Hiding Signatures in Graph Coloring Solutions

Abstract

One way to protect a digital product is to embed an author’s signature into the object in the form of minute errors. However, this technique cannot be directly applied to protect intellectual properties such as solutions to hard problems which must maintain the correct 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 importance in complexity theory and numerous applications in real life. The first adds extra edges between some pairs of vertices and therefore forces them to be colored by different colors. The second pre-colors a set of well-selected vertices according to the watermark. And the last introduces new vertices and edges to the graph. Since credibility and overhead are the most important criteria for any efficient watermarking technique, we derive formulae which explicitly illustrate the trade-off between high credibility and low overhead. Asymptotically we prove that for almost all random graphs G _n,p , an arbitrarily high credibility can be achieved by all proposed watermarking techniques with at most 1-color-overhead. Numerical simulation on random graphs shows that a large amount of information 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 colored with the known minimal number of colors for the original instances with no run-time overhead.