Problem:
Chinese Postman Problem

For a given weighted undirected graph G(V, E), find the distance of the shortest route that meets the following criteria:

It is a closed cycle where it ends at the same point it starts.

The route must go through every edge at least once.

Input

|V| |E|
s0 t0 d0
s1 t1 d1
:
s|E|-1 t|E|-1 d|E|-1

, where |V| is the number of vertices and |E| is the number of edges in the graph. The graph vertices are named with the numbers 0, 1,..., |V|-1 respectively.

si and ti represent source and target verticess of i-th edge (undirected) and di represents the distance between si and ti (the i-th edge).

Note that there can be multiple edges between a pair of vertices.

Output

Print the shortest distance in a line.

Constraints

2 ≤ |V| ≤ 15

0 ≤ |E| ≤ 1,000

0 ≤ di ≤ 1,000

si ≠ ti

The graph is connected

Sample Input 1

4 4
0 1 1
0 2 2
1 3 3
2 3 4

Sample Output 1

10

Sample Input 2

4 5
0 1 1
0 2 2
1 3 3
2 3 4
1 2 5

Sample Output 2

18

Sample Input 3

2 3
0 1 1
0 1 2
0 1 3

Sample Output 3

7

Solution in JAVA (judge verdict: Accepted)

import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.*;
import java.math.BigInteger;
 
public class Main implements Runnable {
	
	static int mod = 1000000007;
	
    public static void main(String[] args) {
    	new Thread(null, new Main(), "", 1024 * 1024 * 1024).start();
    }
    
    public void run() {
    	PrintWriter out = new PrintWriter(System.out);
        FastScanner sc = new FastScanner();
        
        int v = sc.nextInt();
        int e = sc.nextInt();
        WGraph g = new WGraph(v);
        for(int i=0;i<e;i++){
        	g.addEdge(sc.nextInt(), sc.nextInt(), sc.nextInt());
        }

        out.println(chinesePostman(g));
        
        out.flush();
    }
    
	static int chinesePostman(WGraph g){
		int n = g.vnum();
		ArrayList<Integer> odds = new ArrayList<>();	//奇点
		int[][] dist = g.WarshallFloyd();
		int total = g.totalWeight();
		
		for(int i=0;i<n;i++){
			if(g.dig(i)%2 == 1){
				odds.add(i);
			}
		}
		
		if(odds.size()==0){	//オイラーグラフ
			return total;
		}
		else{
			//rem:奇点は必ず偶数個
			int k = odds.size();
			int S = 1<<k;
			int[] dp = new int[S];
			Arrays.fill(dp, Integer.MAX_VALUE);
			dp[0] = 0;
			
			for(int B = 0; B<S; B++){
				for(int i=0; i<k; i++){
					if(((B>>i)&1) == 1){
						continue;
					}
					for(int j=i+1; j<k; j++){
						if(((B>>j)&1) == 1){
							continue;
						}
						if(dist[odds.get(i)][odds.get(j)] == Integer.MAX_VALUE){
							continue;
						}
						if(dp[B] == Integer.MAX_VALUE){
							continue;
						}
						int newstate = B|(1<<i)|(1<<j);
						dp[newstate] = Math.min(dp[newstate], dp[B] + dist[odds.get(i)][odds.get(j)]);
					}
				}
			}

			return total + dp[S-1];
		}
	}

}

class WGraph {
	WEdgeList[] elist;
	
	public WGraph(int vnum){
		this.elist = new WEdgeList[vnum];
		for(int i=0;i<vnum;i++){
			elist[i] = new WEdgeList();
		}
	}

	//同じ辺を複数回追加すると多重辺が生まれてしまう
	public void addEdge(int a, int b, int w){
		elist[a].add(new WEdge(b,w));
		elist[b].add(new WEdge(a,w));
	}
	
	//頂点数を返す
	public int vnum(){
		return elist.length;
	}
	
	//辺数を返す
	public int edgeNum(){
		int sum = 0;
		for(WEdgeList l:elist){
			sum += l.size();
		}
		return sum/2;
	}
	
	//隣接判定
	public boolean isNext(int n, int m){
		return elist[n].contains(m);
	}
	
	//次数
	public int dig(int i){
		return elist[i].size();
	}
	
	//全ての辺の重みの合計
	public int totalWeight(){
		int total = 0;
		for(int i=0;i<vnum();i++){
			for(WEdge e:elist[i]){
				total += e.w;
			}
		}
		return total/2;
	}
	
	//隣接する全ての頂点を返す
	public ArrayList<Integer> nextList(int n){
		ArrayList<Integer> l = new ArrayList<>();
		for(WEdge e:elist[n]){
			l.add(e.to);
		}
		return l;
	}
	
	//隣接する全ての辺を返す
	public WEdgeList EList(int n){
		return elist[n];
	}
	
	//ダイクストラ法でsからの最短距離列を求める。到達不可はMAXVALUE。
	public int Dijkstra(int start, int goal){
		
		PriorityQueue<NextNode> frontier = new PriorityQueue<>((x,y)->Integer.compare(x.g, y.g));
		boolean[] isExplored = new boolean[vnum()];
		int[] g = new int[vnum()];
		Arrays.fill(g, Integer.MAX_VALUE);
		
		g[start] = 0;
		frontier.add(new NextNode(start,0));
		
		while(!frontier.isEmpty()){
			NextNode now = frontier.poll();
			if(now.i == goal){
				return g[now.i];
			}
			
			if(!isExplored[now.i]){
				isExplored[now.i] = true;
				
				for(WEdge e:EList(now.i)){
					if(isExplored[e.to]){
						continue;
					}
					
					int next = e.to;
					if(g[next] <= g[now.i] + e.w){
						continue;
					}
					
					g[next] = now.g + e.w;
					frontier.add(new NextNode(next,g[next]));
				}	
			}
		}
		
		return -1;
	}
	//ダイクストラ法でsからの最短距離列を求める。到達不可はMAXVALUE。
	public int[] DijkstraArray(int start){
		
		PriorityQueue<NextNode> frontier = new PriorityQueue<>((x,y)->Integer.compare(x.g, y.g));
		boolean[] isExplored = new boolean[vnum()];
		int[] g = new int[vnum()];
		Arrays.fill(g, Integer.MAX_VALUE);
		
		g[start] = 0;
		frontier.add(new NextNode(start,0));
		
		while(!frontier.isEmpty()){
			NextNode now = frontier.poll();
			
			if(!isExplored[now.i]){
				isExplored[now.i] = true;
				
				for(WEdge e:EList(now.i)){
					if(isExplored[e.to]){
						continue;
					}
					
					int next = e.to;
					if(g[next] <= g[now.i] + e.w){
						continue;
					}
					
					g[next] = now.g + e.w;
					frontier.add(new NextNode(next,g[next]));
				}	
			}
		}
		
		return g;
	}
	class NextNode{
		int i;
		int g;
		public NextNode(int i, int g){
			this.i=i;
			this.g=g;
		}
	}
	
	//全点対最短距離をワーシャルフロイド法で求める
	public int[][] WarshallFloyd(){
		int[][][] d = new int[vnum()+1][vnum()][vnum()];
		
		for(int k=0;k<vnum()+1;k++){
			for(int i=0;i<vnum();i++){
				Arrays.fill(d[k][i], Integer.MAX_VALUE);
				d[k][i][i] = 0;
			}
		}

		for(int i=0;i<vnum();i++){
			for(WEdge e:elist[i]){
				d[0][i][e.to] = Math.min(d[0][i][e.to], e.w);	//多重辺対策
			}
		}
		
		for(int k=1;k<vnum()+1;k++){
			for(int i=0;i<vnum();i++){
				for(int j=0;j<vnum();j++){
					if((d[k-1][i][k-1] != Integer.MAX_VALUE) && (d[k-1][k-1][j] != Integer.MAX_VALUE)){
						d[k][i][j] = Math.min(d[k-1][i][j], d[k-1][i][k-1] + d[k-1][k-1][j]);
					}
					else{
						d[k][i][j] = d[k-1][i][j];
					}
				}
			}
		}
		
		return d[vnum()];
	}
	
	//木であることを仮定して直径を求める
	public int treeDiameter(){
		int[] d1 = DijkstraArray(0);
		int s = maxIndex(d1);
		int[] d2 = DijkstraArray(s);
		return max(d2);
	}
	//配列の最大値
	private static int max(int[] a){
		int max = Integer.MIN_VALUE;
		for(int i:a){
			if(i>max){
				max = i;
			}
		}
		return max;
	}
	//配列の最大値のインデックス
	private static int maxIndex(int[] a){
		int max = Integer.MIN_VALUE;
		int idx = 0;
		for(int i=0;i<a.length;i++){
			if(a[i]>max){
				max = a[i];
				idx = i;
			}
		}
		return idx;
	}
	
	//木の高さ
	public int[] treeHeight(){
		int[][] dp = new int[vnum()][vnum()]; //iからj方向に向かうときの最大距離
		for(int i=0;i<vnum();i++){
			Arrays.fill(dp[i],Integer.MIN_VALUE);
		}
		int[] height = new int[vnum()];
		
		for(int i=0;i<vnum();i++){
			for(WEdge e:EList(i)){
				dfsth(i,e.to,e.w,dp);
			}
		}
		
		for(int i=0;i<vnum();i++){
			for(WEdge e:EList(i)){
				height[i] = Math.max(height[i],dp[i][e.to]);
			}
		}
		
		return height;
	}
	int dfsth(int i, int j, int w, int[][] dp){
		if(dp[i][j]==Integer.MIN_VALUE){
			int max = 0;
			for(WEdge e:EList(j)){
				if(e.to!=i){
					max = Math.max(max,dfsth(j,e.to,e.w,dp));
				}
			}
			dp[i][j] = max + w;
			return dp[i][j];
		}
		else{
			return dp[i][j];
		}
	}
	
	//木の高さ（再帰なし、若干遅い）
	public int[] treeHeight2(){
		
		int[][] dp = new int[vnum()][vnum()]; //iからj方向に向かうときの最大距離
		for(int i=0;i<vnum();i++){
			Arrays.fill(dp[i],Integer.MIN_VALUE);
		}
		
		int[] height = new int[vnum()];
		
		int[] ord = new int[vnum()];
		int[] ordidx = new int[vnum()]; //そのオーダーを持つ頂点のインデックス
		
		//DFS木を作り、ordを求める
		ArrayDeque<Integer> a = new ArrayDeque<>();
		a.offerFirst(0);
		int noword = 0;
		boolean isVisited[] = new boolean[vnum()];
		
		while(!a.isEmpty()){
			int now = a.pollFirst();
			
			if(isVisited[now]){
				continue;
			}
			isVisited[now] = true;
			
			ord[now] = noword;
			ordidx[noword] = now;
			noword++;
			
			for(WEdge e:EList(now)){
				a.offerFirst(e.to);
			}
		}
		
		WEdge[] par = new WEdge[vnum()];
		
		//親→子の方向をまず埋める
		for(int o=vnum()-1;o>0;o--){
			int dpnum = 0;
			int idx = ordidx[o];
			for(WEdge e:EList(idx)){
				if(ord[e.to]<o){
					par[idx] = e;
				}
				else{
					dpnum = Math.max(dpnum, dp[idx][e.to]);
				}
			}
			dp[par[idx].to][idx] = dpnum + par[idx].w;
		}
		
		//子→親の方向を埋める
		for(int o=1;o<vnum();o++){
			int idx = ordidx[o];
			int dpnum = 0;
			for(WEdge e:EList(par[idx].to)){
				if(e.to != idx){
					dpnum = Math.max(dpnum, dp[par[idx].to][e.to]);
				}
			}
			dp[idx][par[idx].to] = dpnum + par[idx].w;
		}
		
		for(int i=0;i<vnum();i++){
			for(WEdge e:EList(i)){
				height[i] = Math.max(height[i],dp[i][e.to]);
			}
		}
		
		return height;
	}
	
	class WEdgeList extends ArrayList<WEdge>{
		private static final long serialVersionUID = 4901656039324433445L;
	}
	class WEdge{
		int to;
		int w;
		
		public WEdge(int to, int w){
			this.to = to;
			this.w = w;
		}
	}

}


class FastScanner {
	private final InputStream in = System.in;
	private final byte[] buffer = new byte[1024];
	private int ptr = 0;
	private int buflen = 0;
	private boolean hasNextByte() {
		if (ptr < buflen) {
			return true;
		} else {
			ptr = 0;
			try {
				buflen = in.read(buffer);
			} catch (IOException e) {
				e.printStackTrace();
			}
			if (buflen <= 0) {
				return false;
			}
		}
		return true;
	}
	private int readByte() {
		if (hasNextByte())
			return buffer[ptr++];
		else
			return -1;
	}
	private static boolean isPrintableChar(int c) {
		return 33 <= c && c <= 126;
	}
	public boolean hasNext() {
		while (hasNextByte() && !isPrintableChar(buffer[ptr]))
			ptr++;
		return hasNextByte();
	}
	public String next() {
		if (!hasNext())
			throw new NoSuchElementException();
		StringBuilder sb = new StringBuilder();
		int b = readByte();
		while (isPrintableChar(b)) {
			sb.appendCodePoint(b);
			b = readByte();
		}
		return sb.toString();
	}
	public long nextLong() {
		if (!hasNext())
			throw new NoSuchElementException();
		long n = 0;
		boolean minus = false;
		int b = readByte();
		if (b == '-') {
			minus = true;
			b = readByte();
		}
		if (b < '0' || '9' < b) {
			throw new NumberFormatException();
		}
		while (true) {
			if ('0' <= b && b <= '9') {
				n *= 10;
				n += b - '0';
			} else if (b == -1 || !isPrintableChar(b)) {
				return minus ? -n : n;
			} else {
				throw new NumberFormatException();
			}
			b = readByte();
		}
	}
	public int nextInt() {
		long nl = nextLong();
		if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE)
			throw new NumberFormatException();
		return (int) nl;
	}
	public int[] nextintArray(int n){
		int[] a = new int[n];
		for(int i=0;i<n;i++){
			a[i] = nextInt();
		}
		return a;
	}
	public long[] nextlongArray(int n){
		long[] a = new long[n];
		for(int i=0;i<n;i++){
			a[i] = nextLong();
		}
		return a;
	}
	public Integer[] nextIntegerArray(int n){
		Integer[] a = new Integer[n];
		for(int i=0;i<n;i++){
			a[i] = nextInt();
		}
		return a;
	}
	public double nextDouble() {
		return Double.parseDouble(next());
	}
}