Hiding Signatures in Graph Coloring Solutions

One way to protect a digital product is to embed an au thor s signature into the object in the form of minute errors However this technique cannot be directly applied to protect intellectual proper ties such as solutions to hard problems which must maintain the cor rect functionality This paper proposes a constraint based watermarking technique and lays out a theoretical framework to evaluate this type of technique Based on this framework we analyze three watermarking techniques for the graph coloring problem because of its theoretical im portance in complexity theory and numerous applications in real life The rst adds extra edges between some pairs of vertices and therefore forces them to be colored by di erent colors The second pre colors a set of well selected vertices according to the watermark And the last intro duces new vertices and edges to the graph Since credibility and overhead are the most important criteria for any e cient watermarking technique we derive formulae which explicitly illustrate the trade o between high credibility and low overhead Asymptotically we prove that for almost all random graphs Gn p an arbitrarily high credibility can be achieved by all proposed watermarking techniques with at most color overhead Numerical simulation on random graphs shows that a large amount of in formation can be embedded with very low overhead We also watermark several sets of graphs generated from real life benchmarks Almost all the watermarked graphs with a huge watermark embedded can be col ored with the known minimal number of colors for the original instances with no run time overhead