map struct

[waranthem]

.

[icy1]

Perhaps try recursive approach?

#include <iostream>
#include <iomanip>
#include <string>
#include <vector>
#include <fstream>
#include <map>

using namespace std;

void Show(const map<string, vector<pair<string,int>>>& flights)
{
    for(auto& src_map : flights)
    {
        cout << src_map.first << ":" << endl;
        for(auto& dest_price : src_map.second)
            cout << "  =>"<<dest_price.first << " " << dest_price.second << endl;
    }
}

void Fly(map<string, bool> visited, 
         const map<string, vector<pair<string,int>>>& flights,
         pair<string,int> dest_pair,
         const string& final_dest,
         string path,
         vector<pair<int,string>>& all_paths,
         int cost,
         int& best_cost,
         string& best_path
         )
{
    visited[dest_pair.first] = true;
    path += "=>"+dest_pair.first;
    cost += dest_pair.second;
    if (dest_pair.first==final_dest)
    {
        //cout << path << endl;
        all_paths.emplace_back(cost, path);
        if (cost < best_cost)
        {
            best_cost = cost;
            best_path = path;
        }
        return;
    }
    
    for (const auto& dest_pair : flights.at(dest_pair.first))
    {
        if (!visited[dest_pair.first])
        {
            Fly(visited,
                flights,
                dest_pair,
                final_dest,
                path,
                all_paths,
                cost,
                best_cost,
                best_path);
        }
    }
}

void FindBestFlights(const map<string, bool>& sources,
                     const map<string, vector<pair<string,int>>>& flights,
                     const string& start, 
                     const string& final_dest)
{
    if (sources.count(start)==0 || sources.count(final_dest)==0)
    {
        cout << "One or more of ["<<start<<", "<<final_dest<<"] do not exist.\n";
        return;
    }
    string path = start;
    vector<pair<int, string>> all_paths;  // Total cost with total path
    int cost = 0;
    int best_cost = 999999;
    string best_path;
    
    for (const auto& dest_pair : flights.at(start))
    {
        Fly(sources, flights, dest_pair, final_dest,
            path, all_paths, cost, best_cost, best_path);
    }
    
    cout << "best_cost: $" << best_cost << endl;
    cout << "best_path: " << best_path << endl << endl;
    
    cout << "all_paths:"<<endl;
    for (auto& cp : all_paths)
        cout << "  $" << setw(5) << cp.first << "  " << cp.second << endl;
    
}

int main() 
{
    string file_name("flights.txt");
    ifstream ifs(file_name);
    if (!ifs)
    {
        cout << "Unable to open \""<<file_name<<"\".  Please check it exists and not in use.\n";
        return -1;
    }
    
    map<string, vector<pair<string,int>>> flights;
    map<string, bool> sources;
    string src, dst;
    int price;
    while (ifs >> src >> price >> dst )
    {
        flights[src].emplace_back(dst,price);
        sources[src] = false;
    }
    //Show(flights);
    
    src = "LGA";
    dst = "HND";
    sources[src] = true;  // Mark as visited
    FindBestFlights(sources, flights, src, dst);
    
    return 0;
}

can test at https://repl.it/repls/ElectricEmbellishedApi (flights.txt is there too)

best_cost: $582
best_path: LGA=>ORD=>YVR=>ANC=>ICN=>HND

all_paths:
  $ 1159  LGA=>SFO=>ORD=>YVR=>HND
  $ 1003  LGA=>SFO=>ORD=>YVR=>ANC=>ICN=>HND
  $ 1023  LGA=>SFO=>ORD=>YVR=>ICN=>HND
  $  802  LGA=>SFO=>HND
  $  852  LGA=>LAX=>HND
  $  738  LGA=>ORD=>YVR=>HND
  $  582  LGA=>ORD=>YVR=>ANC=>ICN=>HND
  $  602  LGA=>ORD=>YVR=>ICN=>HND
  $  739  LGA=>ORD=>SFO=>HND
  $ 1118  LGA=>YVR=>ORD=>SFO=>HND
  $  759  LGA=>YVR=>HND
  $  603  LGA=>YVR=>ANC=>ICN=>HND
  $  623  LGA=>YVR=>ICN=>HND
  $ 1365  LGA=>YOW=>YVR=>ORD=>SFO=>HND
  $ 1006  LGA=>YOW=>YVR=>HND
  $  850  LGA=>YOW=>YVR=>ANC=>ICN=>HND
  $  870  LGA=>YOW=>YVR=>ICN=>HND

Edit: for some reason to allow const iteration through flights map, it doesn't like operator[] on that map -- i needed to instead use .at() .
Edit2: realized it's because operator[] performs insertion if that key doesn't exist.
Edit3: moved the setting of initial visited state to main() so that all params of FindBestFlights() could be const