1. 介绍
Held-Karp下界基于1tree下界，但是增加了点权重，如下图
通过梯度下降的方法找到最优的 π \pi π。
这里用到的1tree有下面几种：
- 全部点用来生成最小生成树，再加上所有叶子结点第二短的边中数值最大的那个
- 任意选一个点，选取它最短的两边边；然后剩下的点生成最小生成树
- 和2类似，但是枚举所有的点。
2. 代码分析
首先是根据距离列表arr，获得当前节点root最近的两条边
function minimum_two(arr,root)
n = length(arr)
m1=arr[1]
m2=arr[1]
m1_ind=1
m2_ind=1
for i in [1:root-1;root+1:n]
if arr[i]<m1
m2=m1
m1=arr[i]
m2_ind=m1_ind
m1_ind=i
elseif arr[i]<m2
m2=arr[i]
m2_ind=i
end
end
return m1_ind,m2_ind,m1,m2
end
2.1 第一种1tree
function minimum1tree(distmat,pi)
distmat = distmat.+pi.+pi'
mst, c = TravelingSalesmanHeuristics.minspantree(distmat)
x = counter(cat([m[1] for m in mst],[m[2] for m in mst],dims=1))
n = size(distmat)[1]
leaves = []
for i in 1:n
if x[i]==1
append!(leaves,i)
end
end
max_w = 0
max_m1 = 1
max_m2 = 1
for leaf_ind in 1:length(leaves)
leaf = leaves[leaf_ind]
_,m2_ind,_,m2 = minimum_two(distmat[leaf,:],leaf)
w = c+m2
if w > max_w
max_w = w
max_m1 = leaf
max_m2 = m2_ind
end
end
max_v = [x[i] for i in 1:size(distmat)[1]].-2
max_v[max_m1]+=1
max_v[max_m2]+=1
return max_w-2*sum(pi),max_v,max_m1,max_m2
end
2.2 第二种1tree
function minimum1tree(distmat,pi,first_node)
distmat = distmat.+pi.+pi'
m1_ind,m2_ind,m1,m2 = minimum_two(distmat[first_node,:],first_node)
n = size(distmat)[1]
left_nodes = [1:first_node-1;first_node+1:n]
mst, c = TravelingSalesmanHeuristics.minspantree(distmat[left_nodes,left_nodes])
x = counter(cat([left_nodes[m[1]] for m in mst],[left_nodes[m[2]] for m in mst],dims=1))
x[first_node]=2
x[m1_ind]+=1
x[m2_ind]+=1
w = c+m1+m2-2*sum(pi)
v = [x[i] for i in 1:n].-2
distmat = distmat.-pi'.-pi
return w,v
end
2.3 梯度下降代码
这里使用的是第一种minimum1tree，第二种类似。
function ascent(c)
n = size(c)[1]
pi = zeros(n) # 初始化优化参数pi
best_pi = pi
t = 1 # 优化步长
best_w,v,m1,m2 = minimum1tree(c, pi) # 初始化w和v
best_deg = sum(v.*v) # 初始化目标函数
last_v = v
period = max(floor(Int,n/2), 100)
initial_period = period
initialPhase = true
while t > 0.01
p = 1
while p<=period
for i in 1:n;if v[i] == 0;last_v[i] = 0; end;end
pi = pi+t*(0.7*v+0.3*last_v)
last_v = v
w, v, m1, m2 = minimum1tree(c, pi)
deg = sum(v.*v)
if deg == 0
@info("* T = $t, Period = $period, BestW = $w, Norm = $deg, m1 = $m1, m2 = $m2")
return w, pi
elseif (w > best_w && deg <= best_deg)
@info("** T = $t, Period = $period, BestW = $w, Norm = $deg, m1 = $m1, m2 = $m2")
best_w = w
best_pi = pi
best_deg = deg
if initialPhase;t *= 2;end # 增加步长
if p == period;period*=2;end # 增加迭代次数
elseif initialPhase && p > initial_period /2
@info("* T = $t, Period = $period, BestW = $w, Norm = $deg")
initialPhase = false
p = 1
t = t*3/4 # 第一阶段过后，开始逐步收缩步长
end
p+=1
end
t = t/2 # 每个阶段迭代完成后，都收缩步长和迭代次数进行下轮迭代
period = floor(Int,period/2)
end
return best_w, best_pi
end
3. 实测结果
我们使用TSPLIB的att48进行观测，最优解为33522.0。梯度下降打印信息如下：
[ Info: ** T = 1, Period = 100, BestW = 29266.0, Norm = 34, m1 = 2, m2 = 42
[ Info: ** T = 2, Period = 100, BestW = 29333.000000000004, Norm = 34, m1 = 2, m2 = 42
[ Info: ** T = 4, Period = 100, BestW = 29463.4, Norm = 32, m1 = 2, m2 = 42
[ Info: ** T = 8, Period = 100, BestW = 29954.6, Norm = 32, m1 = 2, m2 = 42
[ Info: ** T = 16, Period = 100, BestW = 30398.2, Norm = 26, m1 = 2, m2 = 42
[ Info: ** T = 32, Period = 100, BestW = 31464.399999999998, Norm = 18, m1 = 2, m2 = 42
[ Info: ** T = 64, Period = 100, BestW = 31744.999999999993, Norm = 18, m1 = 2, m2 = 42
[ Info: ** T = 128, Period = 100, BestW = 32487.200000000004, Norm = 18, m1 = 29, m2 = 5
[ Info: * T = 256, Period = 100, BestW = 28475.4, Norm = 62
[ Info: ** T = 96.0, Period = 50, BestW = 32514.400000000012, Norm = 18, m1 = 5, m2 = 29
[ Info: ** T = 96.0, Period = 50, BestW = 32873.40000000001, Norm = 10, m1 = 5, m2 = 29
[ Info: ** T = 96.0, Period = 50, BestW = 33086.20000000001, Norm = 10, m1 = 5, m2 = 29
[ Info: ** T = 96.0, Period = 50, BestW = 33115.8, Norm = 8, m1 = 29, m2 = 5
[ Info: ** T = 48.0, Period = 25, BestW = 33168.00000000001, Norm = 8, m1 = 29, m2 = 34
[ Info: ** T = 48.0, Period = 25, BestW = 33292.399999999994, Norm = 6, m1 = 29, m2 = 34
[ Info: ** T = 24.0, Period = 12, BestW = 33391.399999999994, Norm = 6, m1 = 29, m2 = 34
[ Info: ** T = 24.0, Period = 12, BestW = 33410.6, Norm = 6, m1 = 29, m2 = 34
[ Info: ** T = 12.0, Period = 6, BestW = 33411.200000000004, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 12.0, Period = 12, BestW = 33417.19999999999, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 12.0, Period = 12, BestW = 33421.200000000004, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 12.0, Period = 24, BestW = 33423.6, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 12.0, Period = 24, BestW = 33424.799999999996, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 12.0, Period = 24, BestW = 33424.80000000001, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 12.0, Period = 24, BestW = 33435.200000000004, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 6.0, Period = 12, BestW = 33441.00000000001, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 3.0, Period = 6, BestW = 33442.399999999994, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 3.0, Period = 6, BestW = 33443.4, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 3.0, Period = 12, BestW = 33444.299999999996, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 1.5, Period = 12, BestW = 33444.9, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 1.5, Period = 12, BestW = 33445.350000000006, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 1.5, Period = 12, BestW = 33446.59999999999, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 0.375, Period = 3, BestW = 33447.31249999999, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 0.1875, Period = 3, BestW = 33447.55625, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 0.1875, Period = 6, BestW = 33447.706249999996, Norm = 2, m1 = 29, m2 = 34
[ Info: ** T = 0.09375, Period = 3, BestW = 33447.825, Norm = 2, m1 = 29, m2 = 34
此时结果如下：
最优结果如下，已非常接近。