The Complexity of Optimal Design of Temporally Connected Graphs

Abstract

We study the design of small cost temporally connected graphs, under various constraints. We mainly consider undirected graphs of n vertices, where each edge has an associated set of discrete availability instances (labels). A journey from vertex u to vertex v is a path from u to v where successive path edges have strictly increasing labels. A graph is temporally connected iff there is a (u, v)-journey for any pair of vertices u, v, u = v. We first give a simple polynomial-time algorithm to check whether a given temporal graph is temporally connected. We then consider the case in which a designer of temporal graphs can freely choose availability instances for all edges and aims for temporal connectivity with very small cost; the cost is the total number of availability instances used. We achieve this via a simple polynomialtime procedure which derives designs of cost linear in n. We also show that the above procedure is (almost) optimal when the underlying graph is a tree, by proving a lower bound on the cost for any tree. However, there are pragmatic cases where one is not free to design a temporally connected graph anew, but is instead given a temporal graph design with the claim that it is temporally connected, and wishes to make it more cost-efficient by removing labels without destroying temporal connectivity (redundant labels). Our main technical result is that computing the maximum number of redundant labels is APX-hard, i.e., there is no PTAS unless P = NP . On the positive side, we show that in dense graphs with random edge availabilities, there is asymptotically almost surely a very large number of redundant labels. A temporal design may, however, be minimal, i.e., no redundant labels exist. We show the existence of minimal temporal designs with at least n log n labels.

Sliding Window Temporal Graph Coloring

“…x i form a (not necessarily induced) P 3 in every snapshot, that is {v (1) x i , v (2) x i } ∈ E t and {v (2) x i , v (3) x i } ∈ E t for all 1 ≤ t ≤ 3. Furthermore, we connect v (1) x i and v (3) x i in the second snapshot, that is, {v (1) x i , v (3) x i } ∈ E 2 . Lastly, we create a full C 5 in snapshot three, that is, {v…”

“…Connection of variable and clause gadgets: The clause gadgets and variable gadgets are connected in the third snapshot. Let clause c i = (ℓ i,1 ∨ ℓ i,2 ∨ ℓ i,3 ) with 1 ≤ i ≤ m have literals ℓ i,1 , ℓ i,2 , and ℓ i, 3 . Let x i,j with 1 ≤ i ≤ m and 1 ≤ j ≤ 3 be the variable of the jth literal in clause c i .…”

“…If variable x i is set to false in the satisfying assignment, then we color the triangle of the corresponding variable gadget in a way that leaves edge {v (2) x i , v (3) x i } monochromatic. To be specific, we color vertices v (2) x i and v (3) x i in yellow and vertices v (1) x i , v (4) x i , and v (5) x i in blue. For each clause c i with 1 ≤ i ≤ m we choose one of its literals that satisfies the clause.…”

“…Then we construct a satisfying assignment for φ in the following way: Note that in the second snapshot each variable gadget contains a triangle with exactly one monochromatic edge. The edge {v (1) x i , v (3) x i } only exists in the second snapshot and hence is colored properly by any proper sliding 2-window temporal 2-coloring. This means that either edge {v…”

“…Now the classical computational problems have to be appropriately redefined in the temporal setting in order to properly capture the notion of time. Motivated by the fact that, due to causality, information in temporal graphs can "flow" only along sequences of edges whose time-labels are increasing, most temporal graph parameters and optimization problems that have been studied so far are based on the notion of temporal paths and other "path-related" notions, such as temporal analogues of distance, reachability, separators, diameter, exploration, and centrality [2,3,14,15,18,28,32,40].…”