使用动态规划的 TSP

TSP using Dynamic Programming

本文关键字：TSP 动态规划更新时间：2023-10-16

我正在学习 TSP，并找到了这个 TSP 的递归解决方案

int compute(int start,int set)
{   int masked,mask,result=INT_MAX,temp,i;//result stores the minimum 
    if(g[start][set]!=-1)//memoization DP top-down,check for repeated subproblem
        return g[start][set];
    for(i=0;i<n;i++)
        {   //npow-1 because we always exclude "home" vertex from our set
            mask=(npow-1)-(1<<i);//remove ith vertex from this set
            masked=set&mask;
            if(masked!=set)//in case same set is generated(because ith vertex was not present in the set hence we get the same set on removal) eg 12&13=12
            {   
                temp=adj[start][i]+compute(i,masked);//compute the removed set
                if(temp<result)
                    result=temp,p[start][set]=i;//removing ith vertex gave us minimum
            }
        }
        return g[start][set]=result;//return minimum
}

我不明白屏蔽是如何工作的，如何在不使用递归的情况下将其更改为动态编程解决方案，请帮助我。

这是一个传统的TSP问题，这是它的解决方案。我认为这可能对您有所帮助。

int map[15][15];
int dp[(1<<12)+5][12];
int main() {
    int i,j,n,ans,k,p;
    while(1) {
        scanf("%d",&n);
        if (n==0) break;
        n++;
        for (i=0; i<n; i++) {
            for (j=0; j<n; j++) {
                scanf("%d",&map[i][j]);
            }
        }
        //floyd algorithm, get any two points's minimum distance
        for (k=0; k<n; k++) {
            for (i=0; i<n; i++) {
                for (j=0; j<n; j++) {
                    if (i!=j && i!=k && j!=k) map[i][j]=min(map[i][k]+map[k][j],map[i][j]);
                }
            }
        }
        memset(dp,-1,sizeof(dp));
        dp[1][0]=0;
        // TSP solution here,bitmask and DP
        for (i=1; i<(1<<n); i++) {// the current state
            for (j=0; j<n; j++) {// during the current state,the last station is j
                if (dp[i][j]==-1) continue;
                for (k=1; k<n; k++) {//the next state is k
                    if ((i & (1<<k))!=0) continue;
                    p=(i | (1<<k));// the new state(join k)
                    if (dp[p][k]==-1) dp[p][k]=dp[i][j]+map[j][k];
                    dp[p][k]=min(dp[p][k],dp[i][j]+map[j][k]);
                }
            }
        }
        ans=INF;
        // get answer
        for (i=1; i<n; i++) {
            if (dp[(1<<n)-1][i]>0) ans=min(ans,dp[(1<<n)-1][i]+map[i][0]);
        }
        printf("%dn",ans);
    }
    return 0;
}