Abstract
In this paper, we study the problem of computing a min-color path in an edge-colored graph. More precisely, we are given a graph \(G=(V, E)\), source s, target t, an assignment \(\chi :E \rightarrow 2^{\mathcal {C}}\) of edges to a set of colors in \(\mathcal {C}\), and we want to find a path from s to t such that the number of unique colors on this path is minimum over all possible \(s-t\) paths. We show that this problem is hard (conditionally) to approximate within a factor \(O(n^{1/8})\) of optimum, and give a polynomial time \(O(n^{2/3})\)-approximation algorithm. We translate the ideas used in this approximation algorithm into two simple greedy heuristics, and analyze their performance on an extensive set of synthetic and real world datasets. From our experiments, we found that our heuristics perform significantly better than the best previous heuristic algorithm for the problem on all datasets, both in terms of path quality and the running time.
This work was supported by NSF under Grant CCF-1814172.
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Notes
- 1.
This is because if each edge has at most one color, the pruning stage in algorithm from [11] can be phrased as a maximum coverage problem, for which constant approximations are known. However even if number of colors is exactly two, the pruning stage becomes a variant of the densest k-subgraph problem which is hard to approximate within a factor of \(\varOmega (n^{1/4})\) of optimum.
- 2.
We assume that the random graph is constructed under G(n, p) model, that is an edge exists between a pair of vertices with probability p.
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Kumar, N. (2019). Computing a Minimum Color Path in Edge-Colored Graphs. In: Kotsireas, I., Pardalos, P., Parsopoulos, K., Souravlias, D., Tsokas, A. (eds) Analysis of Experimental Algorithms. SEA 2019. Lecture Notes in Computer Science(), vol 11544. Springer, Cham. https://doi.org/10.1007/978-3-030-34029-2_3
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