The Power of Recourse: Better Algorithms for Facility Location in Online and Dynamic Models

Our main algorithmic technique for proving Theorems 1 and 2 is local search, which is one of the powerful algorithm design paradigms. Indeed, for both results, the competitive (approximation) ratio we achieve is $1+\sqrt{2}+\epsilon$, which matches the best approximation ratio for offline facility location obtained using local search [3]. Both of our results are based on the following key lemma. Suppose we maintain local optimum solutions at every time step in our algorithm. When a new client $j_{t}$ comes at time $t$, we add it to our solution using a simple operation, and let $\Delta_{t}$ be the increase of our cost due to the arrival of $j_{t}$. The key lemma states that the sum of $\Delta_{t}$ values in the first $T^{\prime}$ time steps can be bounded in terms the optimum cost at time $T^{\prime}$. With a simple modification to the local search algorithm, in which we require each local operation decreases enough cost for every client it reconnects, one can bound the total client recourse.

The straightforward way to implement the local search algorithm takes time $\Omega(n^{3})$. To derive a better running time, we leverage the randomized local search idea of Charikar and Guha [10]. At every iteration, we randomly choose a facility $i$ or a closing operation, and then perform the best operation that opens or swaps in $i$, or closes a facility if that is what we choose. By restricting the facility $i$ and with the help of the heap data structure, an iteration of the algorithm can be implemented in time $O(|C|\log|F|)$. As in [10] we can also show that each iteration can make a reasonable progress in expectation, leading to a bound of $\tilde{O}(|F|)$ on the number of iterations for the success of the algorithm with high probability. We remark that the algorithm in [10] used a different local search framework. Therefore, our result shows that the classic algorithm of [3] can also be made fast.

However, directly replacing the randomized local search procedure with a deterministic one does not work: The solution at the end of each time might not be a local optimum as we did not enumerate all possible local operations. Thus the key lemma does not hold any more. Nevertheless we show that applying a few local operations around $j_{t}$ upon its arrival can address the issue. With the key lemma, one can bound the number of times we perform the iterative randomized local search procedure, and thus the overall running time.

Our proof for Theorem 3 is based on a generalization of the greedy algorithm for facility location on HST metrics, which was developed in [16] in the context of differential privacy but only for the case of uniform facility cost. The intuition of the algorithm is as follows: If for some vertex $v$ of the HST $T$, the number of clients in the tree $T_{v}$ (the sub-tree of $T$ rooted at $v$) times the length of parent edge of $v$ is big compared to the cost of the cheapest facility in $T_{v}$, then we should open that facility. Otherwise, we should not open it and let the clients in $T_{v}$ be connected to outside $T_{v}$ through the parent edge. This intuition can be made formal: We mark $v$ in the former case; then simply opening the cheapest facility in $T_{v}$ for all lowest marked vertices $v$ leads to a constant approximation for facility location.

The above offline algorithm leads to a dynamic data structure that maintains $O(1)$-approximate solutions, supports insertion and deletion of clients, and reports the connecting facility of a client in $O(\log D)$ time. This is the case since each time a client arrives or departs, only its ancestors will be affected. However, in a dynamic algorithm setting, we need to maintain the assignment vector in memory, so that when the connecting facility of a client changes, it needs to be notified. This requires that the number of reconnections made by our algorithm to be small. To achieve the goal, we impose two constants for each $v$ when deciding whether $v$ should be marked and the cheapest facility in $T_{v}$ should be open. When a vertex $v$ changes its marking/opening status, we update the constants in such a way that it becomes hard for the status to be changed back.

The classic results by Bartal [4] and Fakcharoenphol, Rao and Talwar [17] state that we can embed any $N$-point metric $(X,d)$ (with minimum non-zero distance being $1$) to a distribution $\pi$ of expanding²²2A metric $(X,d_{T})$ is expanding w.r.t $(X,d)$ if for every $u,v\in X$, we have $d_{T}(u,v)\geq d(u,v)$. HST metrics $(X,d_{T})$ with distortion $O(\log N)$: For every $u,v\in X$, we have $d_{T}(u,v)\geq d(u,v)$ and $\operatorname*{\mathbb{E}}_{u,v}[d_{T}(u,v)]\leq O(\log N)d(u,v)$. Moreover, there is an efficient randomized algorithm [7] that outputs a sample of the tree $T$ from $\pi$. Thus applying standard arguments, Theorem 3 implies Theorem 4.

In this section we specify how the input sequence is given. For the online and dynamic facility location problem, we assume the facility locations $F$, their costs $(f_{i})_{i\in F}$, and the metric $d$ restricted to $F$ are given upfront, and they take $O(|F|^{2})$ space. Whenever a client $j\in C$ arrives, it specifies its distance to every facility $i\in F$ (notice that the connection cost of an assignment $\sigma\in S^{C}$ does not depend on distances between two clients and thus they do not need to be given). Thus the whole input contains $O(n|F|)$ words.

For Theorems 3 and 4, as we do not try to optimize the constants, we do not need that a client specifies its distance to every facility. By losing a multiplicative factor of $2$ and an additive factor of $1$ in the approximation ratio, we can assume that every client $j$ is collocated with its nearest facility in $F$ (See Appendix C). Thus, we only require that when a client $j$ comes, it reports the position of its nearest facility. For Theorem 3, the HST $T$ over $F$ is given at the beginning using $O(|F|)$ words. For Theorem 4, the metric $d$ over $F$ is given at the beginning using $O(|F|^{2})$ words. Then, we use an efficient algorithm [7] to sample a HST $T$.

The local-search technique has been used to obtain the classic $(1+\sqrt{2})$-approximation offline algorithm for facility location [3]. We now give an overview of the algorithm, which will be the baseline of our online and dynamic algorithms for facility location. One can obtain a (tight) $3$-approximation for facility location without scaling facility costs. Scaling the facility costs by a factor of $\lambda:=\sqrt{2}$ when deciding whether an operation can decrease the cost, we can achieve a better approximation ratio of $\alpha_{\mathsf{FL}}:=1+\sqrt{2}$. Throughout, we fix the constants $\lambda=\sqrt{2}$ and $\alpha_{\mathsf{FL}}=1+\sqrt{2}$. For a solution $(S^{\prime},\sigma^{\prime})$ to a facility location instance, we use $\mathsf{cost}_{\lambda}(S^{\prime},\sigma^{\prime})=\lambda f(S^{\prime})+\mathsf{cc}(\sigma^{\prime})$ to denote the cost of the solution $(S^{\prime},\sigma^{\prime})$ with facility costs scaled by $\lambda=\sqrt{2}$. We call $\mathsf{cost}_{\lambda}(S^{\prime},\sigma^{\prime})$ the scaled cost of $(S^{\prime},\sigma^{\prime})$.

Given the current solution $(S,\sigma)$ for a facility location instance defined by $F,C,d$ and $(f_{i})_{i\in F}$, we can apply a local operation that changes the solution $(S,\sigma)$. A valid local operation is one of the following.

• •

  An $\mathsf{open}$ operation, in which we open some facility $i\in F$ and reconnect a subset $C^{\prime}\subseteq C$ of clients to $i$. We allow $i$ to be already in $S$, in which case we simply reconnect $C^{\prime}$ to $i$. This needs to be allowed since our $\sigma$ does not connect clients to their nearest open facilities.

• •

  A $\mathsf{close}$ operation, we close some facility $i^{\prime}\in S$ and reconnect the clients in $\sigma^{-1}(i^{\prime})$ to facilities in $S\setminus\{i^{\prime}\}$.

• •

  In a $\mathsf{swap}$ operation, we open some facility $i\notin S$ and close some facility $i^{\prime}\in S$, reconnect the clients in $\sigma^{-1}(i^{\prime})$ to facilities in $S\setminus\{i^{\prime}\}\cup\{i\}$, and possibly some other clients to $i$. We say $i$ is swapped in and $i^{\prime}$ is swapped out by the operation.

Thus, in any valid operation, we can open and/or close at most one facility. A client can be reconnected if it is currently connected to the facility that will be closed, or it will be connected to the new open facility. After we apply a local operation, $S$ and $\sigma$ will be updated accordingly so that $(S,\sigma)$ is always the current solution.

For the online algorithm with recourse model, since we need to bound the number of reconnections, we apply a local operation only if the scaled cost it decreases is large compared to the number of reconnections it makes. This motivates the following definition:

The following two theorems can be derived from the analysis for the local search algorithms for facility location. We include their proofs in Appendix A for completeness.

In particular, if we apply the theorem with $\phi=0$, then we obtain that $(S,\sigma)$ is a $(\alpha_{\mathsf{FL}}=1+\sqrt{2})$-approximation for the instance.

The following theorem will be used to analyze our randomized local search procedure.

The following lemmas will be used repeatedly in our analysis and thus we prove them separately in Appendix B.

For any $\epsilon>0$, let $\epsilon^{\prime}=\Theta(\epsilon)$ be a parameter that is sufficiently small so that the approximation ratio $\alpha_{\mathsf{FL}}+O(\epsilon^{\prime})=1+\sqrt{2}+O(\epsilon^{\prime})$ achieved by our algorithm is at most $\alpha_{\mathsf{FL}}+\epsilon$. Our algorithm for online facility location is easy to describe. Whenever the client $j_{t}$ comes at time $t$, we use a simple rule to connect $j_{t}$, as defined in the procedure $\mathsf{initial\mathchar 45\relax connect}$ in Algorithm 1: either connecting $j_{t}$ to the nearest facility in $S$, or opening and connecting $j_{t}$ to its nearest facility in $F\setminus S$, whichever incurs the smaller cost. Then we repeatedly perform $\phi$-efficient operations (Definition 6), until no such operations can be found, for $\phi=\frac{\epsilon^{\prime}\cdot\mathsf{cost}(S,\sigma)}{\alpha_{\mathsf{FL}}|C|}$. ³³3There are exponential number of possible operations, but we can check if there is a $\phi$-efficient one efficiently. $\mathsf{close}$ operations can be handled easily. To check if we can open a facility $i$, it suffices to check if $\sum_{j\in C:d(j,i)+\phi<d(j,\sigma_{j})}(d(j,\sigma_{j})-d(j,i)-\phi)>\lambda f_{i}\cdot 1_{i\notin S}$. $\mathsf{swap}$ operations are more complicated but can be handled similarly.

We can show that the algorithm gives an $(\alpha_{\mathsf{FL}}+\epsilon)$-approximation with amortized recourse $O(\log D\log n)$; recall that $D$ is the aspect ratio of the metric. To remove the dependence on $D$, we divide the algorithm into stages, and freeze the connections of clients that arrived in early stages. The final algorithm is described in Algorithm 3, and Algorithm 2 gives one stage of the algorithm.

In Algorithm 2, we do as described above, with two modifications. First, we are given an initial set $C$ of clients and a solution $(S,\sigma)$ for $C$ which is $O(1)$-approximate. Second, the stage will terminate if the cost of our solution increases by a factor of more than $1/\epsilon^{\prime}$. The main algorithm (Algorithm 3) is broken into many stages. Since we shall focus on one stage of the algorithm for most part of our analysis, we simply redefine the time so that every stage starts with time 1. The improved recourse comes from the freezing operation: at the end of each stage, we permanently open one copy of each facility in $S^{\circ}$, and permanently connect clients in $C^{\circ}$ to copies of $S^{\circ}$ according to $\sigma^{\circ}$, where $C^{\circ}$ and $(S^{\circ},\sigma^{\circ})$ are the client set and solution at the beginning of the stage. Notice that we assume the original facilities in $S^{\circ}$ will still participate in the algorithm in the future; that is, they are subject to opening and closing. Thus each facility may be opened multiple times during the algorithm and we take the facility costs of all copies into consideration. This assumption is only for the sake of analysis; the actual algorithm only needs to open one copy and the costs can only be smaller compared to the described algorithm.

From now on, we focus on one stage of the algorithm and assume that the solution given at the beginning of each stage is $O(1)$-approximate. In the end we shall account for the loss due to the freezing of clients and facilities. Within a stage, the approximation ratio follows directly from Theorem 7: Focus on the moment after the while loop at time step $t$ in Algorithm 2. Since there are no $\frac{\epsilon^{\prime}\cdot\mathsf{cost}(S,\sigma)}{\alpha_{\mathsf{FL}}|C|}$-efficient local operations on $(S,\sigma)$, we have by the theorem that $\mathsf{cost}(S,\sigma)\leq\alpha_{\mathsf{FL}}\left(\mathsf{opt}+|C|\cdot\frac{\epsilon^{\prime}\cdot\mathsf{cost}(S,\sigma)}{\alpha_{\mathsf{FL}}|C|}\right)=\alpha_{\mathsf{FL}}\mathsf{opt}+\epsilon^{\prime}\cdot\mathsf{cost}(S,\sigma)$, where $\mathsf{opt}$ is the cost of the optimum solution for $C$. Thus, at the end of each time, we have $\mathsf{cost}(S,\sigma)\leq\frac{\alpha_{\mathsf{FL}}}{1-\epsilon^{\prime}}\cdot\mathsf{opt}$.

We then bound the amortized recourse in a stage; we assume that $\mathsf{cost}(S,\sigma)>0$ at the beginning of the stage since otherwise there will be no recourse involved in the stage (since we terminate the stage when the cost becomes non-zero). We use $T$ to denote the last time of the stage. For every time $t$, let $C_{t}$ be the set $C$ at the end of time $t$, and $\mathsf{opt}_{t}$ to be the cost of the optimum solution for the set $C_{t}$. For every $t\in[T]$, we define $\Delta_{t}$ to be the value of $\mathsf{cost}(S,\sigma)$ after Step 5 at time step $t$ in Algorithm 2, minus that before Step 5. We can think of this as the cost increase due to the arrival of $j_{t}$.

The key lemma we can prove is the following:

With Lemma 11, we can now bound the amortized recourse of one stage. In time $t$, $\mathsf{cost}(S,\delta)$ first increases by $\Delta_{t}$ in Step 5. Then after that, it decreases by at least $\frac{\epsilon^{\prime}\mathsf{cost}(S,\sigma)}{\alpha_{\mathsf{FL}}|C|}\geq\frac{\epsilon^{\prime}\mathsf{opt}_{t}}{\alpha_{\mathsf{FL}}|C|}\geq\frac{\epsilon^{\prime}\mathsf{opt}_{t}}{\alpha_{\mathsf{FL}}|C_{T}|}$ for every reconnection we made. Let $\Phi_{T^{\prime}}=\sum_{t=1}^{T^{\prime}}\Delta_{t}$; Lemma 11 says $\Phi_{t}\leq\alpha\mathsf{opt}_{t}$ for some $\alpha=O(\log T)$ and every $t\in[T]$. Noticing that $(\mathsf{opt}_{t})_{t\in T}$ is a non-decreasing sequence, the total number of reconnections is at most

Notice that $\mathsf{init}\leq O(1)\mathsf{opt}_{0}\leq O(1)\mathsf{opt}_{1}$. Applying Lemma 9 with $T$ replaced by $T-1$, $b_{t}=\Delta_{t},B_{t}=\Phi_{t}$ and $a_{t}=\mathsf{opt}_{t}$ for every $t$, we have that $\sum_{t=1}^{T-1}\frac{\Delta_{t}}{\mathsf{opt}_{t}}\leq\alpha\left(\ln\frac{\mathsf{opt}_{T-1}}{\mathsf{opt}_{1}}+1\right)=O\left(\log T\log\frac{1}{\epsilon^{\prime}}\right)$, since we have $\mathsf{opt}_{T-1}\leq O(1/\epsilon^{\prime})\cdot\mathsf{opt}_{1}$. Notice that $\Delta_{T}\leq\mathsf{opt}_{T}$ since $\mathsf{opt}_{T}\geq\min_{i\in F}(f_{i}+d(i,j_{T}))\geq\Delta_{T}$. So, the total number of reconnections is at most $O\left(\frac{\log T}{\epsilon^{\prime}}\log\frac{1}{\epsilon^{\prime}}\right)\cdot|C_{T}|$. The amortized recourse per client is $O\left(\frac{\log T}{\epsilon^{\prime}}\log\frac{1}{\epsilon^{\prime}}\right)\leq O\left(\frac{\log n}{\epsilon^{\prime}}\log\frac{1}{\epsilon^{\prime}}\right)$, where in the amortization, we only considered clients involved in the stage. Recall that $n$ is the total number of clients arrived.

As each client appears in at most 2 stages, the overall amortized recourse is $O\left(\frac{\log n}{\epsilon^{\prime}}\log\frac{1}{\epsilon^{\prime}}\right)$. Finally we consider the loss in the approximation ratio due to freezing of clients. Suppose we are in the $p$-th stage. Then the clients arrived at and before $(p-2)$-th stage has been frozen and removed. Let $\overline{\mathsf{opt}}$ be the cost of the optimum solution for all clients arrived at or before $(p-1)$-th stage. Then the frozen facilities and clients have cost at most $\overline{\mathsf{opt}}\cdot O\left(\epsilon^{\prime}+\epsilon^{\prime 2}+\epsilon^{\prime 2}+\cdots\right)=O(\epsilon^{\prime})\overline{\mathsf{opt}}$. In any time in the $p$-th stage, the optimum solution taking all arrived clients into consideration has cost $\overline{\mathsf{opt}}^{\prime}\geq\overline{\mathsf{opt}}$, and our solution has cost at most $(\alpha_{\mathsf{FL}}+O(\epsilon^{\prime}))\overline{\mathsf{opt}}^{\prime}$ without considering the frozen clients and facilities. Thus, our solution still has approximation ratio $\frac{(\alpha_{\mathsf{FL}}+O(\epsilon^{\prime}))\overline{\mathsf{opt}}^{\prime}+O(\epsilon^{\prime})\overline{\mathsf{opt}}}{\overline{\mathsf{opt}}^{\prime}}=\alpha_{\mathsf{FL}}+O(\epsilon^{\prime})$ when taking the frozen clients into consideration.

Unlike the online algorithm for facility location in Section 3, in the dynamic algorithm, we guarantee that the clients are connected to their nearest open facilities. That is, we always have $\sigma_{j}=\arg\min_{i\in S}d(j,i)$; we still keep $\sigma$ for convenience of description. We maintain $|C|$ min-heaps, one for each client $j\in C$: The min-heap for $j$ will contain the facilities in $S\setminus\{\sigma_{j}\}$, with priority value of $i$ being $d(j,i)$. This allows us to efficiently retrieve the second nearest open facility to each $j$: This is the facility at the top of the heap for $j$ and we use the procedure $\mathsf{heap\mathchar 45\relax top}(j)$ to return it.

We define four simple procedures $\mathsf{\Delta\mathchar 45\relax open},\mathsf{try\mathchar 45\relax open},\mathsf{\Delta\mathchar 45\relax swap\mathchar 45\relax in}$ and $\mathsf{\Delta\mathchar 45\relax close}$ that are described in Algorithms 4, 5, 6 and 7 respectively. Recall that we use the scaled cost for the local search algorithm; so we are working on the scaled cost function in all these procedures. $\mathsf{\Delta\mathchar 45\relax open}(i)$ for any $i\notin S$ returns $\Delta$, the increment of the scaled cost that will be incurred by opening $i$. (For it to be useful, $\Delta$ should be negative, in which case $|\Delta|$ indicates the cost decrement of opening $i$). This is just one line procedure as in Algorithm 4; $\mathsf{try\mathchar 45\relax open}$ will open $i$ if it can reduce the scaled cost. $\mathsf{\Delta\mathchar 45\relax swap\mathchar 45\relax in}(i)$ for some $i\notin S$ returns a pair $(\Delta,i^{\prime})$, where $\Delta$ is the smallest scaled cost increment we can achieve by opening $i$ and closing some facility $i^{\prime}\in S$, and $i^{\prime}$ gives the facility achieving the smallest value. (Again, for $\Delta$ to be useful, it should be negative, in which case $i^{\prime}$ is the facility that gives the maximum scaled cost decrement $|\Delta|$.) Similarly, $\mathsf{\Delta\mathchar 45\relax close}$ returns a pair $(\Delta,i^{\prime})$, which tells us the maximum scaled cost decrement we can achieve by closing one facility and which facility can achieve the decrement. Notice that in all the procedures, the facility we shall open or swap in is given as a parameter, while the facility we shall close is chosen and returned by the procedures.

With the heaps, the procedures $\mathsf{\Delta\mathchar 45\relax open},\mathsf{\Delta\mathchar 45\relax swap\mathchar 45\relax in}$ and $\mathsf{\Delta\mathchar 45\relax close}$ can run in $O(|C|)$ time. We only analyze $\mathsf{\Delta\mathchar 45\relax swap\mathchar 45\relax in}(i)$ as the other two are easier. First, we define $C^{\prime}$ to be the set of clients $j$ with $d(j,i)<d(j,\sigma_{j})$; these are the clients that will surely be reconnected to $i$ once $i$ is swapped in. Let $\Psi=\lambda f_{i}-\sum_{j\in C^{\prime}}(d(j,\sigma_{j})-d(j,i))$ be the net scaled cost increase by opening $i$ and connecting $C^{\prime}$ to $i$. The computation of $C^{\prime}$ and $\Psi$ in Step 1 takes $O(|C|)$ time. If additionally we close some $i^{\prime}\in S$, we need to reconnect each client in $\sigma^{-1}(i^{\prime})\setminus C^{\prime}$ to either $i$, or the top element in the heap for $j$, whichever is closer to $j$. Steps 2 and 3 compute and return the best scaled cost increment and the best $i^{\prime}$. Since $\sum_{i^{\prime}\in S}|\sigma^{-1}(i^{\prime})|=|C|$, the running time of the step can be bounded by $O(|C|)$.

The running time for $\mathsf{try\mathchar 45\relax open}$, swapping two facilities and closing a facility (which are not defined explicitly as procedures, but used in Algorithms 8) can be bounded by $O(|C|\log|F|)$. The running times come from updating the heap structures: For each of the $|C|$ heaps, we need to delete and/or add at most $2$ elements; each operation takes time $O(\log|F|)$.

With the support of the heaps, we can design a fast algorithm to implement randomized local search. $\mathsf{sampled\mathchar 45\relax local\mathchar 45\relax search}$ in Algorithm 8 gives one iteration of the local search. We first decide which operation we shall perform randomly. With probability $1/3$, we perform the $\mathsf{close}$ operation that will reduce the scaled cost the most (if it exists). With the remaining probability $2/3$, we perform either an $\mathsf{open}$ or a $\mathsf{swap}$ operation. To reduce the running time, we randomly choose a facility $i\in F\setminus S$ and find the best operation that opens or swaps in $i$, and perform the operation if it reduces the cost. One iteration of $\mathsf{sampled\mathchar 45\relax local\mathchar 45\relax search}$ calls the procedures in Algorithms 4 to 7 at most once and performs at most one operation, and thus has running time $O(|C|\log|F|)$.

In the procedure $\mathsf{FL\mathchar 45\relax iterate}(M)$ described in Algorithm 9, we run the $\mathsf{sampled\mathchar 45\relax local\mathchar 45\relax search}$ $M$ times. It returns the best solution obtained in these iterations, according to the original (non-scaled) cost, which is not necessarily the solution given in the last iteration. So we have