Delaying Decisions and Reservation Costs

by   Elisabet Burjons, et al.

We study the Feedback Vertex Set and the Vertex Cover problem in a natural variant of the classical online model that allows for delayed decisions and reservations. Both problems can be characterized by an obstruction set of subgraphs that the online graph needs to avoid. In the case of the Vertex Cover problem, the obstruction set consists of an edge (i.e., the graph of two adjacent vertices), while for the Feedback Vertex Set problem, the obstruction set contains all cycles. In the delayed-decision model, an algorithm needs to maintain a valid partial solution after every request, thus allowing it to postpone decisions until the current partial solution is no longer valid for the current request. The reservation model grants an online algorithm the new and additional option to pay a so-called reservation cost for any given element in order to delay the decision of adding or rejecting it until the end of the instance. For the Feedback Vertex Set problem, we first analyze the variant with only delayed decisions, proving a lower bound of 4 and an upper bound of 5 on the competitive ratio. Then we look at the variant with both delayed decisions and reservation. We show that given bounds on the competitive ratio of a problem with delayed decisions impliy lower and upper bounds for the same problem when adding the option of reservations. This observation allows us to give a lower bound of min{1+3α,4} and an upper bound of min{1+5α,5} for the Feedback Vertex Set problem. Finally, we show that the online Vertex Cover problem, when both delayed decisions and reservations are allowed, is min{1+2α, 2}-competitive, where α∈ℝ_≥ 0 is the reservation cost per reserved vertex.


page 1

page 2

page 3

page 4


Set Cover and Vertex Cover with Delay

The set cover problem is one of the most fundamental problems in compute...

An Improved Deterministic Algorithm for the Online Min-Sum Set Cover Problem

We study the online variant of the Min-Sum Set Cover (MSSC) problem, a g...

A modified greedy algorithm to improve bounds for the vertex cover number

In any attempt at designing an efficient algorithm for the minimum verte...

Hardness and approximation of the Probabilistic p-Center problem under Pressure

The Probabilistic p-Center problem under Pressure (Min PpCP) is a varian...

Tight Bounds for Online Matching in Bounded-Degree Graphs with Vertex Capacities

We study the b-matching problem in bipartite graphs G=(S,R,E). Each vert...

Online learning with graph-structured feedback against adaptive adversaries

We derive upper and lower bounds for the policy regret of T-round online...

The Batched Set Cover Problem

We introduce the batched set cover problem, which is a generalization of...

Please sign up or login with your details

Forgot password? Click here to reset