On the fine-grained complexity of rainbow coloring

The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in $k$ colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any $k\ge 2$, there is no algorithm for Rainbow k-Coloring running in time $2^{o(n^{3/2})}$, unless ETH fails. Motivated by this negative result we consider two parameterized variants of the problem. In Subset Rainbow k-Coloring problem, introduced by Chakraborty et al. [STACS 2009, J. Comb. Opt. 2009], we are additionally given a set $S$ of pairs of vertices and we ask if there is a coloring in which all the pairs in $S$ are connected by rainbow paths. We show that Subset Rainbow k-Coloring is FPT when parameterized by $|S|$. We also study Maximum Rainbow k-Coloring problem, where we are additionally given an integer $q$ and we ask if there is a coloring in which at least $q$ anti-edges are connected by rainbow paths. We show that the problem is FPT when parameterized by $q$ and has a kernel of size $O(q)$ for every $k\ge 2$ (thus proving that the problem is FPT), extending the result of Ananth et al. [FSTTCS 2011]

Linear kernels for outbranching problems in sparse digraphs

In the $k$-Leaf Out-Branching and $k$-Internal Out-Branching problems we are given a directed graph $D$ with a designated root $r$ and a nonnegative integer $k$. The question is to determine the existence of an outbranching rooted at $r$ that has at least $k$ leaves, or at least $k$ internal vertices, respectively. Both these problems were intensively studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with $O(k^2)$ vertices are known on general graphs. In this work we show that $k$-Leaf Out-Branching admits a kernel with $O(k)$ vertices on $\mathcal{H}$-minor-free graphs, for any fixed family of graphs $\mathcal{H}$, whereas $k$-Internal Out-Branching admits a kernel with $O(k)$ vertices on any graph class of bounded expansion.Comment: Extended abstract accepted for IPEC'15, 27 page

Approximation and Parameterized Complexity of Minimax Approval Voting

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 $\mathcal{O}^\star(\left({3}/{\epsilon}\right)^{2d})$, which is essentially tight assuming ETH. Finally, we get a new polynomial-time randomized approximation scheme for Minimax Approval Voting, which runs in time $n^{\mathcal{O}(1/\epsilon^2 \cdot \log(1/\epsilon))} \cdot \mathrm{poly}(m)$, almost matching the running time of the fastest known PTAS for Closest String due to Ma and Sun [SIAM J. Comp. 2009].Comment: 14 pages, 3 figures, 2 pseudocode