Online facility location with weights and congestion

TL;DR

This work studies online facility location under two real-world constraints: weighted service requests and congestion. It extends Meyerson's randomized framework, introducing WRFL for weights and MRFL for congestion, and provides tight bounds via Selection Process reductions, phase analyses, and cluster-based decompositions. The weighted variant exhibits a stark separation: Ω(n) in the worst case but Θ(log n) in the secretarial model, while the congestion variant achieves a constant-competitive regime for monomial congestion with a bound that depends only on the function g and the opening cost f, namely Θ(log k^*/log log k^*) where k^* = 2·g^{-1}( f/(g(2)-2) ). Together, these results yield tight, n-independent performance in the congestion setting and clarify fundamental differences between the two variants, with practical implications for settings like vaccination-center placement under congestion and heterogeneous demand.

Abstract

The classic online facility location problem deals with finding the optimal set of facilities in an online fashion when demand requests arrive one at a time and facilities need to be opened to service these requests. In this work, we study two variants of the online facility location problem; (1) weighted requests and (2) congestion. Both of these variants are motivated by their applications to real life scenarios and the previously known results on online facility location cannot be directly adapted to analyse them. Weighted requests: In this variant, each demand request is a pair $(x,w)$ where $x$ is the standard location of the demand while $w$ is the corresponding weight of the request. The cost of servicing request $(x,w)$ at facility $F$ is $w\cdot d(x,F)$. For this variant, given $n$ requests, we present an online algorithm attaining a competitive ratio of $\mathcal{O}(\log n)$ in the secretarial model for the weighted requests and show that it is optimal. Congestion: The congestion variant considers the case when there is an additional congestion cost that grows with the number of requests served by each facility. For this variant, when the congestion cost is a monomial, we show that there exists an algorithm attaining a constant competitive ratio. This constant is a function of the exponent of the monomial and the facility opening cost but independent of the number of requests.

Figures (6)

• Figure 1:
• Figure 2: Phase $i$ starts when Facility $F_i$ is opened and ends when Facility $F_{i+1}$ is opened where by definition $F_{i+1}$ should be closer to $c^*$ than $F_i$. All the requests in Phase $i$ are shown in red dots. They may be closer or farther away from $F_{i}$ with respect to $c^*$. The only guarantee that we have is that the first request closer than $F_i$ that results in a facility opening would change the phase to $i+1$.
• Figure 3: Array $A$ of probabilities
• Figure 4: The sequence of input requests will consist of $1$ request at the root node, followed by $h$ requests at the next node, $h^2$ requests at the next node and so on. These nodes where requests will be, are shown in red and form a path from the root node to a leaf node.
• Figure 5: This diagram encapsulates the annuli and the ball at the centre, all concentric with centre at $c^*$. For one annulus, $RFL$ might not open a facility for the first few requests and use other facilities nearby (These requests are shown in Red). After this, on some request $RFL$ will open a facility on it (Shown in Blue). All future requests (Shown in green) on arrival will have the facility opened on Blue request (or other open facilities nearer to the request), where they may be allocated by $RFL$, if needed. Notice that the distance from the blue request to any green request in the annulus can be at most $2\cdot m^2A$ .

Theorems & Definitions (43)

• Definition 2.1: Weighted Online Facility Location
• Theorem 2.2
• Theorem 2.3
• Proposition 3.1
• proof
• Definition 3.2
• proof : Proof of \ref{['secmodel']}
• Claim 3.3
• proof : Proof of Claim
• Lemma 3.4