arXiv subjects

M. Kubale

Publications and source records attributed to M. Kubale.

3 recordsLinked to original sources

arXiv2017

Tight Bounds on the Complexity of Semi-Equitable Coloring of Cubic and Subcubic Graphs

A $k$-coloring of a graph $G=(V,E)$ is called semi-equitable if there exists a partition of its vertex set into independent subsets $V_1,\ldots,V_k$ in such a way that $|V_1| \notin \{\lceil |V|/k\rceil, \lfloor |V|/k \rfloor\}$ and $||V_i|-|V_j|| \leq 1$ for each $i,j=2,\ldots,k$. The color class $V_1$ is called non-equitable. In this note we consider the complexity of semi-equitable $k$-coloring, $k\geq 4$, of the vertices of a cubic or subcubic graph $G$. In particular, we show that, given a $n$-vertex subcubic graph $G$ and constants $ε> 0$, $k \geq 4$, it is NP-complete to obtain a semi-equitable $k$-coloring of $G$ whose non-equitable color class is of size $s$ if $s \geq n/3+εn$, and it is polynomially solvable if $s \leq n/3$.

math.CO↗

arXiv2016

Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines

In the paper we consider the problem of scheduling $n$ identical jobs on 4 uniform machines with speeds $s_1 \geq s_2 \geq s_3 \geq s_4,$ respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree $Δ$, where two incompatible jobs cannot be processed on the same machine. We show that the problem is NP-hard even if $s_1=s_2=s_3$. If, however, $Δ\leq 4$ and $s_1 \geq 12 s_2$, $s_2=s_3=s_4$, then the problem can be solved to optimality in time $O(n^{1.5})$. The same algorithm returns a solution of value at most 2 times optimal provided that $s_1 \geq 2s_2$. Finally, we study the case $s_1 \geq s_2 \geq s_3=s_4$ and give an $O(n^{1.5})$-time $32/15$-approximation algorithm in all such situations.

cs.DS↗

arXiv2015

Scheduling of unit-length jobs with cubic incompatibility graphs on three uniform machines

In the paper we consider the problem of scheduling $n$ identical jobs on 3 uniform machines with speeds $s_1, s_2,$ and $s_3$ to minimize the schedule length. We assume that jobs are subjected to some kind of mutual exclusion constraints, modeled by a cubic incompatibility graph. We show that if the graph is 2-chromatic then the problem can be solved in $O(n^2)$ time. If the graph is 3-chromatic, the problem becomes NP-hard even if $s_1>s_2=s_3$. However, in this case there exists a $4/3$-approximation algorithm running in $O(n^3)$ time. Moreover, this algorithm solves the problem almost surely to optimality if $3s_1/4 \leq s_2 = s_3$.

cs.DM↗