arXiv subjects

Hossein Shokouhi

Publications and source records attributed to Hossein Shokouhi.

2 recordsLinked to original sources

arXiv2026

Parameterized Complexity of Fair Coloring Problem

Given a graph $G=(V,E)$, a (proper) $k$-coloring for $G$ is a vertex coloring with $k$ colors such that every two adjacent vertices receive different colors. Suppose that the vertex set $V$ is partitioned into some groups, a proper coloring is called fair if for every color class, the difference between the number of vertices in any two groups does not exceed a given threshold. In this paper, we investigate the parameterized complexity of the fair coloring problem with respect to the structural parameters of the input graph. In particular, we prove that the problem is W[1]-hard with respect to the number of groups for forests and also graphs of modular-width two, even when the number of colors is equal to two. On the positive side, we prove that when the number of colors is equal to two, then the problem is FPT with respect to neighborhood diversity of the input graph. Moreover, in general, the problem is FPT with respect to neighborhood diversity and the number of groups. As a by-product, we prove that unary vector bin packing problem is W[1]-hard with respect to the dimension.

cs.CC↗

arXiv2024

Parameterized Complexity of Fair Many-to-One Matchings

Given a bipartite graph $G=(U\cup V,E)$, a left-perfect many-to-one matching is a subset $M \subseteq E$ such that each vertex in $U$ is incident with exactly one edge in $M$. If $U$ is partitioned into some groups, the matching is called fair if for every $v\in V$, the difference between the number of vertices matched with $v$ in any two groups does not exceed a given threshold. In this paper, we investigate parameterized complexity of fair left-perfect many-to-one matching problem with respect to the structural parameters of the input graph. In particular, we prove that the problem is W[1]-hard with respect to the feedback vertex number, tree-depth and the maximum degree of $U$, combined. Also, it is W[1]-hard with respect to the path-width, the number of groups and the maximum degree of $U$, combined. In the positive side, we prove that the problem is FPT with respect to the treewidth and the maximum degree of $V$. Also, it is FPT with respect to the neighborhood diversity of the input graph (which implies being FPT with respect to vertex cover and modular-width). Finally, we prove that the problem is FPT with respect to the tree-depth and the number of groups.

cs.CC↗