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]

Known Algorithms on Graphs of Bounded Treewidth Are Probably Optimal

We obtain a number of lower bounds on the running time of algoritluns solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that n-variable m-clause SAT cannot be solved in time (2 - epsilon)(n) m(O(1)), we show that for any epsilon > 0: INDEPENDENT SET cannot be solved ill time (2 - epsilon)(tw(G))vertical bar V(G)vertical bar(O(1)), DOMINATING SET cannot be solved in time (3 - epsilon)(tw(G))vertical bar V(G)vertical bar(O(1)), MAX CUT cannot be solved in time (2 - epsilon)(tw(G))vertical bar V(G)vertical bar(O(1)), ODD CYCLE TRANSVERSAL cannot be solved in lime (3 - epsilon)(tw(G))vertical bar V(G)vertical bar(O(1)) For ally fixed q >= 3, q-COLORING cannot be solved in time (q - epsilon)(tw(G)) vertical bar V(G)vertical bar(O(1)), PARTITION INTO TRIANGLES cannot be solved in time (2 - epsilon)(tw(G))vertical bar V(G)vertical bar(O(1)). Our lower bounds match the running times for the best known algoritluns for the problems, up to the epsilon in the base