Łukasz Kowalik

Patryk Jędrzejczak, Łukasz Kowalik

In 1965, Vizing [Diskret. Analiz, 1965] showed that every planar graph of maximum degree $Δ\ge 8$ can be edge-colored using $Δ$ colors. The direct implementation of the Vizing's proof gives an algorithm that finds the coloring in $O(n^2)$ time for an $n$-vertex input graph. Chrobak and Nishizeki [J. Algorithms, 1990] have shown a more careful algorithm, which improves the time to $O(n\log n)$ time, though only for $Δ\ge 9$. In this paper, we extend their ideas to get an algorithm also for the missing case $Δ=8$. To this end, we modify the original recoloring procedure of Vizing. This generalizes to bounded genus graphs.

Marek Cygan, Łukasz Kowalik, Arkadiusz Socała et al.

We present three results on the complexity of Minimax Approval Voting. First, we study Minimax Approval Voting parameterized by the Hamming distance $d$ from the solution to the votes. We show Minimax Approval Voting admits no algorithm running in time $\mathcal{O}^\star(2^{o(d\log d)})$, unless the Exponential Time Hypothesis (ETH) fails. This means that the $\mathcal{O}^\star(d^{2d})$ algorithm of Misra et al. [AAMAS 2015] is essentially optimal. Motivated by this, we then show a parameterized approximation scheme, running in time $\mathcal{O}^\star(\left({3}/ε\right)^{2d})$, which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for Minimax Approval Voting, which runs in time $n^{\mathcal{O}(1/ε^2 \cdot \log(1/ε))} \cdot \mathrm{poly}(m)$, almost matching the running time of the fastest known PTAS for Closest String due to Ma and Sun [SIAM J. Comp. 2009].