The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erdős, Simonovits and Sós. For given graphs G and H the anti-Ramsey number ar ( G , H ) is defined to be the maximum number k such that there exists an assignment of k colors to the edges of G in which every copy of H in G has at least two edges with the same color. Usually, combinatorists study extremal values of anti-Ramsey numbers for various classes of graphs. There are works on the computational complexity of the problem when H is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number ar ( G , P k ) , where P k is a path of length k . First, we observe that when k is close to n (the number of vertices in G ), the problem is hard; hence, the challenging part is the computational complexity of the problem when k is a fixed constant. We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing ar ( G , P k ) for every integer k ≥ 3 is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant k .

In prior work, Gupta et al. (SPAA 2022) presented a distributed algorithm for multiplying sparse n x n matrices, using n computers. They assumed that the input matrices are uniformly sparse---there are at most d non-zeros in each row and column---and the task is to compute a uniformly sparse part of the product matrix. Initially each computer knows one row of each input matrix, and eventually each computer needs to know one row of the product matrix. In each communication round each computer can send and receive one O(łog n)-bit message. Their algorithm solves this task in O(d^1.907 ) rounds, while the trivial bound is O(d^2).

We present a deterministic algorithm for solving a wide range of dynamic programming problems in trees in O(log D) rounds in the massively parallel computation model (MPC), with O(nδ) words of local memory per machine, for any given constant 0 < δ < 1. Here D is the diameter of the tree and n is the number of nodes---we emphasize that our running time is independent of n.

The anti-Ramsey numbers are a fundamental notion in graph theory, introduced in 1978, by Erdös, Simonovits and Sós. For given graphs G and H the anti-Ramsey number ar(G,H) is defined to be the maximum number k such that there exists an assignment of k colors to the edges of G in which every copy of H in G has at least two edges with the same color. Usually, combinatorists study extremal values of anti-Ramsey numbers for various classes of graphs. There are works on the computational complexity of the problem when H is a star. Along this line of research, we study the complexity of computing the anti-Ramsey number ar(G,P_k), where P_k is a path of length k. First, we observe that when k is close to n, the problem is hard; hence, the challenging part is the computational complexity of the problem when k is a fixed constant. We provide a characterization of the problem for paths of constant length. Our first main contribution is to prove that computing ar(G,P_k) for every integer k > 2 is NP-hard. We obtain this by providing several structural properties of such coloring in graphs. We investigate further and show that approximating ar(G,P₃) to a factor of n^{-1/2 - ε} is hard already in 3-partite graphs, unless P = NP. We also study the exact complexity of the precolored version and show that there is no subexponential algorithm for the problem unless ETH fails for any fixed constant k. Given the hardness of approximation and parametrization of the problem, it is natural to study the problem on restricted graph families. Along this line, we first introduce the notion of color connected coloring, and, employing this structural property, we obtain a linear time algorithm to compute ar(G,P_k), for every integer k, when the host graph, G, is a tree.