Here's a solution `ts.lp` in AnsProlog using the Potassco ASP toolkit:

It requires the following data file% The distance relation is symmetric. dist(B, A, N) :- dist(A, B, N). % Two cities are "adjacent" if there is a defined distance between them. adjacent(A, B) :- dist(A, B, _). % A city exists if it has a distance defined to it from somewhere. city(A) :- adjacent(A, _). % Assign a city to each stop on the path from 1..n. 1 { path(N, C): city(C) } 1 :- N = 1..n. % No city can occur twice in the path. :- path(N, C), path(M, C), N != M. % There must be a defined distance between successive cities on the path. :- path(N, C), path(N+1, D), not adjacent(C, D). % Minimize the sum of distances between successive cities in the path. #minimize { X, N : path(N, C), path(N+1, D), dist(C,D,X), N = 1..n }. #show path/2.

`city.lp`as input:

Here's the output from running it:dist(faerun, norrath, 129). dist(faerun, tristram, 58). dist(faerun, alphacentauri, 13). dist(faerun, arbre, 24). dist(faerun, snowdin, 60). dist(faerun, tambi, 71). dist(faerun, straylight, 67). dist(norrath, tristram, 142). dist(norrath, alphacentauri, 15). dist(norrath, arbre, 135). dist(norrath, snowdin, 75). dist(norrath, tambi, 82). dist(norrath, straylight, 54). dist(tristram, alphacentauri, 118). dist(tristram, arbre, 122). dist(tristram, snowdin, 103). dist(tristram, tambi, 49). dist(tristram, straylight, 97). dist(alphacentauri, arbre, 116). dist(alphacentauri, snowdin, 12). dist(alphacentauri, tambi, 18). dist(alphacentauri, straylight, 91). dist(arbre, snowdin, 129). dist(arbre, tambi, 53). dist(arbre, straylight, 40). dist(snowdin, tambi, 15). dist(snowdin, straylight, 99). dist(tambi, straylight, 70). city(alphacentauri;arbre;faerun;norrath;snowdin;straylight;tambi;tristram). #const n = 8.

So the shortest path is from Faerun-Norrath-Tristram-AlphaCentauri-Arbre-Snowdin-Tambi-Straylight with a length of 207.$ clingo city.lp ts.lp clingo version 4.4.0 Reading from city.lp ... Solving... Answer: 1 path(1,faerun) path(3,norrath) path(8,tristram) path(2,alphacentauri) path(6,arbre) path(5,snowdin) path(4,tambi) path(7,straylight) Optimization: 391 Answer: 2 path(8,faerun) path(3,norrath) path(5,tristram) path(2,alphacentauri) path(6,arbre) path(1,snowdin) path(4,tambi) path(7,straylight) Optimization: 387 Answer: 3 path(5,faerun) path(3,norrath) path(8,tristram) path(2,alphacentauri) path(6,arbre) path(1,snowdin) path(4,tambi) path(7,straylight) Optimization: 341 Answer: 4 path(5,faerun) path(1,norrath) path(6,tristram) path(2,alphacentauri) path(8,arbre) path(3,snowdin) path(4,tambi) path(7,straylight) Optimization: 308 Answer: 5 path(5,faerun) path(1,norrath) path(8,tristram) path(2,alphacentauri) path(6,arbre) path(3,snowdin) path(4,tambi) path(7,straylight) Optimization: 274 Answer: 6 path(7,faerun) path(5,norrath) path(1,tristram) path(6,alphacentauri) path(8,arbre) path(3,snowdin) path(2,tambi) path(4,straylight) Optimization: 269 Answer: 7 path(7,faerun) path(5,norrath) path(8,tristram) path(6,alphacentauri) path(3,arbre) path(1,snowdin) path(2,tambi) path(4,straylight) Optimization: 248 Answer: 8 path(2,faerun) path(5,norrath) path(1,tristram) path(6,alphacentauri) path(3,arbre) path(7,snowdin) path(8,tambi) path(4,straylight) Optimization: 218 Answer: 9 path(5,faerun) path(8,norrath) path(1,tristram) path(4,alphacentauri) path(6,arbre) path(3,snowdin) path(2,tambi) path(7,straylight) Optimization: 207 OPTIMUM FOUND Models : 9 Optimum : yes Optimization : 207 Calls : 1 Time : 0.050s (Solving: 0.04s 1st Model: 0.00s Unsat: 0.03s) CPU Time : 0.040s

Changing `#minimize` to `#maximize` in `ts.lp` yields a longest path of Norrath-Tristram-AlphaCentauri-Arbre-Snowdin-Tambi-Straylight, with a length of 804.