Optimizing Health Coverage in Ethiopia: A Learning-augmented Approach and Persistent Proportionality Under an Online Budget

TL;DR

This paper introduces HARP, an optimization tool for Ethiopia's health system expansion that maximizes coverage under budget constraints, incorporating learning-augmented and greedy algorithms for improved planning.

Contribution

The paper presents a novel decision-support framework and algorithms for sequential health facility planning under budget uncertainty, with proven empirical effectiveness in Ethiopia.

Findings

Algorithms outperform expert recommendations in coverage maximization.

The approach ensures proportionality targets are met at each planning step.

Empirical tests show significant improvements across multiple regions.

Abstract

As part of nationwide efforts aligned with the United Nations' Sustainable Development Goal 3 on Universal Health Coverage, Ethiopia's Ministry of Health is strengthening health posts to expand access to essential healthcare services. However, only a fraction of this health system strengthening effort can be implemented each year due to limited budgets and other competing priorities, thus the need for an optimization framework to guide prioritization across the regions of Ethiopia. In this paper, we develop a tool, Health Access Resource Planner (HARP), based on a principled decision-support optimization framework for sequential facility planning that aims to maximize population coverage under budget uncertainty while satisfying region-specific proportionality targets at every time step. We then propose two algorithms: (i) a learning-augmented approach that improves upon expert recommendations at any single-step; and (ii) a greedy algorithm for multi-step planning, both with strong worst-case approximation estimation. In collaboration with the Ethiopian Public Health Institute and Ministry of Health, we demonstrated the empirical efficacy of our method on three regions across various planning scenarios.

Figures

Tables

Table 1. Table 1 : Mapping the Ethiopian context to MOPGP .

Parameters for a region	Component in MOPGP
5-year planning horizon	$h=5$ time steps
Districts within a region	$r$ types
Annual building budget	Online budgets $b^{\left(t\right)}$
Location, at year $t$	The set ${𝐕}^{\left(t\right)}$
Posts built for year $t$	${𝐒}^{\left(t\right)}\subseteq {𝐕}^{\left(t\right)}$ with $\left|{𝐒}^{\left(t\right)}\right|\le b^{\left(t\right)}$
2-hour accessibility goal	Determines $c\in \text{covered}\left({𝐒}^{\left(1:t\right)}\right)$
Forecast for cell $c$ in year $t$	Population weight $w_c^{\left(t\right)}$
Coverage objective	$f$ defined in Eq. 1
District prioritization	Proportions $p_1,\dots ,p_r$
Persistent proportionality	Type feasibility in Definition 1

Equations

$$\mathbf{Q}({\mathbf{S}}^{(1:t)})=\{q\in [r]:{\alpha }_q({\mathbf{S}}^{(1:t)})=\beta (|{\mathbf{S}}^{(1:t)}|)\text{ and }{\mathbf{V}}^{(t)}\cap {\mathbf{T}}_q\ne \varnothing \}$$

$$f(\mathbf{S})=\sum _{t=1}^h\sum _{c\in \text{textsc}covered({\mathbf{S}}^{(1:t)})}{w}_c^{(t)}$$

$$f(\mathbf{U})\ge \underset{i\in \{0,1,\dots ,b\}}{\max }f({\mathbf{A}}_i)+\frac{1-\frac{1}{e}}{\lceil \frac{|\text{textsc}OPT\setminus {\mathbf{A}}_i|}{b-i}\rceil }\cdot g(\text{textsc}OPT,{\mathbf{A}}_i)$$

$$x_q^{(t)}=\#(q,\mathbf{s}(b^{(1:t)},b^{(1:t)}))-\#(q,\mathbf{s}(b^{(1:t-1)},b^{(1:t-1)}))$$

$$f({\mathbf{S}}^{(1)}\uplus \dots \uplus {\mathbf{S}}^{(t)})\ge \frac{1}{2}\cdot f({\text{textsc}OPT}_{\sigma }(b^{(1)},\dots ,b^{(t)}))$$

$$\frac{f_{\text{region}}(\text{textsc}MP(\text{DP}0)\cup {\mathbf{S}}_{\text{existing}})-f_{\text{region}}({\mathbf{S}}_{\text{existing}})}{f_{\text{region}}(\text{textsc}MP(\text{DP}i)\cup {\mathbf{S}}_{\text{existing}})-f_{\text{region}}({\mathbf{S}}_{\text{existing}})},$$

$$f(\mathbf{S})=\sum _{t=1}^h\sum _{c\in \text{textsc}covered({\mathbf{S}}^{(1:t)})}{w}_c^{(t)}$$

$$f(\mathbf{A}\cup \{e\})-f(\mathbf{A})$$

$$\ge \sum _{\tau \le t\le h}\left(\sum _{c\in \text{textsc}covered(\{e\}\setminus {\mathbf{B}}^{(1:t)})}{w}_c^{(t)}\right)$$

$$\ge f(\mathbf{B}\cup \{e\})-f(\mathbf{B})$$

$$\frac{1}{\alpha }\cdot \mathrm{obj}({\mathbf{S}}_{\text{textsc}ALG}(f),\text{textsc}MOPGP(f))$$

$$=\underset{\begin{array}{c}\mathbf{S}\subseteq \mathbf{V}\\ \text{S is type-feasible}\end{array}}{\max }\mathrm{obj}(\mathbf{S},\text{textsc}MOPGP(f))$$

$$\alpha \cdot \mathrm{obj}({\mathbf{S}}_{\text{textsc}OPT}(f),\text{textsc}MOPGP(f))$$

$$\le \alpha \cdot \mathrm{obj}\left({\mathbf{S}}_{\text{textsc}OPT}\left(\frac{\tilde{f}}{1-\epsilon }\right),\text{textsc}MOPGP\left(\frac{\tilde{f}}{1-\epsilon }\right)\right)$$

$$\le \mathrm{obj}\left({\mathbf{S}}_{\text{textsc}ALG}\left(\frac{\tilde{f}}{1-\epsilon }\right),\text{textsc}MOPGP\left(\frac{\tilde{f}}{1-\epsilon }\right)\right)$$

$$\le \mathrm{obj}\left({\mathbf{S}}_{\text{textsc}ALG}\left(\frac{\tilde{f}}{1-\epsilon }\right),\text{textsc}MOPGP\left(\frac{1+\epsilon }{1-\epsilon }\cdot f\right)\right)$$

$$=\mathrm{obj}\left({\mathbf{S}}_{\text{textsc}ALG}(\tilde{f}),\text{textsc}MOPGP\left(\frac{1+\epsilon }{1-\epsilon }\cdot f\right)\right)$$

$$=\frac{1+\epsilon }{1-\epsilon }\cdot \mathrm{obj}\left({\mathbf{S}}_{\text{textsc}ALG}(\tilde{f}),\text{textsc}MOPGP\left(\frac{1+\epsilon }{1-\epsilon }\cdot f\right)\right)$$

$$f(\mathbf{B}\cup \mathbf{S})=\underset{\begin{array}{c}\mathbf{Z}\subseteq \text{textsc}OPT\setminus \mathbf{S}\\ |\mathbf{Z}|=k-|\mathbf{S}|\end{array}}{\max }f(\mathbf{Z}\cup \mathbf{S})$$

$$f(\mathbf{C}\cup \mathbf{S})-f(\mathbf{S})\le \lceil \frac{|\mathbf{C}|}{|\mathbf{B}|}\rceil \cdot \left(f(\mathbf{B}\cup \mathbf{S})-f(\mathbf{S})\right)$$

$$f(\mathbf{C}\cup \mathbf{S})-f(\mathbf{S})$$

$$\le \sum _{i=1}^{\ell }(f(\mathbf{B}\cup \mathbf{S})-f(\mathbf{S}))$$

$$=\lceil \frac{|\mathbf{C}|}{|\mathbf{B}|}\rceil \cdot (f(\mathbf{B}\cup \mathbf{S})-f(\mathbf{S}))$$

$$f({\mathbf{B}}^{\prime }\cup {\mathbf{A}}^{\prime })\ge f({\mathbf{A}}^{\prime })+\frac{1-\frac{1}{e}}{\lceil \frac{|\text{textsc}OPT\setminus {\mathbf{A}}^{\prime }|}{k-|{\mathbf{A}}^{\prime }|}\rceil }\cdot (f(\text{textsc}OPT\cup {\mathbf{A}}^{\prime })-f({\mathbf{A}}^{\prime }))$$

$$f(\mathbf{B}\cup {\mathbf{A}}^{\prime })=\underset{\begin{array}{c}\mathbf{Z}\subseteq \text{textsc}OPT\setminus {\mathbf{A}}^{\prime }\\ |\mathbf{Z}|=k-|{\mathbf{A}}^{\prime }|\end{array}}{\max }f(\mathbf{Z}\cup {\mathbf{A}}^{\prime })$$

$$f(\text{textsc}OPT\cup {\mathbf{A}}^{\prime })-f({\mathbf{A}}^{\prime })$$

$$\le \lceil \frac{|\text{textsc}OPT\setminus {\mathbf{A}}^{\prime }|}{k-|{\mathbf{A}}^{\prime }|}\rceil \cdot (f({\text{textsc}OPT}^{\prime }\cup {\mathbf{A}}^{\prime })-f({\mathbf{A}}^{\prime }))$$

$$\le \frac{1}{1-\frac{1}{e}}\cdot \lceil \frac{|\text{textsc}OPT\setminus {\mathbf{A}}^{\prime }|}{k-|{\mathbf{A}}^{\prime }|}\rceil \cdot (f({\mathbf{B}}^{\prime }\cup {\mathbf{A}}^{\prime })-f({\mathbf{A}}^{\prime }))$$

$$f({\mathbf{U}}_i)\ge f({\mathbf{A}}_i)+\frac{1-\frac{1}{e}}{\lceil \frac{|\text{textsc}OPT\setminus {\mathbf{A}}_i|}{k-i}\rceil }\cdot (f(\text{textsc}OPT\cup {\mathbf{A}}_i)-f({\mathbf{A}}_i))$$

$$f({\mathbf{B}}^{\prime }\cup {\mathbf{A}}^{\prime })\ge f({\mathbf{A}}^{\prime })+\frac{1-\frac{1}{e}}{\underset{q\in [r]}{\max }\lceil \frac{|(\text{textsc}OPT\setminus {\mathbf{A}}^{\prime })\cap {\mathbf{T}}_q|}{x_q-|{\mathbf{A}}^{\prime }\cap {\mathbf{T}}_q|}\rceil }\cdot (f(\text{textsc}OPT\cup {\mathbf{A}}^{\prime })-f({\mathbf{A}}^{\prime }))$$

$$f(\mathbf{B}\cup {\mathbf{A}}^{\prime })=\underset{\begin{array}{c}\mathbf{Z}\subseteq \text{textsc}OPT\setminus {\mathbf{A}}^{\prime }\\ \mathbf{Z}\cup {\mathbf{A}}^{\prime }\in \mathcal{I}\end{array}}{\max }f(\mathbf{Z}\cup {\mathbf{A}}^{\prime })$$

$$f((\mathbf{B}\cup {\mathbf{A}}^{\prime })\cap {\mathbf{T}}_q)=\underset{\begin{array}{c}\mathbf{Z}\subseteq \text{textsc}OPT\setminus {\mathbf{A}}^{\prime }\\ |\mathbf{Z}\cap {\mathbf{T}}_q|\le x_q-|{\mathbf{A}}^{\prime }\cap {\mathbf{T}}_q|\end{array}}{\max }f((\mathbf{Z}\cup {\mathbf{A}}^{\prime })\cap {\mathbf{T}}_q)$$

$${\square }_q=\lceil \frac{|(\text{textsc}OPT\setminus {\mathbf{A}}^{\prime })\cap {\mathbf{T}}_q|}{x_q-|{\mathbf{A}}^{\prime }\cap {\mathbf{T}}_q|}\rceil$$

$$f(\text{textsc}OPT\cup {\mathbf{A}}^{\prime })-f({\mathbf{A}}^{\prime })$$

$$\le \sum _{q=1}^r{\square }_q\cdot \left(f\right((\mathbf{B}\cup {\mathbf{A}}^{\prime })\cap {\mathbf{T}}_q)-f({\mathbf{A}}^{\prime }\cap {\mathbf{T}}_q\left)\right)$$

$$\le \left(\underset{q\in [r]}{\max }{\square }_q\right)\cdot (f(\mathbf{B}\cup {\mathbf{A}}^{\prime })-f({\mathbf{A}}^{\prime }))$$

$$\le \left(\underset{q\in [r]}{\max }{\square }_q\right)\cdot (f({\text{textsc}OPT}^{\prime }\cup {\mathbf{A}}^{\prime })-f({\mathbf{A}}^{\prime }))$$

$$\le \frac{1}{1-\frac{1}{e}}\cdot \left(\underset{q\in [r]}{\max }{\square }_q\right)\cdot (f({\mathbf{B}}^{\prime }\cup {\mathbf{A}}^{\prime })-f({\mathbf{A}}^{\prime }))$$

Full text

1 Introduction

Ethiopia, the second most populous country in Africa, is home to over 130 million people across 12 regional states and two chartered cities, further divided into zones, woredas (districts), and kebeles (the smallest administrative units) [UNF22, WOR22]. Roughly 76% of the population lives in rural areas, where access to healthcare is limited compared to urban settings [HWT+23]. While the physical accessibility of health facilities has significantly improved, the issue is ensuring the quality of essential healthcare service delivery.

The Ethiopian healthcare system follows a three-tier structure comprising Primary Health Care Units (PHCUs), general hospitals, and specialized hospitals. PHCUs, which consist of primary hospitals, health centers, and health posts and are responsible for 70–80% of essential health services, have been expanded under the Ministry of Health’s (MOH) flagship Health Extension Program (HEP), launched in 2003/04 [WTR+16].

Unfortunately, persistent challenges remain as the evolution of the HEP is not aligned with the ever increasing demand of the community to get a comprehensive essential healthcare services (health promotion-rehabilitative care) at the health post. This resulted in inadequate staffing and equipment, and declining and/or stagnating performance in maternal and child health indicators [TAM+20].

To address these gaps, the MOH introduced the HEP Optimization Roadmap (2020–2035) [MIN20]. A central reform is the reclassification of health posts into three categories: mixed, basic, and comprehensive. Comprehensive health posts are designed to deliver a broader suite of services, including childbirth, postnatal care, and chronic disease treatment, and require substantial investments in infrastructure, staffing, and medical supplies.

The roadmap calls for the construction (or upgrading) of over 2,000 comprehensive health posts, with an estimated cost of USD 300k–400k per facility excluding staffing and equipment. As of now, 260 are operational and 183 are under construction. While initial decisions used WHO’s AccessMod platform (https://www.accessmod.org/), they did not necessarily incorporate region-specific prioritization preferences (see below) or account for geographic and demographic factors known to drive service disparities [GFT+25].

Given limited resources and growing health demands, there is a pressing need for a strategic, data-driven approach to guide equitable facility placement (e.g., where to build the next comprehensive health post given limited annual budget to augment the existing network of healthcare facilities). This requires integrating diverse datasets, including geolocation of existing facilities, high-resolution population estimates, topographic and official district administrative shapefiles, and health service coverage indicators like skilled birth attendance and postnatal care. Planning is further complicated by uncertainty in budgets and population forecasts, and the need to ensure fairness at every decision point. Different regions may also impose varying distributional preferences: some may prioritize population density while others may target poorer districts.

Our contributions.

In collaboration with the Ethiopian Public Health Institute and Ministry of Health, we formalize the problem and develop a Health Access Resource Planner (HARP) tool to help regional planners navigate these challenges and customize allocation strategies to local priorities.

In Section 2, we propose and study the new abstract problem of maximizing a non-decreasing submodular function under online partition and global proportional constraints (MOPGP), which captures the real-world challenge of prioritizing health facility upgrades in Ethiopia. Crucially, our framework accommodates region-specific distributional goals by encoding proportionality constraints, making it a flexible, shared decision-support tool for planners with possibly differing regional priorities. 2. 2.

We design algorithms to solve MOPGP with provable worst-case guarantees. These include a learning-augmented algorithm that integrates expert-provided selections, supporting regional planning workflows that start with partial plans and seek principled refinement. 3. 3.

In  Section 5, Appendix F and in Appendix G, we evaluate the HARP tool on some regions of Ethiopia using two possible prioritization rules. We show that our approach successfully prioritizes coverage according to the specified criteria while maintaining reasonable overall coverage.

HARP enables regional planners to solve their specific instance of MOPGP using a unified, principled approach. While our empirical results offer insight into practical performance, we emphasize that all findings are intended as a proof of concept and not official policy recommendations. 111

2 Problem formulation

Notation. We use lowercase letters for scalars, boldface for sets and sequences, and calligraphic letters for higher-order objects (e.g., sets of sets, graphs, matroids). For a set $\mathbf{A}$, let $|\mathbf{A}|$ denote its cardinality, and $2^{\mathbf{A}}$ its power set. The disjoint union of disjoint sets $\mathbf{A}$ and $\mathbf{B}$ is denoted $\mathbf{A}\uplus\mathbf{B}$. We use $\mathbb{R}_{\geq 0}$ for non-negative reals, $\mathbb{N}$ for natural numbers, $\mathbb{N}_{>0}=\mathbb{N}\setminus\{0\}$, and $[n]=\{1,\ldots,n\}$ for any $n\in\mathbb{N}_{>0}$. For any sequence $\mathbf{s}\in[r]^{b}$ and type $q\in[r]$, let $\#(q,\mathbf{s})$ denote the number of times $q$ appear in $\mathbf{s}$, e.g., for $\mathbf{s}=(1,2,1,3)\in[3]^{4}$, we have $\#(1,\mathbf{s})=2$, $\#(2,\mathbf{s})=1$, and $\#(3,\mathbf{s})=1$.

In the following, we will formally define MOPGP in full generality and then contextualize to how it assists in facility location within any particular region of Ethiopia.

We model MOPGP as a sequential subset selection task, where the goal is to maximize a non-decreasing submodular function (see Section 3 for a formal definition) subject to two constraints: (i) an online budget constraint modeling incremental budget arrivals, and (ii) a global proportional constraint enforcing type-level proportionality. Let $\mathbf{V}=\mathbf{V}^{(1)}\uplus\ldots\uplus\mathbf{V}^{(h)}$ denote the ground set partitioned over $h$ rounds. Each element $e\in\mathbf{V}$ is associated with a type $\mathrm{type}(e)\in[r]$, inducing an second partitioning $\mathbf{V}=\mathbf{T}_{1}\uplus\ldots\uplus\mathbf{T}_{r}$, where $\mathbf{T}_{q}=\{e\in\mathbf{V}:\mathrm{type}(e)=q\}$. At each round $t\in[h]$ (a year), a budget $b^{(t)}\leq|\mathbf{V}^{(t)}|$ is revealed and we must irrevocably select a subset $\mathbf{S}^{(t)}\subseteq\mathbf{V}^{(t)}$ of size $|\mathbf{S}^{(t)}|\leq b^{(t)}$. We define the cumulative budget and selection up to round $t$ as $b^{(1:t)}=\sum_{\tau=1}^{t}b^{(\tau)}$ and $\mathbf{S}^{(1:t)}=\uplus_{\tau=1}^{t}\mathbf{S}^{(\tau)}$ respectively. The objective is to maximize $f:2^{\mathbf{V}}\to\mathbb{R}$, a non-decreasing submodular function over the final selection $\mathbf{S}=\mathbf{S}^{(1:h)}\subseteq\mathbf{V}$, subject to the follow two constraints:

Online budget constraint: $|\mathbf{S}^{(t)}|\leq b^{(t)}$ for all $t\in[h]$. 2. 2.

Global proportional constraint: For given proportions $p_{1},\ldots,p_{r}\in[0,1]$ with $\sum_{q=1}^{r}p_{q}\leq 1$, the selection should satisfy $|\mathbf{S}\cap\mathbf{T}_{q}|\geq p_{q}\cdot|\mathbf{S}|$ for all types $q\in[r]$.

Unfortunately, it may not be feasible to always meet proportional targets. For instance, if $b^{(1)}=1$ and we require each of $r\geq 2$ types to have at least $\frac{1}{10r}$ proportion, then no single-element selection in the first time step can meet this condition. As such, we define the notion of satisfaction ratio as $\alpha_{q}(\mathbf{S})=\frac{|\mathbf{S}\cap\mathbf{T}_{q}|}{p_{q}\cdot|\mathbf{S}|}$ where $\alpha_{\min}(\mathbf{S})=\min_{q\in[r]}\alpha_{q}(\mathbf{S})$ and define the best possible such ratio under budget $b$ as $\beta(b)=\max_{\mathbf{A}\subseteq\mathbf{V},|\mathbf{A}|=b}\alpha_{\min}(\mathbf{A})$. Note that the proportional constraints on all types can be satisfied when $\beta(b)\geq 1$.

To ensure long-term proportional balance, we insist that any feasible solution is a type feasible selection.

Definition 1** (Type feasible selection).**

A subset $\mathbf{S}^{(1:h)}\subseteq\mathbf{V}$ of size $b^{(1:h)}$ is type feasible if, for all time steps $t\in[h]$:

$\alpha_{\min}(\mathbf{S}^{(1:t)})=\beta(b^{(1:t)})$ 2. 2.

Among all such sets, it minimizes $|\mathbf{Q}(\mathbf{S}^{(1:t)})|$ where

[TABLE]

Intuitively, the set $\mathbf{Q}(\mathbf{S}^{(1:t)})$ contains all types with the minimum satisfaction ratio. Since improving the satisfication ratio $\alpha_{\min}(\cdot)$ requires one to select at least one at least element of each type in $\mathbf{Q}(\cdot)$, minimizing $|\mathbf{Q}(\cdot)|$ serves to hedge against potential budget uncertainty.

We are now ready to formally define MOPGP.

Definition 2** (MOPGP).**

Given a non-decreasing submodular function $f:2^{\mathbf{V}}\rightarrow\mathbb{R}$, time horizon $h\in\mathbb{N}$, number of types $r\in\mathbb{N}_{>0}$, ground set $\mathbf{V}=\mathbf{V}^{(1)}\uplus\ldots\uplus\mathbf{V}^{(h)}=\mathbf{T}_{1}\uplus\ldots\uplus\mathbf{T}_{r}$, online budgets $\mathbf{b}=\{b^{(1)},\ldots,b^{(h)}\}$, and type proportions $\mathbf{p}=\{p_{1},\ldots,p_{r}\}$, the MOPGP$(f,h,r,\mathbf{V},\mathbf{b},\mathbf{p})$ problem seeks to compute a type feasible selection $\mathbf{S}\subseteq\mathbf{V}$ such that $|\mathbf{S}^{(t)}|\leq b^{(t)}$ for all time steps $t\in[h]$.

Observe that MOPGP is a generalization of the well-studied NP-hard maximum coverage problem for a suitably defined set function $f$ when $t=1$, $r=1$, and $p_{1}=0$ (i.e., no proportionality constraints). As such, one cannot expect to be able to compute an optimum selection $\mathbf{S}^{*}\subseteq\mathbf{V}$ in polynomial time in general. Instead, we will design and analyze efficient approximation algorithms that can compute some selection $\mathbf{S}\subseteq\mathbf{V}$ such that $f(\mathbf{S})\geq\alpha\cdot f(\mathbf{S}^{*})$ for some approximation ratio $0\leq\alpha\leq 1$.

Contextualizing MOPGP to the Ethiopian setting.

For our problem, we use the 1km-by-1km grid population forecasts for each region in Ethiopia [WOR25], as well as geographical information that enables computation of travel time [WNV+20].

Let $\mathbf{U}$ denote the space of grid cells considered. For each year $t\in[h]$, the set $\mathbf{V}^{(t)}$ correspond to all grid cells in the region for which we can build a comprehensive health post while types correspond to the districts within the different regions in Ethiopia. Note that $\mathbf{V}^{(1)},\ldots,\mathbf{V}^{(h)}$ are all defined over the same set of grid cells $\mathbf{U}$ but correspond to building a facility at different time steps. For any facility located at grid cell $c\in\mathbf{U}$, let ${\textsc{covered}}{}(c)\subseteq\mathbf{U}$ be the set of grid cells that are reachable from $c$ within 2-hours of travel time, which depends on the topological characteristics of the terrain and may vary across locations and directions. For a set of facility locations (i.e., subset of grid cells) $\mathbf{S}$, we write ${\textsc{covered}}{}(\mathbf{S})=\bigcup_{c\in\mathbf{S}}{\textsc{covered}}{}(c)\subseteq\mathbf{U}$ to mean the union of the covered cells.

As the planning process occurs over a multi-year horizon where facilities continue to provide coverage in all future time steps once it has been built, we define the objective function $f$ is the number of people who can access a comprehensive health post within a walking time of at most two hours under the proposed selection. To be precise,

[TABLE]

where $w^{(t)}_{c}$ is the population forecast for year $t$ for grid cell $c$. We formally show that the above objective Eq. 1 is submodular in Section B.1. Table 1 summarizes the mapping while Example 3 illustrates our formulation visually.

Example 3*.*

Consider the $5\times 5$ population grid in Fig. 2 over $h=2$ time steps, with $b^{(1)}=2$ and $b^{(2)}=1$. Suppose a facility at any cell provides coverage for any cell that is of Manhattan distance 1 from it. If we build two health posts at coordinate $\mathbf{S}^{(1)}=\{(2,2),(4,3)\}$ at the first time step and another at $\mathbf{S}^{(2)}=\{(2,4)\}$ at the second time step, then we see that $f(\mathbf{S})=f(\mathbf{S}^{(1)}\uplus\mathbf{S}^{(2)})=110$, where we obtain a coverage of $46$ from Year $1$ and $64$ in Year $2$. Observe that the population can change over time and health posts continue to provide coverage in future time steps once built.

3 Key concepts and related work

After relating our formulation of MOPGP to the known concepts of submodular optimization and matroid theory, we review related work on optimization under these structures, highlighting limitations of existing methods in addressing the sequential and proportional constraints of MOPGP. Finally, we briefly introduce learning-augmented algorithms. For more related work, see Appendix A

Submodular Functions. Let $\mathbf{V}$ be a finite ground set (e.g., the pairs (location, year) for building health posts) and $f:2^{\mathbf{V}}\to\mathbb{R}$ be a set function, e.g., a function that assigns a coverage value to any given set of locations where health posts are built. We say that $f$ is non-decreasing if $f(\mathbf{A})\leq f(\mathbf{B})$ for all $\mathbf{A}\subseteq\mathbf{B}\subseteq\mathbf{V}$, and $f$ is submodular if for all $\mathbf{A}\subseteq\mathbf{B}\subseteq\mathbf{V}$ and $e\in\mathbf{V}\setminus\mathbf{B}$, $f(\mathbf{A}\cup\{e\})-f(\mathbf{A})\geq f(\mathbf{B}\cup\{e\})-f(\mathbf{B})$. It is known that a greedy selection gives a $1-1/e$ approximation for maximizing non-decreasing submodular functions under a cardinality constraint [NWF78]. For notational convenience, we will write the marginal gain function $g$ with respect to $f$ as $g(\mathbf{A},\mathbf{B})=f(\mathbf{A}\uplus\mathbf{B})-f(\mathbf{A})$ for any two disjoint subsets $\mathbf{A},\mathbf{B}\subseteq\mathbf{V}$.

Matroids. Matroids encompass a wide range of constraints that enable efficient optimization algorithms that admit provable approximation guarantees. In this work, we are particularly interested in the special class of partition matroid [REC73] which has applications like load balancing, scheduling, and constrained subset selection.

Definition 4** (Partition matroid).**

Let $\mathbf{V}=\mathbf{V}_{1}\uplus\ldots\uplus\mathbf{V}_{r}$ be a partition of $\mathbf{V}$ and $x_{1},\ldots,x_{r}\in\mathbb{N}$ be upper bounds. Then, the independence set $\mathcal{I}$ of a partition matroid $\mathcal{M}=(\mathbf{V},\mathcal{I})$ is defined as $\mathcal{I}=\{\mathbf{S}\subseteq\mathbf{V}:|\mathbf{S}\cap\mathbf{V}_{q}|\leq x_{q},\forall q\in[r]\}$.

In our Ethiopian setting, budget and proportionality constraints each induce a partition matroid over the set of candidate facilities, e.g., budget at time step $t$ imposes the constraint that any valid selection $\mathbf{S}$ satisfies $|\mathbf{S}^{(t)}\cap\mathbf{V}^{(t)}|\leq b^{(t)}$.

Online maximization under matroid constraints. A closely related problem to MOPGP is that of Problem (1.6) of [FNW78], which we present using our notation in Definition 5.

Definition 5** (Maximizing a non-decreasing submodular function subject to $k$ matroid intersections over $h$ rounds).**

Let $\mathbf{V}=\mathbf{V}^{(1)}\uplus\ldots\uplus\mathbf{V}^{(h)}$ be a set of $h$ disjoint ground sets, $f:2^{\mathbf{V}}\to\mathbb{R}$ be a non-decreasing submodular set function, and $\mathcal{M}_{1}=(\mathbf{V},\mathcal{I}_{1}),\ldots,\mathcal{M}_{k}=(\mathbf{V},\mathcal{I}_{k})$ be $k$ matroids over $\mathbf{V}$. The goal is to find a subset $\mathbf{S}\subseteq\mathbf{V}$ that maximizes $f(\mathbf{S})$ subject to $\mathbf{S}\cap\mathbf{V}^{(t)}\in\mathcal{I}_{1}\cap\ldots\cap\mathcal{I}_{k}$ for all $t\in[h]$.

If budget uncertainty were absent and only final-round proportional constraints mattered, the problem could be modeled using two partition matroids ($k=2$), allowing the local greedy algorithm of [FNW78] to achieve a $1/(k+1)=1/3$ approximation. However, MOPGP differs fundamentally from Definition 5 in that matroidal constraints – driven by online budget arrivals – are revealed incrementally over time. This renders direct application of their method infeasible.

For the single-matroid ($k=1$) case, [CCP+11] achieve a $1-1/e$ approximation via continuous greedy methods, while [GS07] show that local greedy retains a $1/(\alpha+1)$ approximation even when optimizing a proxy function $g$ that approximates the true objective $f$ within a factor of $\alpha\geq 1$.

To address evolving constraints, one may consider online optimization frameworks. The submodular secretary problem [BIK07, BHZ13] assumes fixed matroids and randomly arriving elements, whereas our setting reverses this: the ground set is fixed, but feasibility constraints (budgets) are revealed over time. Recent work by [CDK+24, SSZ25] explores online feasibility arrival but focuses on linear objectives and does not support global, type-based fairness constraints.

Finally, fairness-aware submodular optimization [CHK+19, TWR+19] has addressed proportionality constraints, but these methods typically require full input access and use continuous relaxations or local exchanges, making them unsuitable for our setting of irrevocable, sequential decisions under uncertainty.

Representation-constrained coverage and online constraints Representation constraints have been studied in coverage problems. For example, [ABD+23] consider a max-$k$-coverage problem with group fairness constraints but enforce rigid group-wise coverage parity, which can lead to infeasibility. In contrast, our model allows flexible type-level bounds, maintaining feasibility and realism. Recently, ([CDK+24]; [SSZ25]) explored online problems where constraints, rather than elements, arrive over time, similar to our evolving feasibility model. However, they focus on maximizing simple linear objectives and do not directly extend to submodular objectives or type-based distributional constraints.

Learning-augmented algorithms. In practice, regional governments may already have existing facility selection plans (for any single fixed time step with budget $b$) that are constructed using domain knowledge, heuristics, or political considerations. Such an advice selection often encodes valuable local preferences but may not be optimal under the formal objective. Learning-augmented algorithms provide a formal framework to improve decision quality by incorporating advice or predictions, while maintaining worst-case guarantees when the advice is poor [LV21]. The goal is to design algorithms that are both consistent (optimal when the advice is perfect) and robust (competitive with the best advice-free baseline), ideally with performance degrading gracefully as advice quality worsens. See [MV22] for a survey and Appendix A for further examples. Closest to us are the works of [LS24, AB24], which aim to improve update times in dynamic settings. In contrast, we aim to improve the approximation ratio achievable by any polynomial-time algorithm in a static selection setting.

4 Algorithmic contributions

We present a learning-augmented222While “learning-augmented” originated from using machine-learned predictions, the framework more broadly encompasses any source of advice, such as expert-curated selections in our setting. approach for optimizing facility locations at any single time step in Section 4.1 and analyze a greedy approach for multi-step planning under uncertainty in Section 4.2. While our learning-augmented approach applies at every decision point, we only provide guarantees for a single fixed time step.333Single-step guarantees offer interpretable, modular assurances and are especially relevant in dynamic policy settings, where decisions are made incrementally over multiple time steps. In  Section 4.3, we highlight the importance of Definition 1 by showing that one cannot achieve non-trivial guarantees for MOPGP under online budgets without additional structure on the problem instance. All proofs are deferred to the appendix.

4.1 Learning-augmented single-step planning

Our goal in this section is to design a learning-augmented algorithm that uses such advice to produce selections from $\mathbf{V}$ that are provably no worse and potentially better. Here, we present the algorithm for the simple case of $r=1$ type (Algorithm 1) and show how to extend it to the more general partition matroid setting with $r\geq 1$ in Appendix C.

A key advantage of the learning-augmented approach is its practicality in real-world decision-making environments. Rather than discarding or overriding expert input, our method builds on it — treating existing recommendations as a starting point and refining them in a principled way. This makes it particularly attractive to policymakers and planners, as it respects domain expertise accumulated over years of fieldwork, local knowledge, and institutional memory. Importantly, our approach offers formal performance guarantees: it matches the expert plan when it is already optimal, and improves upon it when possible, without ever doing worse than the best advice-free alternative. This balance between respecting human judgment and augmenting it with algorithmic rigor makes the method especially compelling in collaborative, policy-driven settings.

At the high-level, our algorithmic approach is to augment subsets of the advice selection with greedy selections444MOPGP generalizes the NP-hard problem of maximum coverage, and greedy selections attain tight approximation guarantees. For any subset $\mathbf{A}^{\prime}\subseteq\mathbf{A}$, we fill the remaining $b-|\mathbf{A}^{\prime}|$ slots by greedily selecting elements from $\mathbf{V}\setminus\mathbf{A}^{\prime}$ to maximize marginal gain with respect to the non-decreasing submodular set function $g(\cdot,\mathbf{A}^{\prime})=f(\cdot\cup\mathbf{A}^{\prime})-f(\mathbf{A}^{\prime})$. That is, we greedily choose elements conditioned on already chosen $\mathbf{A}^{\prime}$.

Theorem 6**.**

Consider Algorithm 1 and let ${\textsc{OPT}}\subseteq\mathbf{V}$ be an optimal selection. Then,

[TABLE]

Algorithm 1 is essentially an algorithm for size-constrained non-decreasing submodular maximization. Observe that $\mathbf{U}_{0}$ is the advice-free greedy selection $\mathbf{S}$, and $\mathbf{U}_{b}$ is the advice selection $\mathbf{A}$, so we always have $f(\mathbf{U}_{i^{*}})\geq\max\{f(\mathbf{U}_{0}),f(\mathbf{A})\}$ and thus the output $\mathbf{U}_{i^{*}}\subseteq\mathbf{V}$ of our learning-augmented algorithm is at least $(1-\frac{1}{e})$-robust and $1$-consistent. This is because $f(\mathbf{U}_{i^{*}})\geq f(\mathbf{U}_{0})\geq(1-\frac{1}{e})\cdot f({\textsc{OPT}})$ by submodularity of $f$, and $f(\mathbf{U}_{i^{*}})=f({\textsc{OPT}})$ when $f(\mathbf{A})=f({\textsc{OPT}})$. Our approach of Theorem 6 naturally extends to $r\geq 1$ types for type-decomposable objective functions, but with the base guarantee of $f(\mathbf{U}_{0})\geq(1-\frac{1}{e})\cdot f({\textsc{OPT}})$ degrading to $f(\mathbf{U}_{0})\geq\frac{1}{2}\cdot f({\textsc{OPT}})$ due to the matroidal constraints; see Appendix C for details.

As a remark, we would like to point out that there is nothing inherently special about the choice of $\mathbf{A}_{0},\ldots,\mathbf{A}_{b}$. From the proof of Theorem 6 in the appendix, one can see that Algorithm 1 can be modified to work with any polynomial number of subsets of $\mathbf{A}$ while still being polynomial time.

Example 7*.*

Fig. 3 shows three possible selections of $b=3$ facilities on the first grid of Example 3. As before, suppose a facility at any cell provides coverage for any cell that is of Manhattan distance 1 from it. Greedily selecting based on marginal gain produces the selection $\mathbf{G}=[(3,2),(0,3),(1,1)]$ with $f(\mathbf{G})=62$. Meanwhile, an advice selection of $\mathbf{A}=[(2,1),(1,3),(3,3)]$ has $f(\mathbf{A})=60$. Running Algorithm 1 with the advice facility at coordinate $(2,1)$ produces the selection $\mathbf{U}=[(2,1),(3,3),(0,3)]$ with $f(\mathbf{U})=67$, which also happens to be an optimum selection.

4.2 Multi-step planning under budget uncertainty

We now show that a variant of the local greedy algorithm from [FNW78], adapted to satisfy type feasibility, is both practical and provably robust. A key difficulty in solving MOPGP lies in the abstract nature of type feasibility: Definition 1 specifies high-level desiderata but does not provide a directly actionable definition. While the objective $f$ is non-decreasing and submodular, the proportional constraints across types introduce combinatorial complexity that does not decompose neatly.

To make the notion of type feasibility operational, we introduce a fixed tie-breaking rule $\sigma:[r]\to[r]$ over types and restrict our attention to a structured subclass of solutions, which we call $\sigma$-type feasible selections.

Definition 8** ($\sigma$-type feasible selection).**

Let $\sigma:[r]\to[r]$ be an arbitrary total preference ordering over types. A subset $\mathbf{S}\subseteq\mathbf{V}$ of size $b$ is said to be $\sigma$-type feasible if it is type feasible and for $\mathbf{Q}(\mathbf{S}^{(1:t)})$, favors types according to $\sigma$: if $\sigma(q)<\sigma(q^{\prime})$, then type $q$ is preferred over $q^{\prime}$.

This reduction justifies restricting our attention to the class of $\sigma$-type feasible solutions: they are not only easier to operationalize algorithmically, but also provably approximate any type-feasible solution up to a factor that depends on the granularity of type availability.

Our next result tells us that any optimum $\sigma$-type feasible selection ${\textsc{OPT}}_{\sigma}$ is competitive against any optimum type feasible selection OPT if at least $k$ elements of each type is chosen per round when applying MultistepPlanning.

Theorem 9**.**

Let OPT be an optimal type-feasible selection and let ${\textsc{OPT}}_{\sigma}$ be an optimal $\sigma$-type feasible selection. Fix an integer $k\in\mathbb{N}_{>0}$. If $|{\textsc{OPT}}_{\sigma}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|\geq k$ for all $t\in[h]$ and $q\in[r]$, then $f({\textsc{OPT}})\leq\frac{k+1}{k}\cdot f({\textsc{OPT}}_{\sigma})$.

Note that we typically have $k\geq 1$: the Ethiopian Ministry of Health is currently building $2,000$ comprehensive health posts across \sim$$670 rural districts [OCH23], and we expect more facilities to be built in the future.

Our algorithm MultistepPlanning is presented in Algorithm 2. At each round $t\in[h]$, MultistepPlanning determines how many comprehensive health posts $x^{(t)}_{q}\in\mathbb{N}$ from each district $q\in[r]$ should be selected (line 3), based on the desired global proportion and elements selected so far in $\mathbf{S}^{(1:t-1)}$. Note that $\sum_{q\in[r]}x^{(t)}_{q}=b^{(t)}\leq|\mathbf{V}^{(t)}|$. These type-specific quotas define a partition matroid (line 4), over which the algorithm performs greedy selection (lines 5-7) by iteratively adding the element with highest marginal gain while respecting the matroid. Theorem 10 shows the formal guarantees of MultistepPlanning.

Theorem 10**.**

Fix an arbitrary tie-breaking ordering $\sigma:[r]\to[r]$. Assuming the objective function $f$ can be evaluated in constant time given any subset, MultistepPlanning runs in $\mathcal{O}(\sum_{t=1}^{h}b^{(t)}\cdot(r+|\mathbf{V}^{(t)}|))$ time and outputs $\mathbf{S}=\uplus_{t=1}^{h}\mathbf{S}^{(t)}$ such that

For all $t\in[h]$, $\mathbf{S}^{(t)}\subseteq\mathbf{V}^{(t)}$ and $|\mathbf{S}^{(t)}|=b^{(t)}$. 2. 2.

The output $\mathbf{S}$ is a $\sigma$-type feasible selection. 3. 3.

For all $t\in[h]$, we have

[TABLE]

where ${\textsc{OPT}}_{\sigma}(b^{(1)},\ldots,b^{(t)})$ is an optimum $\sigma$-type feasible selection restricted to budgets $b^{(1)},\ldots,b^{(t)}$.

4.3 Our theoretical bounds are tight in general

Proposition 11 shows that any solution to MOPGP violating Definition 1 is brittle against future budget uncertainties while Proposition 12 show that the approximation ratio terms for MultistepPlanning are essentially tight.

Proposition 11**.**

Let ALG be any deterministic algorithm to MOPGP aiming to produce selections $\mathbf{S}^{(1)},\ldots,\mathbf{S}^{(h)}$ that maximizes $\beta(b^{(1:t)})$ at each time step $t\in[h]$. If ALG does not minimize $|\mathbf{Q}(\mathbf{S}^{(t)})|$ for each $t\in[h]$, then there exists instances where it fails to minimize $\beta(b^{(1:t)})$ at each $t\in[h]$.

Proposition 12**.**

There exists instances where MultistepPlanning achieves an approximation ratio of $\leq\frac{1}{2}$ due to budget uncertainties and $\leq\frac{k}{k+1}$ due to $\sigma$-ordering.

5 Experiments

We empirically evaluate the performance of the individual components of our proposed HARP tool. See Appendix G for additional experimental details and access to source code and scripts.

5.1 Experimental setup

5.1.1 Data.

To model facility coverage, we use point-to-point walking distance estimates between 1km-by-1km grid cells across Ethiopia, based on the global friction surface from [WNV+20], and population forecasts from WorldPop projections for the years 2026–2030 [WOR25].555Population forecasts are inherently uncertain and the true objective function $f$ is not directly accessible. In Section B.2, we show that the impact on facility coverage is not substantial. We are also given a set of existing facilities $\mathbf{S}_{\text{existing}}$ and available budgets were chosen based on expert consultation.

5.1.2 Regions analyzed.

We focus on rural regions where access remains limited, i.e., major cities such as Addis Ababa are excluded. Specifically, we evaluate our algorithms on the regions of Afar, Benishangul Gumuz, and Somali in the main paper, and on the Sidama region in Appendix F. This way we reflect Ethiopia’s two dominant rural livelihood types.666In the main paper, we focus on less populated regions to better highlight the spatial coverage gains achieved by our HARP.

There are around 30 districts in Afar and Sidama, around 20 in Benishangul Gumuz, and around 70 districts in Somali. A larger budget was considered for Somali as it is much larger than the other regions.

5.1.3 Distributional Policies (DP).

To ensure equitable allocation, each region may define a distributional policy by specifying target proportions $p_{1},\ldots,p_{r}$ across $r$ districts. These type-level constraints shape the facility selection process by prioritizing areas with greater unmet needs. In our experiments, we evaluate three policy scenarios:

DP0

No distributional constraint (i.e., $p_{1}=\ldots=p_{r}=0$), corresponding to unconstrained greedy selection.

DP1

Proportions $p_{q}$ are set based on unassisted home birth rates, favoring districts with poor maternal care access.

DP2

Proportions $p_{q}$ are set by the inverse of postnatal care coverage, favoring districts lacking early childhood services.

DP3

Proportions $p_{q}$ are set by the inverse of the average between home-birth rates and postnatal-care coverage, thereby balancing the two scenarios.

Both DP1 and DP2 target existing healthcare service gaps that can be provided by comprehensive health posts. While our framework supports multiple active constraints simultaneously, we focus on one distributional policy at a time to preserve theoretical guarantees and interpretability. In our experiments below, we write $\textsc{MP}(\text{DP}i)$, for $i\in\{0,1,2\}$, to denote the output of Algorithm 2 applied with the proportional constraints imposed by each policy.

5.2 Experiment 1: Impact of budget on coverage

In Fig. 4, the bars show how total coverage improves under different allocation strategies and budget levels over a five-year planning horizon. As all regions are large relative to the available budget, coverage increases approximately linearly with budget in all cases.

For $i\in\{1,2\}$ in Afar, Benishangul Gumuz, and Somali, the dashed and dotted lines in Fig. 4 show the ratios

[TABLE]

quantifying the trade-off incurred by distributional constraints, where $f$ is the coverage function defined in Eq. 1.

Note that we subtracted away the coverage from existing health posts $\mathbf{S}_{\text{existing}}$ to measure the relative gain in coverage as existing health posts are assumed to remain operational throughout. This ratio is similar to the notion of price of fairness [CKK+12]: a ratio close to 1 indicates that enforcing proportional fairness incurs minimal efficiency loss, while higher values reflect stronger trade-offs. As expected, these ratios improve with increasing budgets, suggesting that the proportional constraints come at a lower cost when more resources are available. Note that as the results of MP(DP$1$) and MP(DP$2$) were very similar, we did not report the performance of their average, MP(DP$3$).

5.3 Experiment 2: Evaluating district-level equity

In Experiment 1, we demonstrated that imposing proportional constraints does not significantly degrade coverage. We now show that these constraints improve adherence to the intended proportions. To this end, we examine the minimum satisfaction ratio, $\alpha_{\min}(\mathbf{S})$, achieved under each policy. As shown in Fig. 5, MP(DP$1$) and MP(DP$2$) outperform all other methods in the DP$1$ and DP$2$ metrics respectively, while MP(DP$3$) is somewhere in between and MP(DP[math]) consistently attains $\alpha_{\min}=0$ . Interestingly, in Afar, MP(DP$1$) yields significant gain under DP$1$ than MP(DP$2$) does under DP$2$, whereas the reverse trend is observed in the Benishangul Gumuz and Somali regions.

5.4 Experiment 3: High quality advice can help

To evaluate the practical utility of our learning-augmented approach, we conduct a retrospective analysis using the set of comprehensive health posts $\mathbf{A}$ that were previously constructed by regional planners: we simulate the original decision-making scenario in which $|\mathbf{A}|$ health posts must be selected from scratch. We compare three strategies:

1. Expert selection: Use expert-curated selection $\mathbf{A}$.

2. Greedy baseline: Apply Algorithm 2 to get $\mathbf{G}$.

3. Learning-augmented: Use Algorithm 1 to produce $\mathbf{U}$.

Although planning decisions are made at the regional level, our analysis is conducted at the district level to capture localized effects. Across all districts in the Afar, Benishangul Gumuz, and Somali regions, the greedy baseline $\mathbf{G}$ consistently outperforms the expert selection $\mathbf{A}$. Interestingly, we found $25$ districts within these regions in which our learning-augmented selection $\mathbf{U}$ outperforms both, i.e., $f(\mathbf{U})>f(\mathbf{G})>f(\mathbf{A})$. These results were obtained by selecting the best performing $\mathbf{U}$ from ten runs of Algorithm 1, each using a different random permutation of the original advice selection $\mathbf{A}$.

Fig. 6 presents a representative example, showing a zoomed-in subarea of a district where all three strategies select two health posts (the rest of the district is relatively sparse), with $\frac{f(\mathbf{U})-f(\mathbf{G})}{f(\mathbf{G})}\approx 2\%$ and $\frac{f(\mathbf{U})-f(\mathbf{A})}{f(\mathbf{A})}\approx 20\%$. The greedy algorithm, having made its first choice (the left location in $\mathbf{G}$) based solely on immediate marginal gain, is subsequently limited in its second selection. In contrast, Algorithm 1 could leverage on the leftward choice in $\mathbf{A}$ to produce a more globally effective selection $\mathbf{U}$.

6 Conclusion

In partnership with Ethiopia’s Ministry of Health and Public Health Institute, we developed a decision-support tool (HARP) to guide health facility planning. HARP allows regional planners to solve their own instance of MOPGP within a unified, principled framework that models the planning task as a constrained submodular optimization with online budget and global proportionality constraints. On the technical front, we are the first to integrate learning-augmented algorithms with multi-step submodular optimization, enabling principled refinement of expert plans with worst-case guarantees. Using real-world health and demographic data from Ethiopia, we show that our methods are effective in supporting equitable, data-driven infrastructure decisions across diverse regional contexts.

Acknowledgements

The findings and conclusions in this report are those of the authors and do not represent any recommendations or official positions.

Appendix A Further related work

In this section, we provide additional related work that was not mentioned in Section 3 due to space constraints.

A.1 Submodular maximization under constraints

In Section 3, we also mentioned a locally greedy algorithm of [FNW78] which yields a tight $\frac{1}{k+1}$ approximation to the problem of maximizing a non-decreasing submodular function subject to $k$ matroid intersections. We provide the pseudocode of this locally greedy algorithm in Algorithm 3.

Maximizing monotone submodular functions under matroid constraints has a rich history in offline settings [NWF78, CCP+11, CVZ11]. While work on submodular maximization under multiple matroid constraints such as [LMN+10] hints at possible modeling of our setting (one matroid for budgets and one for distributional constraints), these are offline formulations with static feasibility and do not capture our problem’s adaptivity or temporal uncertainty. In the online setting, one canonical problem that of the submodular secretary problem [BIK07, BHZ13], but it assumes that the matroid is fixed and known, while elements arrive sequentially in random or adversarial order. In contrast, our setting inverts the uncertainty: the ground set is known in advance, but feasibility constraints (budgets) are revealed over time. Moreover, secretary-style methods typically do not support global type-based constraints across time and assumes that arrivals are in random order. More recent work such as those by [CHK+19] and [TWR+19] introduce global representation constraints into this framework in the form of fairness constraints. The state-of-the-art approaches for solving the above-mentioned problems typically rely on full knowledge of the input, employ continuous relaxations, or local exchange techniques. As such, one cannot directly apply their strategies in our settings where decisions are irrevocable and constraints evolve over time.

A.2 Facility location under temporal and representation constraints

Classical and robust facility location problems are extensively studied (see e.g., surveys by [AF12, OD98]), including in dynamic and uncertain environments [SNY06, CAR21, AZ07, BMN11]. However, most of these models aim to optimize a final objective after uncertainty is resolved. Our approach departs from this paradigm by seeking robust decisions at each time step, enabling responsiveness to time-varying demand. Healthcare-specific studies, such as [GMS22], introduce incremental facility placement under multiple normative criteria, including fairness. However, their work also lacks support for temporally evolving feasibility constraints. Additionally, they assume static facility needs and a metric distance space, which may not hold in contexts like rural Ethiopia. Another work by [GJ13] proposes a budgeted healthcare access model in Iran, but does not consider representation or proportionality constraints across population types.

A.3 Measure distance to facilities

[TGH06] proposed a model for measuring the actual distances to health facilities. [NZH+03] incorporated equity considerations into their model to enhance the planning and monitoring of health facilities. Using the AccessMod [RE08] tool for computing the coverage of health facilities, [HOR+23] analyze the coverage in Ethiopia, and specifically in the Somali region. They suggest that the coverage in the Somali region is insufficient. Recently, [WNV+20] proposed a method to quantify access to health facilities globally. Using However, these works emphasize the importance of locating the facilities appropriately, but do not provide algorithmic methods for doing so. Our model explicitly incorporates time-sensitive population variation and provides algorithmic support for adaptive placement subject to budget and distributional considerations.

A.4 Representation-constrained coverage and online constraints

Representation constraints have also been studied in coverage problems. For example, [ABD+23] consider a max-$k$-coverage problem with group fairness constraints but enforce rigid group-wise coverage parity, which can lead to infeasibility. In contrast, our model allows flexible type-level bounds, maintaining feasibility and realism. [DRS+25] explore proportionality in set cover, applying constraints to the sets rather than the elements. Their formulation targets minimum-cost full coverage, which diverges from our goal of budget-constrained partial coverage under temporal uncertainty. More recently, [CDK+24] and [SSZ25] explore online problems where constraints, rather than elements, arrive over time, similar to our evolving feasibility model. However, these papers focus on maximizing simple linear (weight-based) objectives and do not directly extend to submodular objectives or type-based distributional constraints.

A.5 AI For Social Impact

Recently, many works are focused on developing AI tools for social impact (see e.g., [SWF20] for a broad survey). The aforementioned survey emphasizes that the healthcare domain has received the most attention among other application domains. In the context of vaccination, [NPW+22] introduced a framework that optimizes the allocation of health interventions under uncertainty, with the aim of improving vaccination uptake in Nigeria. Another work by [OSK+22], proposed an algorithm for optimizing mobile interventions in India. However, many of these works focus on dynamic/reactive intervention allocation problems while we focus on proactive, infrastructure-level investment planning. Another line of research examines health-facility location. [SMO12, GJ13] develop models for optimizing health-care coverage in Malaysia and Iran, respectively. However, these studies do not incorporate representation or proportionality constraints across different population groups.

A.6 Learning-augmented algorithms

Since the seminal work of [LV21], there has been growing interest in designing algorithms that incorporate unreliable advice, with performance guarantees that degrade gracefully with advice quality. This framework has been especially influential in online optimization, where advice can serve as a proxy for unknown future inputs. Notable applications include ski-rental [GP19, WLW20, ADJ+20], non-clairvoyant scheduling [PSK18], job scheduling [LLM+20, BMR+20, AJS22], data structures with predictions (e.g., learned indexes [KBC+18], Bloom filters [MIT18]), online selection and matching [AGK+20, DLP+21, CGL+24, CJS25], and online TSP [BLM+22, GLS23]. More recent work has expanded this paradigm to graph algorithms [DIL+21, CSV+22], causal discovery [CGB23], mechanism design [ABG+22, GKS+22], improving approximation guarantees in offline NP-hard problems [EFS+22, BDS+24, CdG+24], and distribution learning [BCG+25a, BCG+25b]. For a broader overview, we refer the reader to the survey by [MV22] and the website https://algorithms-with-predictions.github.io/ which tracks papers on this research topic.

Appendix B Deferred technical details for Section 2

B.1 Showing that objective function Eq. 1 is non-decreasing and submodular

Proposition 13**.**

The objective function defined in Eq. 1 is non-decreasing and submodular.

Proof.

Recall from Eq. 1 that

[TABLE]

Since population counts are non-negative, we see that $f(\mathbf{A})\leq f(\mathbf{B})$ for any two selections $\mathbf{A}\subseteq\mathbf{B}$, so $f$ is non-decreasing. Meanwhile, to see that $f$ is submodular, fix any $\mathbf{A}\subseteq\mathbf{B}\subseteq\mathbf{V}$ and $e\in\mathbf{V}\setminus\mathbf{B}$. Let $\tau\in[h]$ be the time step for which $e\in\mathbf{V}^{(\tau)}$. Then,

[TABLE]

Since $f(\mathbf{A}\cup\{e\})-f(\mathbf{A})\geq f(\mathbf{B}\cup\{e\})-f(\mathbf{B})$, we see that $f$ is submodular. ∎

B.2 Noisy estimate of the true objective function

Suppose we do not have access to the true objective function $f$ in MOPGP but are instead given a proxy function $\widetilde{f}$ such that $(1-\varepsilon)\cdot f(\mathbf{S})\leq\widetilde{f}(\mathbf{S})\leq(1+\varepsilon)\cdot f(\mathbf{S})$ for all $\mathbf{S}\subseteq\mathbf{V}$, for some multiplicative error $\varepsilon\in[0,1]$.

Given an algorithm ALG that has provable guarantees for MOPGP, the next theorem shows that one can obtain provable approximation guarantees with respect to $\varepsilon$ by running ALG in a blackbox manner while treating $\widetilde{f}$ as the true objective function. In particular, the $\alpha$ in Theorem 14 is $\frac{1}{2}\cdot\frac{k}{k+1}$ if at least $k$ elements of each type is chosen per round when applying MultistepPlanning.

Theorem 14**.**

Suppose we do not have access to $f$ for an ${\textsc{MOPGP}}{}(f,h,r,\mathbf{V},\mathbf{b},\mathbf{p})$ instance. Instead, we are given a proxy function $\widetilde{f}:2^{\mathbf{V}}\to\mathbb{R}$ with known bounded multiplicative error $\varepsilon\in[0,1]$ such that $(1-\varepsilon)\cdot f(\mathbf{S})\leq\widetilde{f}(\mathbf{S})\leq(1+\varepsilon)\cdot f(\mathbf{S})$ for all $\mathbf{S}\subseteq\mathbf{V}$. If ${\textsc{ALG}}(f)$ is an algorithm that achieves $\alpha$-approximation to ${\textsc{MOPGP}}{}(f,h,r,\mathbf{V},\mathbf{b},\mathbf{p})$, for some $\alpha\in[0,1]$, then ${\textsc{ALG}}(\frac{\widetilde{f}}{1-\varepsilon})$ achieves an approximation ratio of $\alpha\cdot\frac{1-\varepsilon}{1+\varepsilon}$ to ${\textsc{MOPGP}}{}(f,h,r,\mathbf{V},\mathbf{b},\mathbf{p})$.

Proof.

Besides the objective function $f$, let us fix all other parameters $h$, $q$, $\mathbf{V}$, $\mathbf{b}$, and $\mathbf{p}$ arbitrarily. So, for notational convenience, we simply write ${\textsc{MOPGP}}{}(f)$ to mean ${\textsc{MOPGP}}{}(f,h,q,\mathbf{V},\mathbf{b},\mathbf{p})$ in the rest of this proof.

For any function $f$, let $\mathbf{S}_{{\textsc{OPT}}}(f)$ denote the optimum type-feasible selection for ${\textsc{MOPGP}}{}(f)$ and $\mathbf{S}_{{\textsc{ALG}}}(f)$ denote type-feasible selection produced by ${\textsc{ALG}}(f)$. For any type-feasible selection $\mathbf{S}\subseteq\mathbf{V}$, we write $\mathrm{obj}(\mathbf{S},{\textsc{MOPGP}}{}(f))$ to denote the objective value attained by $\mathbf{S}$ with respect to the objective function $f$. Using this notation, we have

[TABLE]

Therefore, using the proxy function $\widetilde{f}$, we see that

[TABLE]

That is, running ALG on the proxy function $\widetilde{f}$ yields an approximation ratio of $\alpha\cdot\frac{1-\varepsilon}{1+\varepsilon}$. ∎

Appendix C Deferred technical details for Section 4.1

In this section, we first prove Theorem 6 where there is only $k=1$ type, i.e., we are essentially solving size-constrained submodular maximization, and then show how to adapt to the setting with $k\geq 1$ types Theorem 17. For convenience, we reproduce Algorithm 1 below.

The proof of Theorem 6 follows from the following two lemmas: Lemma 15 and Lemma 16.

Lemma 15**.**

Fix any optimal selection ${\textsc{OPT}}\subseteq\mathbf{V}$ of size $|{\textsc{OPT}}|=k$ and any arbitrary selection $\mathbf{S}\subseteq\mathbf{V}$ of size $|\mathbf{S}|\leq k$. Define $\mathbf{B}$ as the best selection in ${\textsc{OPT}}\setminus\mathbf{S}$ such that

[TABLE]

Then, for any non-empty subset $\mathbf{C}\subseteq{\textsc{OPT}}\setminus\mathbf{S}$, we have

[TABLE]

Proof.

Break up $\mathbf{C}$ into $\ell=\left\lceil\frac{|\mathbf{C}|}{|\mathbf{B}|}\right\rceil$ partitions $\mathbf{C}=\mathbf{C}_{1}\uplus\ldots\uplus\mathbf{C}_{\ell}$, each of size at most $|\mathbf{B}|$. Then,

[TABLE]

The claim follows by rearranging the terms. ∎

Lemma 16**.**

Fix any optimal selection ${\textsc{OPT}}\subseteq\mathbf{V}$ and advice selection $\mathbf{A}\subseteq\mathbf{V}$ of size $|{\textsc{OPT}}|=|\mathbf{A}|=k$. Fix an arbitrary subset $\mathbf{A}^{\prime}\subseteq\mathbf{A}$ of $\mathbf{A}$. Let $\mathbf{B}^{\prime}\subseteq\mathbf{V}\setminus\mathbf{A}^{\prime}$ be the output produced from greedy selection and ${\textsc{OPT}}^{\prime}\subseteq\mathbf{V}\setminus\mathbf{A}^{\prime}$ be the optimum selection with respect to the set function $f(\mathbf{S}\cup\mathbf{A}^{\prime})-f(\mathbf{A}^{\prime})$ for any subset $\mathbf{S}\subseteq\mathbf{V}\setminus\mathbf{A}^{\prime}$, where $|\mathbf{B}^{\prime}|=|{\textsc{OPT}}^{\prime}|=k-|\mathbf{A}^{\prime}|$. Then,

[TABLE]

Proof.

Define $\mathbf{B}\subseteq{\textsc{OPT}}\setminus\mathbf{A}^{\prime}$ such that

[TABLE]

Then, we have

[TABLE]

where $(\ast)$ is via Lemma 15 while $({\ddagger})$ is due to the greedy construction of $\mathbf{B}^{\prime}$ while satisfying the size constraint $k$; see [NWF78]. The claim follows by rearranging the terms. ∎

See 6

Proof.

By Lemma 16, we have

[TABLE]

for all $i\in\{0,1,\ldots,k\}$. The claim follows since $\mathbf{U}$ is the best selection out of $\mathbf{U}_{0},\mathbf{U}_{1},\ldots,\mathbf{U}_{k}$. ∎

To extend Theorem 6 to $r\geq 1$ types, consider an alternative objective set function $f^{\prime}$ which computes $f$ additively across each type: $f^{\prime}(\mathbf{S})=\sum_{q=1}^{r}f(\mathbf{S}\cap\mathbf{T}_{q})$ for any $\mathbf{S}\subseteq\mathbf{V}$. By construction, $f^{\prime}$ never underestimates $f$. That is, $f^{\prime}(\mathbf{S})\geq f(\mathbf{S})\geq\beta_{f,\mathbf{S}}\cdot f^{\prime}(\mathbf{S})$ for some $\beta_{f,\mathbf{S}}\in[0,1]$ that depends on $f$ and $\mathbf{S}$. Operating under the assumption that $f$ decomposes with respect to the types simplifies analysis while worsening the approximation ratio by a multiplicative factor proportional to $\max_{\mathbf{S}\subseteq\mathbf{V}}\beta_{f,\mathbf{S}}$; we can invoke the result of Section B.2 to $f^{\prime}$. Using $f^{\prime}$ as a proxy of $f$ is justified because each facility primarily serves the population of that same district. For instance, in the Sidama region of Ethiopia, an average of 81% of the population within a two-hour walking distance from a cell belongs to the same district.

Theorem 17**.**

Suppose the objective $f$ is type-decomposable and we need to produce a selection subject to a partition matroid over the types with $\mathcal{I}=\{\mathbf{S}\subseteq\mathbf{V}:|\mathbf{S}\cap\mathbf{T}_{q}|\leq x_{q},\forall q\in[r]\}$, for some $x_{1},\ldots,x_{q}\in\mathbb{N}$. Let ${\textsc{OPT}}\subseteq\mathbf{V}$ and $\mathbf{A}\subseteq\mathbf{V}$ be any optimal and advice selection such that $|{\textsc{OPT}}\cap\mathbf{T}_{q}|=|\mathbf{A}\cap\mathbf{T}_{q}|=x_{q}$ for all types $q\in[r]$. Then, for any subset $\mathbf{A}^{\prime}\subseteq\mathbf{A}$, there is a polynomial time algorithm that producing $\mathbf{B}^{\prime}\subseteq\mathbf{V}\setminus\mathbf{A}^{\prime}$ such that $\mathbf{B}^{\prime}\cup\mathbf{A}^{\prime}\in\mathcal{I}$ and

[TABLE]

Proof.

Define $\mathbf{B}\subseteq{\textsc{OPT}}\setminus\mathbf{A}$ such that $\mathbf{B}\cup\mathbf{A}\in\mathcal{I}$ and

[TABLE]

Since $f$ decomposes additively across types, we get

[TABLE]

for each type $q\in[r]$. We define $\mathbf{B}^{\prime}$ to be the greedy selection produced by LocalGreedy (Algorithm 3) over the matroid $\mathcal{I}$ having already selected $\mathbf{A}^{\prime}$. For readability, let us define

[TABLE]

for each type $q\in[r]$. Then, we have

[TABLE]

where $(\ast)$ is via Lemma 15 while $({\ddagger})$ is due to the greedy construction of $\mathbf{B}^{\prime}$ while satisfying the matroid constraint $\mathcal{I}$; see [FNW78]. The claim follows by rearranging the terms. ∎

Motivated by our analyses, one can define the following learning-augmented variant of MultistepPlanning in Algorithm 5. While one can use similar approaches to obtain theoretical guarantees for Algorithm 5, the resulting guarantees are unwieldly and do not provide any intuition. As such, we do not make any theoretical claims and simply present Algorithm 5 as a heuristic algorithm.

Appendix D Deferred technical details for Section 4.2

In this section, we work towards proving Theorem 10. We begin by proving Theorem 9, along with some additional results about $\sigma$-type feasible selections (Definition 8) that will be useful for proving algorithmic guarantees subsequently.

Note that any $\sigma$-type feasible selection (Definition 8) is also type feasible (Definition 1), though the latter can tie-break arbitrarily when selecting items from the set of types with minimal satisfaction ratio. We now show that a $\sigma$-type feasible optimum selection ${\textsc{OPT}}_{\sigma}$ is competitive against a type feasible optimum selection OPT if at least $k$ elements of each type is chosen per round when applying MultistepPlanning.

The following lemma about $\sigma$-type feasible subsets of a type feasible selections will be useful for proving our main result for this section (Theorem 9).

Lemma 18**.**

Fix an MOPGP problem instance $\mathcal{P}$. Let ${\textsc{OPT}}_{\sigma}$ be a $\sigma$-type feasible optimal selection to $\mathcal{P}$ and OPT be a type feasible optimal selection to $\mathcal{P}$. Let $\mathcal{I}\subseteq 2^{\mathbf{V}}$ be the independence set corresponding to the tie-broken type feasibility of ${\textsc{OPT}}_{\sigma}$ and define $\mathbf{W}\subseteq{\textsc{OPT}}$ to minimize the marginal gain:

[TABLE]

Then, $|\mathbf{W}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|\leq 1$ for any $t\in[h]$ and $q\in[r]$.

Proof.

Note that the minimality of $\mathbf{W}$ would try to make ${\textsc{OPT}}\setminus\mathbf{W}$ overlap with ${\textsc{OPT}}_{\sigma}$, in terms of element types at each time step, as much as possible. More formally, for any time step $t\in[h]$ and type $q\in[r]$, we have

[TABLE]

Case 1: $|{\textsc{OPT}}_{\sigma}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|>|{\textsc{OPT}}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|$.

By Eq. 2, we have $|({\textsc{OPT}}\setminus\mathbf{W})\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|=|{\textsc{OPT}}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|$. So,

[TABLE]

Case 2: $|{\textsc{OPT}}_{\sigma}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|\leq|{\textsc{OPT}}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|$.

Suppose, for a contradiction, that $|\mathbf{W}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|\geq 2$. By Eq. 2, we have $|({\textsc{OPT}}\setminus\mathbf{W})\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|=|{\textsc{OPT}}_{\sigma}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|$. So,

[TABLE]

Let $e$ be an arbitrary element in $\mathbf{W}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}$. Note that $\mathbf{W}\subseteq{\textsc{OPT}}$, $\alpha_{\min}(({\textsc{OPT}}\cap\mathbf{V}^{(1:t)})\setminus\{e\})\leq\alpha_{\min}({\textsc{OPT}}\cap\mathbf{V}^{(1:t)})$, and the inequality is strict only when removing $e$ causes the min-ratio to worsen. As ${\textsc{OPT}}_{\sigma}$ and OPT are both type feasible, we have $\alpha_{\min}({\textsc{OPT}}_{\sigma}\cap\mathbf{V}^{(1:t)})=\alpha_{\min}({\textsc{OPT}}\cap\mathbf{V}^{(1:t)})$ for any time step $t\in[h]$, and so

[TABLE]

for all $t\in[h]$ and $q\in[r]$. In particular, we have

[TABLE]

at time step $t-1$ and any $q\in[r]$. Now, since we have $|{\textsc{OPT}}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|\geq 2+|{\textsc{OPT}}_{\sigma}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|$ from above, so it must be the case that removing $e$ of $\mathrm{type}(e)=q$ does not worsen the min-ratio. So,

[TABLE]

Let $e_{\min}\in\mathbf{V}^{(1:t)}\setminus{\textsc{OPT}}$ be an element with $\mathrm{type}(e_{\min})\in\mathbf{Q}({\textsc{OPT}}\cap\mathbf{V}^{(1:t)})$. Then, the set $\mathbf{S}=({\textsc{OPT}}\cap\mathbf{V}^{(1:t)})\setminus\{e\}\cup\{e_{\min}\}$ satisfies $|\mathbf{S}|=|{\textsc{OPT}}\cap\mathbf{V}^{(1:t)}|$ and

[TABLE]

We should not have $\alpha_{\min}(\mathbf{S})>\alpha_{\min}({\textsc{OPT}}\cap\mathbf{V}^{(1:t)})$ since OPT is supposed to be type feasible. However, if $\alpha_{\min}(\mathbf{S})=\alpha_{\min}({\textsc{OPT}}\cap\mathbf{V}^{(1:t)})$, then we have $|\mathbf{Q}(\mathbf{S})|<|\mathbf{Q}({\textsc{OPT}}\cap\mathbf{V}^{(1:t)})|$ because $\mathbf{S}$ has the additional element $e_{\min}$ with $\mathrm{type}(e_{\min})\in\mathbf{Q}({\textsc{OPT}}\cap\mathbf{V}^{(1:t)})$. This contradicts the lemma assumption that OPT is type feasible. So, we cannot have $|\mathbf{W}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|\geq 2$, i.e., $|\mathbf{W}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|\leq 1$. ∎

See 9

Proof.

Let $\mathcal{I}\subseteq 2^{\mathbf{V}}$ be the independence set corresponding to the type feasibility of ${\textsc{OPT}}_{\sigma}$ and define $\mathbf{W}\subseteq{\textsc{OPT}}$ to minimize the marginal gain:

[TABLE]

Since $f$ is non-decreasing, the optimization will try to pick a $\mathbf{W}$ that is as small as possible, subject to ${\textsc{OPT}}\setminus\mathbf{W}$ satisfying the independence set constraints of $\mathcal{I}$.

Fix a time step $t\in[h]$ and type $q\in[r]$. By minimality of $\mathbf{W}$, we know that $|\mathbf{W}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|\leq|{\textsc{OPT}}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|-|{\textsc{OPT}}_{\sigma}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|$. Putting together, we get

[TABLE]

By Lemma 18, we know that $|\mathbf{W}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|\leq 1$, and therefore there exist $k$ disjoint subsets $\mathbf{Y}_{1},\ldots,\mathbf{Y}_{k}\subseteq({\textsc{OPT}}\setminus\mathbf{W})\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}$ such that $|\mathbf{Y}_{i}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|=|\mathbf{W}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|$ and ${\textsc{OPT}}\setminus\mathbf{Y}_{i}\in\mathcal{I}$ for each index $i\in[k]$. Writing $\mathbf{Z}={\textsc{OPT}}\setminus\mathbf{W}$, observe that

[TABLE]

where $(\ast)$ is because $\uplus_{i=1}^{k}\mathbf{Y}_{i}\subseteq\mathbf{Z}$ and $f$ is non-decreasing, and $({\ddagger})$ is because $\uplus_{j=1}^{i-1}\mathbf{Y}_{j}\subseteq{\textsc{OPT}}\setminus\mathbf{Y}_{i}$ and $f$ is submodular. Therefore,

[TABLE]

where the last inequality is due to optimality of ${\textsc{OPT}}_{\sigma}$ and since $\mathbf{Z}={\textsc{OPT}}\setminus\mathbf{W}\in\mathcal{I}$ by choice of $\mathbf{W}$. That is, $\mathbf{Z}$ satisfies the type feasible constraints of ${\textsc{OPT}}_{\sigma}$. ∎

We now prove some additional properties about $\sigma$-type feasible selections (Definition 8) that will be useful in our proofs in later sections.

Definition 19** ($b$-minimal $\sigma$-type feasible set).**

For any $b\in\mathbb{N}$, we say that a subset $\mathbf{S}\subseteq\mathbf{V}$ is a $b$-minimal $\sigma$-type feasible set if $1\leq|\mathbf{S}|\leq b$ and

[TABLE]

Lemma 20**.**

Let $\mathbf{S}$ be a $\sigma$-type feasible set of size $|\mathbf{S}|=b$. Then, we have $\mathbf{S}^{\prime}\subseteq\mathbf{S}$ for any $b$-minimal $\sigma$-type feasible set $\mathbf{S}^{\prime}$ such that

[TABLE]

Proof.

Suppose, for a contradiction, that there is some $(a^{\prime},b)$-minimal $\sigma$-type feasible set $\mathbf{S}^{\prime}$ such that $\mathbf{S}^{\prime}\not\subseteq\mathbf{S}$. Then, there exists some element $e\in\mathbf{S}^{\prime}\setminus\mathbf{S}$ such that $q^{*}=\mathrm{type}(e)$ and $|\mathbf{S}^{\prime}\cap\mathbf{T}_{q^{*}}|>|\mathbf{S}\cap\mathbf{T}_{q^{*}}|$. So, $|(\mathbf{S}^{\prime}\setminus\{e\})\cap\mathbf{T}_{q^{*}}|\geq|\mathbf{S}\cap\mathbf{T}_{q^{*}}|$. Then, we see that

[TABLE]

However, this is a contradiction to Eq. 4. ∎

Lemma 21 implies that any $\sigma$-type feasible solution of length $b$ is an extension of another $\sigma$-type feasible solution of length $b-1$.

Lemma 21**.**

For any length $b\in\mathbb{N}_{>0}$, let $\mathbf{S}_{b}$ be a $\sigma$-type feasible selection of length $b$ and $\mathbf{S}_{b-1}$ be a $\sigma$-type feasible selection of length $b-1$. Then, $|\mathbf{S}_{b}\setminus\mathbf{S}_{b-1}|=1$.

Proof.

We first show that

[TABLE]

Suppose, for a contradiction, that Eq. 5 is false. Then, any strict superset of $\mathbf{A}\supset\mathbf{S}_{b-1}$ of size $b$ will have

[TABLE]

since adding elements can only increase the numerator. This contradicts the assumption that $\mathbf{S}_{b}$ is a $\sigma$-type feasible selection of size $b$, so Eq. 5 must be true.

Now, suppose that $\min_{q\in[r]}\frac{|\mathbf{S}_{b-1}\cap\mathbf{T}_{q}|}{p_{q}\cdot b}=0$. Then,

[TABLE]

Since $\mathbf{S}_{b-1}$ is $\sigma$-type feasible, it must be the case that $\mathbf{S}_{b-1}\subseteq\mathbf{Q}(\emptyset)$ and $\mathbf{S}_{b-1}$ comprises of elements following the $\sigma$ ordering. If $\alpha_{\min}(\mathbf{S}_{b})=0$ as well, then so does $\mathbf{S}_{b}$. Otherwise, if $\alpha_{\min}(\mathbf{S}_{b})>0$, then we must have $\mathbf{Q}(\emptyset)\subseteq\mathbf{S}_{b}$. Both cases implies that $\mathbf{S}_{b-1}\subset\mathbf{S}_{b}$ and so $|\mathbf{S}_{b}\setminus\mathbf{S}_{b-1}|=1$. Note that we cannot have $\alpha_{\min}(\mathbf{S}_{b-1})>\alpha_{\min}(\mathbf{S}_{b})$ due to Eq. 5.

Given the argument above, we may assume that $\min_{q\in[r]}\frac{|\mathbf{S}_{b-1}\cap\mathbf{T}_{q}|}{p_{q}\cdot b}>0$ in the remaining of the proof. This implies that there will exist non-empty $b$-minimal $\sigma$-type feasible sets which is a necessary condition for us to invoke Eq. 5.

Suppose $\mathbf{S}_{b-1}$ is a $b$-minimal $\sigma$-type feasible set. Then, under Eq. 5, Lemma 20 yields $\mathbf{S}_{b-1}\subseteq\mathbf{S}$, so $|\mathbf{S}_{b}\setminus\mathbf{S}_{b-1}|=1$.

Now, suppose that $\mathbf{S}_{b-1}$ is not a $b$-minimal $\sigma$-type feasible set.

Case 1: Suppose $\min_{q\in[r]}\frac{|\mathbf{S}_{b-1}\cap\mathbf{T}_{q}|}{p_{q}\cdot b}=\min_{q\in[r]}\frac{|\mathbf{S}_{b}\cap\mathbf{T}_{q}|}{p_{q}\cdot b}$.

Let $\mathbf{A}$ be a $b$-minimal $\sigma$-type feasible subset such that

[TABLE]

By Lemma 20, we have both $\mathbf{A}\subseteq\mathbf{S}_{b-1}$ and $\mathbf{A}\subseteq\mathbf{S}_{b}$. Since $\mathbf{S}_{b-1}$ is $\sigma$-type feasible, it maximizes $\beta(b-1)$ where

[TABLE]

Since $\frac{b}{b-1}$ is a constant independent of $q\in[r]$, we see that $\mathbf{A}$ also maximizes $\beta(b-1)$. Since a $\sigma$-type feasible selection must minimize $|\mathbf{Q}(\mathbf{S})|$ and $\mathbf{A}$ is a minimal subset, we see that $\mathbf{S}^{\prime}\setminus\mathbf{A}\subset\mathbf{Q}(\mathbf{A})$. Crucially, note that $\mathbf{Q}(\mathbf{A})\setminus\mathbf{S}^{\prime}\neq\emptyset$, otherwise $\min_{q\in[r]}\frac{|\mathbf{A}\cap\mathbf{T}_{q}|}{p_{q}\cdot b}<\min_{q\in[r]}\frac{|\mathbf{S}^{\prime}\cap\mathbf{T}_{q}|}{p_{q}\cdot b}$. Following a similar argument, one can show that $\mathbf{S}_{b}\setminus\mathbf{A}\subset\mathbf{Q}(\mathbf{A})$. Now, since both $\mathbf{S}_{b-1}$ and $\mathbf{S}_{b}$ are $\sigma$-type feasible, they must select from $\mathbf{Q}(\mathbf{A})$ in the ordering of $\sigma$. In other words, we have $\mathbf{S}_{b-1}\setminus\mathbf{A}\subseteq\mathbf{S}_{b}\setminus\mathbf{A}$, so $\mathbf{S}_{b-1}\subseteq\mathbf{S}_{b}$, and thus $|\mathbf{S}_{b}\setminus\mathbf{S}_{b-1}|=1$.

Case 2: Suppose $\min_{q\in[r]}\frac{|\mathbf{S}_{b-1}\cap\mathbf{T}_{q}|}{p_{q}\cdot b}<\min_{q\in[r]}\frac{|\mathbf{S}_{b}\cap\mathbf{T}_{q}|}{p_{q}\cdot b}$.

Let $\mathbf{B}$ be a $b$-minimal $\sigma$-type feasible subset such that

[TABLE]

By Lemma 20, we have both $\mathbf{B}\subseteq\mathbf{S}_{b-1}$ and $\mathbf{B}\subseteq\mathbf{S}_{b}$. Following a similar argument as Case 1, we see that $\mathbf{S}_{b-1}\setminus\mathbf{B}\subset\mathbf{Q}(\mathbf{B})$. However, since $\min_{q\in[r]}\frac{|\mathbf{B}\cap\mathbf{T}_{q}|}{p_{q}\cdot b}<\min_{q\in[r]}\frac{|\mathbf{S}_{b}\cap\mathbf{T}_{q}|}{p_{q}\cdot b}$, we must have $\mathbf{Q}(\mathbf{B})\subseteq\mathbf{S}$. In other words, we get $\mathbf{S}_{b-1}\subseteq\mathbf{S}_{b}$ and thus $|\mathbf{S}_{b}\setminus\mathbf{S}_{b-1}|=1$. ∎

To operationalize the abstract requirements of $\sigma$-type feasible selections (Definition 8), let us define the notion of min-ratio type sequences (Definition 22).

Definition 22** (Min-ratio type sequence).**

Let $\mathbf{p}=(p_{1},\ldots,p_{r})\in\mathbb{R}_{\geq 0}^{r}$ be a vector of lower distributional bounds, $\sigma:[r]\to[r]$ be a total order over types, $b$ be a size and let $d\in\mathbb{N}$ be a fixed denominator. The min-ratio type sequence $\mathbf{s}(\mathbf{p},\sigma,b,d)\in[r]^{\mathbb{N}}$ is defined by starting with the empty sequence and repeating the following steps $b$ times: append to the end of $\mathbf{s}(\mathbf{p},\sigma,b,d)$ the type $q\in[r]$ that minimizes $\frac{\#(q,\mathbf{s}(\mathbf{p},\sigma,b,d))}{p_{q}\cdot d}$, breaking ties according to the preference ordering $\sigma$. Since $p$ and $\sigma$ are fixed inputs to any problem instance, we simply write $\mathbf{s}(b,d)$ to mean $\mathbf{s}(\mathbf{p},\sigma,b,d)$.

Lemma 23 shows that Definition 22 is independent of the denominator, which will be useful in our analysis later.

Lemma 23**.**

For fixed $\mathbf{p}\in\mathbb{R}_{\geq 0}^{r}$, $\sigma:[r]\to[r]$, and $b\in\mathbb{N}$, we have $\mathbf{s}(b,d_{1})=\mathbf{s}(b,d_{2})$ for any $d_{1},d_{2}\in\mathbb{N}_{>0}$.

Proof.

We will prove this by induction over $b\in\mathbb{N}$.

Base case ($b=0$): When $b=0$, we trivially have both $\mathbf{s}(0,d_{1})$ and $\mathbf{s}(0,d_{2})$ being the empty sequence.

Inductive case ($b>0$): By induction hypothesis, we know that $\mathbf{s}(b-1,d_{1})=\mathbf{s}(b-1,d_{2})$, so $\#(q,\mathbf{s}(b-1,d_{1}))=\#(q,\mathbf{s}(b-1,d_{2}))$ for any $q\in[r]$. Since $\frac{\#(q,\mathbf{s}(b-1,d_{1}))}{p_{q}\cdot d_{1}}=\frac{d_{2}}{d_{1}}\cdot\frac{\#(q,\mathbf{s}(b-1,d_{2}))}{p_{q}\cdot d_{2}}$ for any $q\in[r]$ and $\frac{d_{2}}{d_{1}}$ is a constant independent of $q$, we have $\operatorname*{argmin}_{q\in[r]}\frac{\#(q,\mathbf{s}(b-1,d_{1}))}{p_{q}\cdot d_{1}}=\operatorname*{argmin}_{q\in[r]}\frac{\#(q,\mathbf{s}(b-1,d_{2}))}{p_{q}\cdot d_{2}}$, and thus both sequences will append the same $b$-th type, resulting in $\mathbf{s}(b,d_{1})=\mathbf{s}(b,d_{2})$. ∎

Observe that Definition 8 and Definition 22 are both defined independently of the budget constraints across time. The former is an abstract axiomatic definition of the desired properties motivated from our Ethiopian application while the latter is a specific concrete way of choosing a sequence of types. Interestingly, Lemma 24 tells us that multisets of types produced by the two definitions are the same.

Lemma 24**.**

For any length $b\in\mathbb{N}$, there is a unique length $b$ $\sigma$-type feasible selection $\mathbf{S}\subseteq\mathbf{V}$ satisfying

[TABLE]

Proof.

The claim trivially holds when $r=1$, so we shall consider $r>1$. We will prove this by induction over $b\in\mathbb{N}$.

Base case ($b=1$): Let $\mathbf{S}$ be any type feasible solution of size $|\mathbf{S}|=1$ according to Definition 1. Since $r>1$, we have $\beta(1)=0$, there exists some $q\in[r]$ such that $\mathbf{S}\cap\mathbf{T}_{q}=\emptyset$, and $|\mathbf{Q}(\mathbf{S})|=r-1$. Since $\mathbf{S}$ is type feasible, it has to break ties according to $\sigma$. Thus, $\mathbf{S}=\{q\in[r]:\sigma(q)=1\}$ is the unique type feasible solution of size $1$. Meanwhile, Definition 22 produces $\mathbf{s}(1,1)=(q^{*})$, where $\sigma(q^{*})=1$. This is because all ratios are [math] initially and ties are broken according to $\sigma$. So, $\mathbf{S}=\mathbf{s}(1,1)$ as desired.

Inductive case ($b>1$): Let $\mathbf{S}_{b-1}$ be any type feasible solution of length $|\mathbf{S}_{b-1}|=b-1$. By induction hypothesis, $\mathbf{S}_{b-1}$ is unique and $\mathbf{S}_{b-1}=\mathbf{s}(b-1,b-1)$. By Lemma 23, we know that $\mathbf{s}(b-1,b-1)=\mathbf{s}(b-1,b)$. Since $\mathbf{S}_{b-1}=\mathbf{s}(b-1,b)$, we have $|\mathbf{S}_{b-1}\cap\mathbf{T}_{q}|=\#(q,\mathbf{s}(b-1,b))$ for all types $q\in[r]$. By definition, both $\mathbf{s}(b-1,b)$ will extend by choosing something from $\mathbf{Q}(\mathbf{S}_{b-1})$ and break ties according to $\sigma$. Meanwhile, we know from Lemma 21 that $\mathbf{S}_{b-1}\subset\mathbf{S}_{b}$, so we can also obtain $\mathbf{S}_{b}$ by extending $\mathbf{S}_{b-1}$ by choosing something from $\mathbf{Q}(\mathbf{S}_{b-1})$ and break ties according to $\sigma$. As such, both sequences will extend by appending the same type, we have $\mathbf{S}=\mathbf{s}(b,b)$ as desired. ∎

For convenience, we reproduce MultistepPlanning below, which was designed with Definition 22 in mind. Lemma 25 below tells us that any choice of independent set of size $b^{(t)}$ at each time step by MultistepPlanning results in the same multiset of types as the prefix of the min-ratio type sequence of length $\sum_{t=1}^{h}b^{(t)}$. Note that MultistepPlanning may not choose the elements whose type sequence agrees exactly with Definition 22, but the total counts will agree for any given budget.

Lemma 25**.**

For any time step $t\in[h]$, let $b^{(1:t)}=\sum_{\tau=1}^{t}b^{(t)}$ and $\mathcal{M}^{(t)}=(\mathbf{V}^{(t)},\mathcal{I}^{(t)})$ be the matroid defined in MultistepPlanning. Then, we have $\mathbf{m}^{t}_{1}=\mathbf{m}^{t}_{2}$ for all $t\in[h]$, where multisets $\mathbf{m}^{t}_{1},\mathbf{m}^{t}_{2}\in[r]^{b^{(1:t)}}$ are defined as

[TABLE]

and

[TABLE]

Proof.

We will prove this claim via an induction over $t\in\{1,\ldots,h\}$.

Base case ($t=1$): By definition of $\mathcal{M}^{(1)}$, any independent set of size $b^{(1)}$ in $\mathcal{I}^{(1)}$ involves $x^{(1)}_{q}=\#(q,\mathbf{s}(b^{(1)},b^{(1)}))$ items of type $q\in[r]$ such that $\sum_{q\in[r]}x^{(1)}_{q}=b^{(1)}$. So, the multisets $\mathbf{m}^{(t)}_{2}=\mathbf{m}^{(t)}_{1}$ both count the types appearing in $\mathbf{s}(b_{1},b_{1})$.

Inductive case ($t>1$): By definition of $\mathcal{M}^{(t)}$, any independent set of size $b^{(t)}$ in $\mathcal{I}^{(t)}$ involves

[TABLE]

items of type $q\in[r]$ such that $\sum_{q\in[r]}x^{(t)}_{q}=b^{(t)}$. Let $\mathbf{x}^{(t)}$ be the multiset of size $b^{(t)}$ such that $q\in[r]$ appears $x^{(t)}_{q}$ times. That is, the multiset $\mathbf{x}^{(t)}$ counts the types appearing in the length $b^{(t)}=b^{(1:t)}-b^{(1:t-1)}$ suffix of $\mathbf{s}(b^{(1:t)},b^{(1:t)})$.

By Lemma 23, we also know that $\mathbf{s}(b^{(1:t-1)},b^{(1:t-1)})=\mathbf{s}(b^{(1:t-1)},b^{(1:t)})$, so $\mathbf{s}(b^{(1:t-1)},b^{(1:t-1)})$ is simply the length $b^{(1:t-1)}$ prefix of $\mathbf{s}(b^{(1:t)},b^{(1:t)})$ according to Definition 22. By induction hypothesis, this means that $\mathbf{m}^{(t-1)}_{2}$ is the multiset counting the types appearing in the length $b^{(1:t-1)}$ prefix of $\mathbf{s}(b^{(1:t)},b^{(1:t)})$. So, the multisets $\mathbf{m}^{(t)}_{2}=\mathbf{m}^{(t-1)}_{2}\cup\mathbf{x}^{(t)}=\mathbf{m}^{(t)}_{1}$ both count the types appearing in $\mathbf{s}(b^{(1:t)},b^{(1:t)})$. ∎

We are now ready to prove our Theorem 10 which states the theoretical guarantees of MultistepPlanning with respect to $\sigma$-type feasible selections.

See 10

Proof.

In each time step $t\in[h]$, MultistepPlanning can compute the $x^{(t)}_{q}$ values in $\mathcal{O}(r\cdot b^{(t)})$ time by appropriate bookkeeping, then perform $b^{(t)}$ greedy selections, each requiring the calculation of at most $|\mathbf{V}^{(t)}|$ marginal gains. The overall running time across all time steps is $\mathcal{O}(\sum_{t=1}^{h}b^{(t)}\cdot(r+|\mathbf{V}^{(t)}|))$.

We now show the three items separately.

1. We always choose an independent set containing elements from $\mathbf{V}^{(t)}$ at each time step $t\in[h]$, up to size $b^{(t)}$.

2. Lemma 24 and Lemma 25 jointly ensures that any output of MultistepPlanning is a $\sigma$-type feasible selection.

3. Lemma 24 and Lemma 25 jointly ensures that any $\sigma$-type feasible solution is a possible output of Algorithm 2. As $\mathbf{V}^{(1)},\ldots,\mathbf{V}^{(h)}$ are disjoint, $\mathcal{I}_{1},\ldots,\mathcal{I}_{h}$ jointly induce a single partition matroid which enforces $\sum_{q=1}^{r}x^{(t)}_{q}=b^{(t)}$ at each time step $t\in[h]$. Since MultistepPlanning is performs greedy selection at each time step over a single matroid, it yields a $\frac{1}{2}$-approximation to the offline optimum selection at any time step $t\in[h]$; just like LocalGreedy with $k=1$. ∎

Appendix E Deferred technical details for Section 4.3

Here, we provide the full proofs of the impossibility results stated in Section 4.3.

See 11

Proof.

Consider an arbitrary MOPGP instance $\mathcal{M}$ with $r$ types for which there is some timestep $t$ such that ALG maximizes $\beta(b^{(1:t)})$ but did not minimize $|\mathbf{Q}(\mathbf{S}^{(t)})|$. Suppose $|\mathbf{Q}(\mathbf{S}^{(t)})|=k\leq r$ while it could have been of size $\ell<k$. We now define a new MOPGP instance $\mathcal{M}^{\prime}$ obtained by using the prefix of $\mathcal{M}$ up to time $t$, and adding one additional time step $b^{(t+1)}=\ell$. Since $\mathcal{M}$ and $\mathcal{M}^{\prime}$ are indistinguishable up to time step $t$, and ALG is deterministic, ALG must choose the same selections up to time step $t$, resulting in $|\mathbf{Q}(\mathbf{S}^{(t)})|=k>\ell$. Thus, for the final time step $t+1$ for $\mathcal{M}^{\prime}$, ALG would have $\beta(b^{(1:t+1)})=\beta(b^{(1:t)})$ instead of being able to improve the satisfaction ratio if it had minimized $|\mathbf{Q}(\mathbf{S}^{(t)})|$ at time step $t$ to be of size $\ell$. ∎

We prove Proposition 12 (reproduced below for convenience) separately for the bound of $\frac{1}{2}$ and $\frac{k}{k+1}$ in Proposition 26 and Proposition 27 respectively.

See 12

Proof.

See Proposition 26 and Proposition 27. ∎

Proposition 26**.**

There exists instances where MultistepPlanning achieves an approximation ratio of at most $\frac{1}{2}$ due to budget uncertainties.

Proof.

Fix values $x,\varepsilon\geq 0$, where $\varepsilon$ is to be treated as a vanishingly small value. Suppose $r=1$, $h=2$, $b^{(1)}=b^{(2)}=1$, $\mathbf{V}^{(1)}=\{a,b\}$, $\mathbf{V}^{(2)}=\{c\}$, and

[TABLE]

One can verify that $f$ is a non-decreasing submodular function. MultistepPlanning will greedily select $\mathbf{S}^{(1)}=\{a\}$ and $\mathbf{S}^{(2)}=\{c\}$, obtaining an objective value of $f(\{a,c\})=x+2\varepsilon$. Meanwhile, an optimal selection that operates without budget uncertainty will select $\mathbf{S}^{(1)}=\{b\}$ and $\mathbf{S}^{(2)}=\{c\}$, obtaining an objective value of $f(\{a,c\})=2x+2\varepsilon$. We see that the competitive ratio is $\lim_{\varepsilon\to 0}\frac{x+2\varepsilon}{2x+2\varepsilon}=\frac{1}{2}$. ∎

Proposition 27**.**

There exists instances where MultistepPlanning achieves an approximation ratio of at most $\frac{k}{k+1}$ due to $\sigma$-ordering.

Proof.

Fix $k\in\mathbb{N}$ and the ordering $\sigma:[r]\to[r]$ which favors smaller indexed types. That is, when tie-breaking for satisfaction ratio, the unselected type(s) have large indices. Suppose $|{\textsc{OPT}}_{\sigma}\cap\mathbf{V}^{(t)}\cap\mathbf{T}_{q}|\geq k$ for all $t\in[h]$ and $q\in[r]$. Now suppose $h=1$, $b^{(1)}=rk+1$, $f(\mathbf{S})=|\mathbf{S}\cap\mathbf{T}_{r}|$, and there are more than $b^{(1)}$ elements of each type. One can check that that $f$ is a non-decreasing submodular function. Then, MultistepPlanning will select $k$ elements of each type and then one element of type $1$ due to $\sigma$, obtaining an objective value of $k$. Meanwhile, an optimal selection will select $k$ elements of each type and then one element of type $r$, obtaining an objective value of $k+1$. Thus, the competitive ratio is $\frac{k}{k+1}$. ∎

Appendix F Results for Sidama Region

F.1 Experiment 1: Impact of budget on coverage

In Fig. 7, the bars show how total coverage improves under different allocation strategies and budget levels over a five-year planning horizon. Since the initial coverage is already high, the measured improvement is marginal.

F.2 Experiment 2: Evaluating district-level equity

Here we examine the minimum satisfaction ratio, $\alpha_{\min}(\mathbf{S})$, achieved under each policy. As shown in Fig. 8, MP(DP$1$) and MP(DP$2$) outperform all other methods in the DP$1$ and DP$2$ metrics respectively, while MP(DP[math]) consistently attains $\alpha_{\min}=0$. Interestingly, although in Fig. 7, building facilities resulted in mild effect on the coverage, the effect on the district-level equity is more noticeable.

F.3 Experiment 3: High quality advice can help

The results in this section are similar to the results in the main paper. Out of around $30$ districts in Sidama, we found $11$ districts in which our learning-augmented selection $\mathbf{U}$ outperforms both the greedy baseline $\mathbf{G}$ and the expert selection $\mathbf{A}$, i.e., $f(\mathbf{U})>f(\mathbf{G})>f(\mathbf{A})$.

Appendix G Further experimental details

All experiments were conducted using Python 3.12 on two systems: (i) a Linux-based server with 48-core Intel Xeon CPUs and 190GB of RAM, and (ii) a Windows laptop running WSL with 8 cores and 16GB of RAM.

Population projections were obtained from the WorldPop dataset777https://data.worldpop.org/GIS/Population/Global_2015_2030/R2024B/2026/ETH/v1/1km_ua/constrained/eth_pop_2026_CN_1km_R2024B_UA_v1.tif. Friction surface data for walking time computation was sourced from the MalariaAtlas GitHub repository888https://github.com/malaria-atlas-project/malariaAtlas, using an R script we provide. Administrative shapefiles were downloaded from the United Nations Office for the Coordination of Humanitarian Affairs (OCHA)999https://data.humdata.org/dataset/cb58fa1f-687d-4cac-81a7-655ab1efb2d0/resource/63c4a9af-53a7-455b-a4d2-adcc22b48d28/download/eth_adm_csa_bofedb_2021_shp.zip.

For Experiment $3$ (retrospective analysis), we fixed a random seed of [math], and generated $10$ permutations of the original advice set to serve as inputs to our learning-augmented algorithm.

Several data inputs (such as district-level rates of home births and postnatal care, as well as facility registries) request permission and thus cannot be shared. To ensure usability, we include code that generates synthetic examples, allowing others to replicate our methods.

Reproducibility focus.

Our goal is to make our methodological contribution reproducible and reusable. All core components of our tool are fully data-agnostic and will yield the same visualizations and planning outputs when provided with actual Ethiopia-specific data. Once these data become publicly available, all results in the main paper can be replicated by plugging them into our codebase. The code is available at https://github.com/yohayt/OHCE/.