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In this paper, we study the All-Colors problem: given a graph G each of whose vertices is equipped with a button and assigned a color value from the set \(\{0,1, \ldots , m-1\}\) and an integer k, can we reach color value 1 \(\ (\textrm{mod}\ m)\) on every vertex of G by pressing the button at most k times. The rule we follow is the following: if a button of a corresponding vertex is pressed one time, then the color values of the vertex and its neighbors are incremented by 1. This problem is known to be NP-hard on bipartite graphs even when \(m =2\) [Theor. Comput. Sci., 2007], although linear time solvable on trees [SIAM J. Comput., 2004]. In this work, we study this problem in the realm of parameterized complexity with respect to several parameters. In particular, we show the following for All-Colors.

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A Parameterized Perspective of All-Colors

  • 会议论文

  • 发表日期：

    2025年

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  • Václav Blažej,
  • Satyabrata Jana,
  • Peter Strulo

摘要

In this paper, we study the All-Colors problem: given a graph G each of whose vertices is equipped with a button and assigned a color value from the set \(\{0,1, \ldots , m-1\}\) and an integer k, can we reach color value 1 \(\ (\textrm{mod}\ m)\) on every vertex of G by pressing the button at most k times. The rule we follow is the following: if a button of a corresponding vertex is pressed one time, then the color values of the vertex and its neighbors are incremented by 1. This problem is known to be NP-hard on bipartite graphs even when \(m =2\) [Theor. Comput. Sci., 2007], although linear time solvable on trees [SIAM J. Comput., 2004]. In this work, we study this problem in the realm of parameterized complexity with respect to several parameters. In particular, we show the following for All-Colors.