = dp[i, j+1]
             else
@@ -13,12 +13,12 @@ function pp_dp(w::Vector{Int}, v::Vector{Int}, V::Int)
             end
         end
     end
-    return dp[n+1, V+1]
+    return dp[n+1, W+1]
 end
 
-function pp(w::Vector{Int}, v::Vector{Int}, V::Int)
+function pp(w::Vector{Int}, v::Vector{Int}, W::Int)
     @assert length(w) == length(v)
-    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && V >= 0
+    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && W >= 0
     n = length(w)
     res2 = -1
     for k = 0:2^n-1
@@ -28,23 +28,21 @@ function pp(w::Vector{Int}, v::Vector{Int}, V::Int)
             k >>= 1
         end
         val = sum(chosen .* v)
-        if val > res2 && sum(chosen .* w) <= V
+        if val > res2 && sum(chosen .* w) <= W
             res2 = val
         end
     end
     return res2
 end
 
-function pp_fptas(w::Vector{Int}, v::Vector{Int}, V::Int, ε::Float64)
-    @assert findmin(w)[1] >= 0 && findmin(v)[1] >= 0 && V >= 0 && 0 < ε < 1
+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
-    w_scaled = floor.(Int, w ./ c)
     v_scaled = floor.(Int, v ./ c)
-    V_scaled = floor(Int, V / c)
-    return pp_dp(w_scaled, v_scaled, V_scaled) * c
+    return pp_dp(w, v_scaled, W) * c
 end
 
 @testset "pp_fptas" begin
@@ -53,9 +51,9 @@ end
     M = 100
     w = rand(collect(1:M), N)
     v = rand(collect(1:M), N)
-    V = sum((M÷2)*(N÷2))
+    W = sum((M÷2)*(N÷2))
     ε = 0.1
-    res0 = pp(w, v, V)
-    res1 = pp_fptas(w, v, V, ε)
+    res0 = pp(w, v, W)
+    res1 = pp_fptas(w, v, W, ε)
     @test abs(res1 - res0) <= ε * res0
 end
\ No newline at end of file
diff --git a/hw1/kaiwenjin/hw1.md b/hw1/kaiwenjin/hw1.md
index 273db3e..6ed9597 100644
--- a/hw1/kaiwenjin/hw1.md
+++ b/hw1/kaiwenjin/hw1.md
@@ -1,26 +1,32 @@
 ### 9.6
-The algorithm and test refers to [algorithm 9.6](9.6.jl)
+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 not "P=NP": this $T$ depends on $v_{max}$, which may be much larger than $n$.
+Why "P≠NP": this $T$ depends on $v_{max}$, which may be much larger than $n$.
 
 ### 9.7
-The algorithm and test refers to [algorithm 9.7](9.7.jl)
+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 (assume that $v_{max}\leq V$)
+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:
+Original form:
 $$\max(c^Tx)\\
 Ax \leq b\\x \geq 0$$
 
-standard form:
+Standard form $L$:
 $$\max(c^Tx)\\
-[A,I]\left[\begin{array}{c}x\\s\end{array}\right] = b\\x \geq 0, s \geq 0$$
\ No newline at end of file
+[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 $L$.