Graph Exploration with Edge Weight Estimates

TL;DR

The paper introduces Graph Exploration with Edge Weight Estimates (GEEWE), a semi-online model where each edge weight is bounded by known intervals and revealed when a vertex is visited. It proves a general lower bound of $α$ on the competitive ratio for unrestricted graphs when $α≤2$, and shows this bound is achievable by a simple precomputation-based strategy; for uniform weights in $[1,α]$ on restricted graph classes it presents a recalculating algorithm with competitive ratio $(1+α)/2$, proven tight on complete graphs and complete bipartite graphs. It also discusses limitations on other graph classes such as cycles and outlines open problems in extending these results. Overall, the work bridges TSP and graph exploration under weight uncertainty, providing a principled framework and concrete bounds for adaptive routing with estimated edge costs.

Abstract

In the Travelling Salesman Problem, every vertex of an edge-weighted graph has to be visited by an agent who traverses the edges of the graph. In this problem, it is usually assumed that the costs of each edge are given in advance, making it computationally hard but possible to calculate an optimal tour for the agent. Also in the Graph Exploration Problem, every vertex of a given graph must be visited, but here the graph is not known in the beginning - at every point, an algorithm only knows about the already visited vertices and their neighbors. Both however are not necessarily realistic settings: Usually the structure of the graph (for example underlying road network) is known in advance, but the details are not. One usually has a prediction of how long it takes to traverse through a particular road, but due to road conditions or imprecise maps the agent might realize that a road will take slightly longer than expected when arriving on it. To deal with those deviations, it is natural to assume that the agent is able to adapt to the situation: When realizing that taking a particular road is more expensive than expected, recalculating the tour and taking another road instead is possible. We analyze the competitive ratio of this problem based on the perturbation factor $α$ of the edge weights. For general graphs we show that for realistic factors smaller than $2$ there is no strategy that achieves a competitive ratio better than $α$, which can be matched by a simple algorithm. In addition, we prove an algorithm which has a competitive ratio of $\frac{1+α}{2}$ for restricted graph classes like complete graphs with uniform announced edge weights. Here, we present a matching lower bound as well, proving that the strategy for those graph classes is best possible. We conclude with a remark about special graph classes like cycles.

Figures (2)

• Figure 1: An example instance with the announcements of recursion depth $2$ with $k=3$ and $\alpha=2$. An online algorithm needs to traverse from $s_2$ to $t_2$. The edges of level $0$ are just given with the weight $1$, as they will be presented with weight $1$ anyway. Depending on the decisions of the algorithm, the other edges will either have a value of the upper or lower bound.
• Figure 2: Every edge is announced with $[1,\alpha]$ and revealed with the labeled weight. The adaptive exploration algorithm will traverse on the broad edges, and will not take the small ones (so every second layer). An alternative solution would take the continuous ones (and takes some edges twice in the bottom layer), and ignore the dashed ones. The adaptive exploration algorithm therefore walks on a Hamiltonian path, where each edge costs $\alpha$. The alternative solution only pays $1$ for the majority of the edges in the center, but needs more edges.

Theorems & Definitions (18)

• definition thmcounterdefinition: Travelling Salesman Problem (TSP)
• definition thmcounterdefinition: Graph Exploration with Edge Weight Estimates (GEEWE)
• theorem thmcountertheorem
• proof
• remark thmcounterremark
• theorem thmcountertheorem
• proof
• theorem thmcountertheorem
• lemma thmcounterlemma
• proof : of Lemma \ref{['lemma6']}