Complexity of assigning colors in network patterns on spider structures
Colored Interaction-Profile Realization: Complexity of Matching-Match on Spiders
Data Structures and Algorithms
Summary
This paper looks at the challenge of coloring parts of a network so that specific color-based connections appear as desired. The researchers focus on a problem called Matching-Match Puzzle, where you color nodes in a graph to match a set of colored interactions. They find that this problem remains very hard (NP-complete) even with just two colors and simple graphs. They also study special graphs called spiders, showing that some versions are still tough, while others can be solved more efficiently depending on the spider's shape and the number of colors.
What this means in practice
- •For graph algorithm designers: Avoid using the Matching-Match coloring approach for certain spider-structured networks when the problem is NP-hard or W[1]-hard to improve algorithm design efficiency.
- •For network modelers: Choose spider network structures with short legs when modeling colored interactions to enable efficient algorithms based on fixed-parameter tractability results.
A theory result. No direct application yet.
Authors
Ilie Dumitru, Adrian Miclaus, Alexandru Popa
Abstract
Network motifs and colored local interaction patterns provide a useful way to describe the structure of complex networks. Motivated by an inverse realization perspective, we study the problem of assigning colors to the vertices of a fixed graph so that its edges realize a prescribed multiset of colored pairwise interactions. This problem is formalized by the Matching-Match Puzzle, introduced by Iburi and Uehara. We investigate how its computational complexity depends on the number of colors and on the structure of the host graph. We first prove that Matching-Match is NP-complete with only two colors, even when no vertex is precolored and the graph has maximum degree three. We then focus on spiders. We show that the problem is W[1]-hard parameterized by the number of colors even on spiders with only the body precolored. In contrast, for spiders whose legs have length at most two, we give a fixed-parameter tractable algorithm parameterized by the number of colors, allowing arbitrary precoloring. Finally, we prove NP-completeness for spiders whose legs all have length exactly three when precoloring is allowed.