# Optimal Path Discovery Problem with Homogeneous Knowledge

- 67 Downloads

## Abstract

Consider the following problem: given a complete graph *G* = (*V*, *E*), two nodes *s* and *t* in *V*, and a positive hidden value *f*(*e*) for each edge *e* ∈ *E*, discover an *s* − *t*-path *P* that minimizes the value *F*(*P*), for some objective function *F*. The issue is that the edge values *f*(⋅) are hidden, hence, to discover an optimal path, it is required to uncover the value of some edges. The goal then is to discover an optimal path by means of uncovering the least possible amount of edge values. This problem, named the *Optimal Path Discovery* (OPD) problem, is an extension of the well known *Shortest Path Discovery* problem in which *f*(*e*) represents the length of *e*, and *F*(*P*) computes the length of *P*. In this paper, we study the OPD problem when the only previous information known about the *f*(⋅) values is that they fall in the interval (0,*∞*) for all *e* ∈ *E*. We first study the number of uncovered edges as a measure to evaluate algorithms. We see that this measure does not differentiate correctly algorithms according to their performance. Therefore, we introduce the *query ratio*, the ratio between the number of uncovered edges and the least number of edge values required to solve the problem. We prove a 1 + 4/*n* − 8/*n*^{2} lower bound on the query ratio and we present an algorithm whose query ratio, when it finds the optimal path, is upper bounded by 2 − 1/(*n* − 1), where *n* = |*V* |. Finally, we implement different algorithms and evaluate their query ratio experimentally.

## Keywords

Optimal path Query ratio Shortest path discovery problem Lower and upper bounds## Notes

### Acknowledgments

The research leading to these results has received funding from the European Community’s Seventh Framework Programme [FP7/2007-2013] under the PANACEA Project (http://www.panacea-cloud.eu), grant agreement n 610764, from the Marie Sklodowska-Curie grant agreement No 777778, from the Basque Government, Spain, Consolidated Research Group called Mathematical Modeling, Simulation and Industrial Application (MS2I) with the Grant Reference IT649-13 and from the Spanish Ministry of Economy and Competitiveness project with reference MTM2016-76329-R.

## References

- 1.Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications, 1st edn. Prentice Hall (1993)Google Scholar
- 2.Alon, N., Emek, Y., Feldman, M., Tennenholtz, M.: Economical graph discovery. In: ICS, pp. 476–486 (2011)Google Scholar
- 3.Aron, I.D., Van Hentenryck, P.: On the complexity of the robust spanning tree problem with interval data. Oper. Res. Lett.
**32**(1), 36–40 (2004)MathSciNetCrossRefGoogle Scholar - 4.Bellman, R.: On a routing problem. Q. Appl. Math.
**16**, 87–90 (1958)CrossRefGoogle Scholar - 5.Bruce, R., Hoffmann, M., Krizanc, D., Raman, R.: Efficient update strategies for geometric computing with uncertainty. Theory Comput. Syst.
**38**(4), 411–423 (2005)MathSciNetCrossRefGoogle Scholar - 6.Cherkassky, B.V., Goldberg, A.V., Radzik, T.: Shortest paths algorithms: Theory and experimental evaluation. Math. Program.
**73**(2), 129–174 (1996)MathSciNetCrossRefGoogle Scholar - 7.Davis, H.W., Pollack, R.B., Sudkamp, T.: Towards a better understanding of bidirectioanl search. In: AAAI (1984)Google Scholar
- 8.de Champeaux, D.: Bidirectional heuristic search again. J. ACM
**30**(1), 22–32 (1983)MathSciNetCrossRefGoogle Scholar - 9.Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math.
**1**, 269–271 (1959)MathSciNetCrossRefGoogle Scholar - 10.Erlebach, T., Hoffmann, M., Kammer, F.: Query-competitive algorithms for cheapest set problems under uncertainty. Theor. Comput. Sci.
**613**(C), 51–64 (2016)MathSciNetCrossRefGoogle Scholar - 11.Feder, T., Motwani, R., Panigrahy, R., Olston, C., Widom, J.: Computing the median with uncertainty. SIAM J. Comput.
**32**(2), 538–547 (2003)MathSciNetCrossRefGoogle Scholar - 12.Feder, T., Motwani, R., O’Callaghan, L., Olston, C., Panigrahy, R.: Computing shortest paths with uncertainty. J. Algor.
**62**(1), 1–18 (2007)MathSciNetCrossRefGoogle Scholar - 13.Floyd, R.W.: Algorithm 97: Shortest path. Commun. ACM
**5**(6), 345 (1962)CrossRefGoogle Scholar - 14.Ford, L.R.: Network flow theory. Technical Report Paper P-923, RAND Corporation, Santa Monica, California (1956)Google Scholar
- 15.Ghosh, S., Mahanti, A.: Bidirectional heuristic search with limited resources. Inf. Process. Lett.
**40**(6), 335–340 (1991)CrossRefGoogle Scholar - 16.Hart, P. E., Nilsson, N.J., Raphael, B.: A formal basis for the heuristic determination of minimum cost paths. In: IEEE Transactions on Systems Science and Cybernetics, pp. 100–107 (1968)CrossRefGoogle Scholar
- 17.Hoffmann, M., Erlebach, T., Krizanc, D., Mihal’ák, M., Raman, R.: Computing minimum spanning trees with uncertainty. In: 25th International Symposium on Theoretical Aspects of Computer Science, Leibniz International Proceedings in Informatics (LIPIcs), pp. 277–288. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2008)Google Scholar
- 18.Johnson, D.B.: Efficient algorithms for shortest paths in sparse networks. J. ACM
**24**(1), 1–13 (1977)MathSciNetCrossRefGoogle Scholar - 19.Kahan, S.: A model for data in motion. In: Proceedings of the Twenty-third Annual ACM Symposium on Theory of Computing, STOC ’91, pp. 265–277. ACM (1991)Google Scholar
- 20.Kasperski, A., Zieliński, P.: An approximation algorithm for interval data minmax regret combinatorial optimization problems. Inf. Process. Lett.
**97**(5), 177–180 (2006)MathSciNetCrossRefGoogle Scholar - 21.Khanna, S., Tan, W.-C.: On computing functions with uncertainty. In: Proceedings of the Twentieth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, PODS ’01, pp. 171–182. ACM (2001)Google Scholar
- 22.Korf, R.E., Kumar, V.: Optimal path-finding algorithms. In: Kanal, L. (ed.) Search in Artificial Intelligence, Symbolic Computation, pp 223–267. Springer, New York (1988)CrossRefGoogle Scholar
- 23.Lippi, M., Ernandes, M., Felner, A.: Efficient single frontier bidirectional search. In: Proceeding of the Forth International Symposium on Combinatorial Search (2012)Google Scholar
- 24.Luby, M., Ragde, P.: A bidirectional shortest-path algorithm with good average-case behavior. Algorithmica
**4**(1–4), 551–567 (1989)MathSciNetCrossRefGoogle Scholar - 25.Montemanni, R., Gambardella, L.M.: An algorithm for the relative robust shortest path problem with interval data. Technical Report IDSIA-05-02 Dalle Molle Institute for Artificial Intelligence (2002)Google Scholar
- 26.Olston, C., Widom, J.: Offering a precision-performance tradeoff for aggregation queries over replicated data. In: Proceedings of the 26th International Conference on Very Large Data Bases, VLDB ’00, pp. 144–155. Morgan Kaufmann Publishers Inc. (2000)Google Scholar
- 27.Karasan, H.Y.O.E., Pinar, M.C.: The robust shortest path problem with interval data. Technical report, Bilkent University, Department of Industrial Engineering (2001)Google Scholar
- 28.Pohl, I.: Bi-Directional and Heuristics Search in Path Problems. PhD thesis, Standford University (1969)Google Scholar
- 29.Szepesvári, C.: Shortest path discovery problems: A framework, algorithms and experimental results. In: AAAI, pp. 550–555 (2004)Google Scholar
- 30.Yaman, H., Karasan, O.E., Pinar, M.C.: The robust spanning tree problem with interval data. Oper. Res. Lett.
**29**(1), 31–40 (2001)MathSciNetCrossRefGoogle Scholar - 31.Brun, O., Wang, L., Gelenbe, E.: Big data for autonomic intercontinental overlays. IEEE J. Selected Areas Commun.,
**34**(3) (2016)CrossRefGoogle Scholar - 32.Feamster, N., Balakrishnan, H., Rexford, J., Shaikh, A, van der Merwe, J.: The case for separating routing from routers. In: Proceedings of the ACM SIGCOMM Workshop on Future Directions in Network Architecture, pp 5–12. ACM, Portland (2004)Google Scholar