On multi-path routing for reliable communications in failure interdependent complex networks

Abstract

This paper studies several new multiple routing path computation problems in failure-interdependent complex networks such as smart grid communication network, each of which exhibits unique failure interdependency. Despite the difference of the formulation of the problems, we show that each of the problems can be reduced to another within polynomial time, and therefore they are equivalent in terms of hardness. Then, we show that they are not only $\mathcal {NP}$-hard, but also cannot be approximated within a certain bound unless ${\mathcal {P}}=\mathcal {NP}$. Besides, we show that their decision versions to determine if there exist two failure independent paths between two given end nodes are still $\mathcal {NP}$-complete. Finally, a series of heuristic algorithms are proposed to deal with the daunting hardness of the problems. Most importantly, this paper opens a new series of research problems with daunting complexity based on important real world applications.

Appendix: Simulation results

Following is a list the running results of 4 schemes of algorithm on 10 random connected graphs with random coloring. 10 blocks show the simulation results on 10 graphs respectively. The first line of each blocks gives some coefficients of the graph. n represents to the graph order and k refers to the number of candidate colors, while ed and cd tell the link density and color density of this random colored graph. In addition, p, another important coefficient, means the total number of candidate paths. Next 4 lines show the performance of each scheme on the graph. The first number is the number of paths finding by the algorithm and the other number denotes running time with respect to the unit second.

Scheme for collecting candidate paths:

• Scheme 1: Arbitrary initial setting and always resetting.

• Scheme 2: Arbitrary initial setting and always punishing.

• Scheme 3: Color based initial setting and always resetting.

• Scheme 4: Color based initial setting and always punishing.

• 1: $n=90,k=86,ed=0.35,cd=2.4,p=854$;

  24    4.790673

  24    3.467621

  11    23.455156

  26    3.593162

• 2: $n=77,k=40,ed=0.43,cd=2.2,p=676$;

  10    1.834425

  12    1.764007

  3    11.299291

  13    1.921618

• 3: $n=73,k=70,ed=0.36,cd=1.7,p=624$;

  21    1.692405

  21    1.627282

  12    10.490142

  23    1.838357

• 4: $n=72,k=67,ed=0.31,cd=1.9,p=611$;

  27    1.765221

  26    1.611116

  22    10.164275

  28    1.797583

• 5: $n=59,k=46,ed=0.46,cd=1.6,p=453$;

  16    0.750409

  15    0.735994

  8    4.663884

  15    0.842362

• 6: $n=78,k=74,ed=0.62,cd=2.5,p=689$;

  15    1.517225

  15    1.419095

  4    12.148387

  16    1.583640

• 7: $n=97,k=54,ed=0.60,cd=2.2,p=955$;

  18    3.128945

  18    2.909633

  7    25.257773

  20    3.279582

• 8: $n=76,k=75,ed=0.68,cd=2.2,p=663$;

  13    1.181470

  13    1.138946

  4    11.041537

  13    1.267679

• 9: $n=80,k=72,ed=0.43,cd=2.3,p=716$;

  17    2.122116

  18    2.011091

  4    13.872896

  19    2.209266

• 10: $n=55,k=52,ed=0.48,cd=1.7,p=408$;

  12   0.577832

  13   0.594291

  5    3.690573

  13   0.662304