nzy/hw1 9.6 9.7

hw1/ZhongyiNi/answer.md

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+## 9.6
+Follow the hints, we can easily get a $O(l^2v_{max})$ algorithm. But $v_{max}$ can be exponentially large with respect to the input size. Therefore, this will not imply P=NP.
+
+## 9.7
+Let $S$ denote the optimal set.
+$$
+    v = \sum_{i \in S} v_i
+$$
+Let $S'$ denote the optimal set of problem $[v_i/K]$.
+$$
+v' = \sum_{i \in S'} v_i
+$$
+We have $v' \leq v$ and
+$$
+\sum_{i \in S} [\frac{v_i}{K}] \leq \sum_{i \in S'} [\frac{v_i}{K}]
+$$
+Since
+$$
+x-1 < [x] \leq x
+$$
+we have
+$$
+v/K-|S|=\sum_{i \in S} \frac{v_i}{K} - |S|<\sum_{i \in S} [\frac{v_i}{K}] \leq \sum_{i \in S'} [\frac{v_i}{K}]\leq \sum_{i \in S'} \frac{v_i}{K} = v'/K
+$$
+
+Thus,
+$$
+\frac{v'}{v} > 1 - K|S|/v > 1 - Kl/v
+$$
+
+I don't know how to decide $K$ with respect to $\epsilon$.

hw1/ZhongyiNi/dp.jl

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+struct Knapsack{T<:Integer}
+    W::T
+    w::Vector{T}
+    v::Vector{T}
+end
+
+function dp(kp::Knapsack{T}) where T
+    l = length(kp.w)
+    vmax = maximum(kp.v)
+    wbound = sum(kp.w)+1
+    @assert l == length(kp.v)
+    Sw = zeros(T,l,l*vmax)
+    Sset = Matrix{Vector{Int}}(undef,l,l*vmax)
+    for v in 1:l*vmax
+        for j in l:(-1):1
+            if j == l
+                if kp.v[j] == v
+                    Sw[j,v] = kp.w[j]
+                    Sset[j,v] = [j]
+                else
+                    Sw[j,v] = wbound
+                    Sset[j,v] = Int[]
+                end
+            else
+                if v - kp.v[j] < 0
+                    Sw[j,v] = Sw[j+1,v]
+                    Sset[j,v] = Sset[j+1,v]
+                else
+                    if v == kp.v[j]
+                        w_temp = kp.w[j]
+                        set_temp = [j]
+                    else
+                        w_temp = kp.w[j] + Sw[j+1,v - kp.v[j]] 
+                        set_temp = [j] ∪ Sset[j+1,v - kp.v[j]]
+                    end
+                    if w_temp <  Sw[j+1,v] 
+                        Sw[j,v] = w_temp
+                        Sset[j,v] = set_temp
+                    else
+                        Sw[j,v] = Sw[j+1,v]
+                        Sset[j,v] = Sset[j+1,v]
+                    end
+                end
+            end
+        end
+    end
+    W = min(kp.W, wbound-1)
+    pos = findfirst(i->Sw[1,i] <= W ,l*vmax:(-1):1)
+    if isnothing(pos)
+        return zero(T),Int[]
+    else
+        optv = l*vmax-pos+1
+        return Sw[1,optv],Sset[1,optv],optv
+    end
+end
+
+function rescale_kp(kp::Knapsack{T},K) where T
+    return Knapsack(kp.W,kp.w,floor.(T,kp.v ./ K))
+end
+
+function _rescale_dp(kp::Knapsack,K)
+    kp2 = rescale_kp(kp,K)
+    w,set,_ = dp(kp2)
+    return w,set,sum(i -> kp.v[i], set)
+end

hw1/ZhongyiNi/dptest.jl

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+using Test
+include("dp.jl")
+
+@testset "testdp" begin
+    kp = Knapsack(10,[1,2,3,4,5],[2,3,4,5,6])
+    w,set,optv = dp(kp)
+    @test w == 10
+    @test set == [1,2,3,4]
+    @test optv == 14
+end
+
+@testset "rescale_kp" begin
+    kp = Knapsack(10,[1,2,3,4,5],[23,34,47,58,69])
+    K = 10
+    kp2 = rescale_kp(kp,K)
+    @test kp2.v == [2,3,4,5,6]
+end
+
+@testset "rescale_dp" begin
+    kp = Knapsack(10,[1,2,3,4,5],[23,34,47,58,69])
+    K = 10
+    w,set,optv = _rescale_dp(kp,K)
+    @test w == 10
+    @test set == [1,2,3,4]
+    @test optv == 162
+end