SlowGuess/NeurIPS_2023_OCR huggingface dataset:
If in the second condition of PRSC we replace  $\mathrm{OPT}\left(\overline{Q^{\prime}}\right)$  by  $\mathrm{OPT}_S\left(\overline{Q^{\prime}}\right)$ , we say that ALG is  $\beta$ -PRSC using OA w.r.t.  $S$ .
Proposition C.2. A  $\beta$ -PRSC algorithm using OA (w.r.t.  $S$ ) is also Lagrangian  $\beta$ -subset-competitive using OA (w.r.t.  $S$ ).
Proof. We prove this for a general solution, restricting to  $S$  is identical. Consider prize-collecting input  $Q$ , and any subset  $Q' \subseteq Q$ . Let SOL be the optimal solution for  $Q'$ , which pays penalties for  $Q'_p$  and serves  $Q'_b = Q' \backslash Q'_p$  optimally. Then it holds that
$$
\begin{array}{l} \mathbb {E} \bigl [ \mathrm {O A} (Q ^ {\prime} | Q) \bigr ] = \mathbb {E} \Bigl [ \mathrm {O A} (Q _ {b} ^ {\prime} | Q) \Bigr ] + \mathbb {E} \Bigl [ \mathrm {O A} (Q _ {p} ^ {\prime} | Q) \Bigr ] \\ \leq \beta \cdot \operatorname {O P T} \left(\overline {{Q _ {b} ^ {\prime}}}\right) + O (1) \cdot \sum_ {(q, \pi) \in Q _ {p} ^ {\prime}} \pi \\ = \beta \cdot \operatorname {S O L} ^ {b} \left(Q ^ {\prime}\right) + O (1) \cdot \operatorname {S O L} ^ {p} \left(Q ^ {\prime}\right) \\ \end{array}
$$
where the inequality uses both properties of PRSC.
D Proofs of Theorems 1.1, 1.2 and 1.3
We establish these theorems in three steps. First, we combine various constituent prize-collecting algorithms using Theorem 2.5 and explicitly state the guarantees for the resulting algorithms. Then, we use these guarantees to derive the discrete-smoothness property for the individual problems with respect to the size of the prediction (i.e., Equation (6)). Finally, we use Theorem 2.1 to make the competitive ratio depend on  $|S\backslash \mathrm{OPT}|$  rather than on  $|S|$ .
Before proceeding further, we need to precisely define the intersection/difference of a solution with a prediction to make Theorem 1.1, Theorem 1.2, and Theorem 1.3 completely formal.
Definition D.1 (restriction of solution with prediction). Consider an online covering problem with items  $\mathcal{E}$ , let  $S \subseteq \mathcal{E}$  be some prediction. For every solution  $A$  which buys some items from  $\mathcal{E}$ :
- Define  $A|_{S}$  to be the solution which only buys items from  $S$ , to the same amount as  $A$ .
- Define  $A|_{S}$  to be the solution which only buys items outside  $S$ , to the same amount as  $A$ .
Facility Location with Predictions. In order to describe facility location as a covering problem, we must describe the set of items. Here, the set of items comprises an opening item  $b_{\nu}$  for each facility and a connection item  $c_{\nu,q}$  for each (request, facility) pair. When we informally write that  $S$  is a set of possible facilities, this can be formalized to the set of items  $b_{\nu}$  for  $\nu \in S$ , plus the connection items  $c_{\nu,q}$  for all  $q$  in the input and  $\nu \in S$ .
Due to Theorem 3.1 and Theorem B.1, we have that both Algorithm 2 and Algorithm 4 can serve as constituent algorithms in our framework. Combining both algorithms using Theorem 2.5 thus implies the following theorem.
Theorem D.2. For facility location with predictions, there exists a randomized algorithm ALG such that for every input  $Q$ , and for every partition of  $Q$  into  $Q_{1}, Q_{2}$ , we have
$$
\mathbb {E} [ \operatorname {A L G} (Q) ] \leq O (\log | S \backslash \operatorname {O P T} |) \cdot \operatorname {O P T} _ {S} (Q _ {1}) + O (\log | Q _ {2} |) \cdot \operatorname {O P T} (Q _ {2}).
$$
We obtain an additional result, which is useful for small metric spaces, from combining two instances of Algorithm 2, one for the entire metric space  $X$  and one for the predictions  $S$ .
Theorem D.3. For facility location with predictions, there exists a randomized algorithm ALG such that for every input  $Q$ , and for every partition of  $Q$  into  $Q_{1}, Q_{2}$ , we have
$$
\mathbb {E} [ \mathrm {A L G} (Q) ] \leq O (\log | S \backslash \mathrm {O P T} |) \cdot \mathrm {O P T} _ {S} (Q _ {1}) + O (\log | X |) \cdot \mathrm {O P T} (Q _ {2}).
$$
Proof of Theorem 1.1. Consider a solution OPT to facility location on a set of requests  $Q$ . Partition  $Q$  into  $Q_{1}, Q_{2}$  such that  $Q_{1}$  contains all requests from  $Q$  that are connected to a facility in OPT|S (and  $Q_{2}$  is complementary). Using the algorithm ALG from Theorem D.2, we have
$$
\operatorname {A L G} (Q) \leq O (\log | S |) \cdot \operatorname {O P T} _ {S} \left(Q _ {1}\right) + O \left(\log | Q _ {2} |\right) \cdot \operatorname {O P T} \left(Q _ {2}\right). \tag {11}
$$
Now note that  $\mathrm{OPT}|_{S}$  is a solution to  $Q_{1}$  that only uses facility and connection items from  $S$ , and thus  $\mathrm{OPT}_S(Q_1)\leq \mathrm{OPT}|_S$ . Moreover,  $\mathrm{OPT}|_{\overline{S}}$  is a solution to  $Q_{2}$ , and thus  $\mathrm{OPT}_S(Q_2)\leq \mathrm{OPT}|_{\overline{S}}$ . Plugging into Equation (12), and noting that  $|Q_{2}|\leq |Q|$ , we get
$$
\operatorname {A L G} (Q) \leq O (\log | S |) \cdot \operatorname {O P T} | _ {S} + O (\log | Q |) \cdot \operatorname {O P T} | _ {\bar {S}}.
$$
We now plug the above equation into Theorem 2.1, thus replacing the dependence on  $|S|$  with dependence on  $|S\backslash \mathrm{OPT}|$ .
Proof of Theorem 1.2. Identical to the proof of Theorem 1.1, but using Theorem D.3.
Set Cover with Predictions. Theorem 4.1 implies that Algorithm 3 is Lagrangian subset-competitive. In addition, it is easy to see that Algorithm 3 is monotone, as defined in Definition 2.2. Thus, the algorithm can serve as a constituent algorithm in our framework. From combining two instances of Algorithm 3, Theorem 2.5 thus implies the following theorem.
Theorem D.4. For fractional set cover with predictions, with universe  $(E,U)$  and a prediction  $S\subseteq U$  there exists a deterministic algorithm ALG such that for every input  $Q$  ,and for every partition of  $Q$  into  $Q_{1},Q_{2}$  ,we have
$$
\operatorname {A L G} (Q) \leq O (\log | S |) \cdot \operatorname {O P T} _ {S} \left(Q _ {1}\right) + O \left(\log | U |\right) \cdot \operatorname {O P T} \left(Q _ {2}\right).
$$
Using standard rounding techniques (see [3, 15]) for online set cover, we can round the fractional solution online at a loss of  $O(\log |Q|)$ . In addition, we can then apply Theorem 2.1 to replace  $|S|$  with  $|S\backslash \mathrm{OPT}|$ . Thus, Theorem D.4 yields the following corollary.
Corollary D.5. For (integral) set cover with predictions, with universe  $(E,U)$  and a prediction  $S\subseteq U$ , there exists a randomized algorithm ALG such that for every input  $Q$ , and for every partition of  $Q$  into  $Q_{1},Q_{2}$ , we have
$$
\mathbb {E} \left[ \operatorname {A L G} (Q) \right] \leq O \left(\log | Q | \log | S \backslash \mathrm {O P T} |\right) \cdot \operatorname {O P T} _ {S} (Q _ {1}) + O \left(\log | Q | \log | U |\right) \cdot \operatorname {O P T} (Q _ {2}).
$$
Proof of Theorem 1.3. Consider a solution OPT to set cover on a set of requests  $Q$ . Partition  $Q$  into  $Q_{1}, Q_{2}$  such that  $Q_{1}$  contains all requests from  $Q$  that belong to a set in  $\mathrm{OPT}|_{S}$  (and  $Q_{2}$  is complementary). Using the randomized algorithm ALG from Corollary D.5, we have
$$
\operatorname {A L G} (Q) \leq O (\log | Q | \log | S |) \cdot \operatorname {O P T} _ {S} \left(Q _ {1}\right) + O (\log | Q | \log | U |) \cdot \operatorname {O P T} \left(Q _ {2}\right). \tag {12}
$$
Now note that  $\mathrm{OPT}|_{S}$  is a solution to  $Q_{1}$  that only uses sets from  $S$ , and thus  $\mathrm{OPT}_S(Q_1)\leq \mathrm{OPT}|_S$ . Moreover,  $\mathrm{OPT}|_{\overline{S}}$  is a solution to  $Q_{2}$ , and thus  $\mathrm{OPT}_S(Q_2)\leq \mathrm{OPT}|_{\overline{S}}$ . Plugging into Equation (12), we get
$$
\operatorname {A L G} (Q) \leq O (\log | Q | \log | S |) \cdot \operatorname {O P T} | _ {S} + O (\log | Q | \log | U |) \cdot \operatorname {O P T} | _ {\bar {S}}.
$$
E Proof of Theorem 2.1: Reduction from Equation (6) to Equation (1)
In this section, we give the proof of Theorem 2.1 whose goal is to give a reduction from Equation (6) to Equation (1). This replaces  $s$  in the bound of Equation (6) with the term  $s_{\delta}$ , where  $s_{\delta} \coloneqq |S\backslash \mathrm{OPT}|$ , in order to obtain Equation (1).
Proof of Theorem 2.1. Assume, without loss of generality, that the cheapest item in  $\mathcal{E}$  costs 1. Consider the following construction of the algorithm ALG using the algorithm  $\mathrm{ALG}'$ :
1 Initialize  $i\gets 0,S^{\prime}\gets S,B\gets 0$  , and define the item cost function  $c^\prime \leftarrow c$    
2 Let  $A$  be an instance of ALG' with prediction set  $S^{\prime}$  , and cost function  $c^\prime$    
3 for incoming request q do   
4 while True do   
5 Simulate sending  $q$  to  $A$  , and let  $c$  be the resulting cost.   
6 if  $B + c <   2^{i}$  then break   
7 Spend  $2^{i}$  budget in buying the cheapest items in  $S^{\prime}$  , let the bought subset of items be  $T$    
8 Set  $S^{\prime}\gets S^{\prime}\backslash T,B\gets 0,i\gets i + 1$    
9 For every  $e\in T$  , set  $c^{\prime}(e)\gets 0$    
10 Reset  $A$  to be a new instance of ALG', given  $S^{\prime}$  as prediction, and using the (modified) cost function  $c^{\prime}$    
11 Send  $q$  to  $A$  , and set  $B\gets B + c$
For integer  $\ell$ , define phase  $\ell$  to be the subsequence of requests in which variable  $i$  takes value  $\ell$ . The cost of the algorithm can be charged to a constant times  $2^{j}$ , where  $j$  is the penultimate value of  $i$  in the algorithm. If  $2^{j-1} < \mathrm{OPT}$ , then the cost of the algorithm is at most  $O(1) \cdot \mathrm{OPT}$  and we are done. Henceforth, suppose  $\mathrm{OPT} \leq 2^{j-1}$ . Define  $S_{j}', A_{j}, c_{j}'$  to be the values of the variables  $S', A$  and  $c'$  during phase  $j$ . When considering the cost of a solution relative to a cost function, we place that cost function as superscript (e.g.,  $\mathrm{OPT}^{c_{j}'}$ ). Before the beginning of phase  $j$ , the algorithm spent at least OPT budget on buying the cheapest items in the (remaining) prediction; it thus holds that  $\left|S_{j}'\right| \leq |S \backslash \mathrm{OPT}|$ . Let  $q_{1}, \dots, q_{k}$  be the requests of phase  $j$ ; moreover, let  $q_{k+1}$  be the request upon which the variable  $i$  was incremented to  $j+1$ . From the definition of  $q_{k+1}$ , it holds that the cost of the instance of  $A$  in phase  $j$  on  $(q_{1}, \dots, q_{k}, q_{k+1})$  is at least  $2^{j}$ ; thus, the total cost of the algorithm can be charged to this cost, which we denote by  $\alpha$ . But, through Equation (6), and from the fact that
OPT is a solution which serves  $(q_{1},\dots ,q_{k + 1})$  , we have
$$
\begin{array}{l} \alpha \leq O \left(f \left(\left| S ^ {\backslash} \mathrm {O P T} \right|\right)\right) \cdot \mathrm {O P T} ^ {c _ {j} ^ {\prime}} \mid_ {S _ {j} ^ {\prime}} + O (g) \cdot \mathrm {O P T} ^ {c _ {j} ^ {\prime}} \mid_ {\overline {{S _ {j} ^ {\prime}}}} \\ \leq O \left(f \left(\left| S \backslash \mathrm {O P T} \right|\right)\right) \cdot \mathrm {O P T} | _ {S _ {j} ^ {\prime}} + O (g) \cdot \left(\mathrm {O P T} ^ {c _ {j} ^ {\prime}} | _ {\bar {S}} + \mathrm {O P T} ^ {c _ {j} ^ {\prime}} | _ {S \backslash S _ {j} ^ {\prime}}\right) \\ \leq O (f (| S \backslash \mathrm {O P T} |)) \cdot \mathrm {O P T} | _ {S} + O (g) \cdot \mathrm {O P T} | _ {\bar {S}} \\ \end{array}
$$
F Proof of Lemma 4.4
Proof of Lemma 4.4. First, note that  $\mathrm{ALG}(q,\pi)\leq 3y_q$ , where  $y_{q}$  is the final value of the variable of that name: Proposition 4.5 implies that the buying cost is at most  $2y_{q}$ , while a penalty of  $\pi$  is paid only if  $\pi \leq y_{q}$ . We show that  $\sum_{(q,\pi)\in Q^{\prime}}y_{q}\leq O(\log m)\cdot \mathrm{OPT}(\overline{Q^{\prime}})$ ; since  $\mathrm{ALG}(q,\pi)\leq 3\cdot y_q$ , this would complete the proof of the lemma. Consider the (standard) primal and dual LPs for fractional set cover of  $Q^{\prime}$  without penalties (i.e. solving  $\overline{Q^{\prime}}$ ). The primal LP is given by:
$$
\min  \sum_ {s \in U} x _ {s} \cdot c (s) \text {s u c h t h a t} \forall q \in Q ^ {\prime}: \sum_ {s | q \in s} x _ {s} \geq 1 \text {a n d} \forall s \in U: x _ {s} \geq 0.
$$
and the dual LP is given by:
$$
\max  \sum_ {q \in Q ^ {\prime}} y _ {q} \text {s u c h t h a t} \forall s \in U: \sum_ {q | q \in s} y _ {q} \leq c (s) \text {a n d} \forall q \in Q ^ {\prime}: y _ {q} \geq 0.
$$
We claim that the dual solution  $\{y_q\}_{q \in Q'}$  violates dual constraints by at most  $O(\log m)$ ; thus, scaling it down by that factor yields a feasible dual solution, and a lower bound to  $\mathrm{OPT}\left(\overline{Q'}\right)$ .
Consider the dual constraint corresponding to the set  $s$ ; we want to bound the term  $\sum_{q \in Q'|q \in s} y_q$ . Through induction on  $k$ , we can prove that once  $\sum_{q \in Q'|q \in s} y_q = k$  for some integer  $k$ , it holds that  $x_s \geq \frac{1}{m} \left( \left( 1 + \frac{1}{c_s} \right)^k - 1 \right)$ . Thus, once  $k = \Theta(c_s \log m)$  we have  $x_s \geq 1$ , and  $\sum_{q \in s} y_q$  would increase no more. This implies that scaling down  $\{y_q\}_{q \in Q'}$  by  $\Theta(\log m)$  yields a feasible dual solution, which lower bounds OPT  $(\overline{Q'})$ , and completes the proof.
It remains to prove the inductive claim. For the base case where  $k = 0$ , the claim holds trivially. Now, assume that the claim holds for  $k - 1$ , and consider point in which  $\sum_{q \in Q' | q \in s} y_q$  is incremented from  $k - 1$  to  $k$ ; let  $x, x'$  be the old and new amounts by which  $s$  is held in the algorithm. We have
$$
x ^ {\prime} = x \cdot \left(1 + \frac {1}{c (s)}\right) + \frac {1}{U (q) c (s)} \geq \frac {1}{m} \left(\left(1 + \frac {1}{c (s)}\right) ^ {k} - 1 - \frac {1}{c (s)}\right) + \frac {1}{m c (s)} \geq \frac {1}{m} \left(\left(1 + \frac {1}{c (s)}\right) ^ {k} - 1\right) \tag {13}
$$
where the inequality uses the inductive hypothesis as well as the fact that  $|U(q)| \leq m$ .