Cheapest-Feasible Matching (CFM) Overview

• Cheapest-Feasible Matching (CFM) is a real-time minimal-cost matching rule that pairs agents in bipartite markets under strict feasibility constraints.
• The algorithm greedily selects the cheapest available provider for each demand, ensuring a unique stable matching with provable cost and regret bounds.
• CFM exhibits robust performance with analytical guarantees in random graph settings and dynamic perishable-resource markets, demonstrating incentive alignment and efficiency.

Cheapest-Feasible Matching (CFM) is a real-time allocation rule, central to market design and matching theory, that determines the unique minimal-cost, feasible matching between agents on opposite sides of a bipartite interaction—such as compute providers and computational jobs, or vertices of a graph with random costs. CFM appears both in mean-field stable matchings for random bipartite graphs and as a mechanism design primitive in perishable-resource markets, with robust performance and incentive alignment guarantees in each domain (Ahlberg et al., 7 Jun 2024, Zang et al., 20 Nov 2025).

1. Formal Problem Structure

The core problem considers two sets: a supply side $\S$ (e.g., compute providers, vertices) and a demand side $\D$ (e.g., jobs, opposing vertices). Each provider or vertex $s$ has an availability $\tau_s$ and a cost $\hat c_s$, while each job or vertex $d$ demands $w_d$ units of service or matches, subject to deadlines and budgets when formulated as a perishable utility market. The matching must be partial—each provider matches at most once, and each job at most once—with feasibility constraints (e.g., sufficient remaining provider time, or match-respecting blocking-pair avoidance) (Zang et al., 20 Nov 2025, Ahlberg et al., 7 Jun 2024).

In the canonical random matching case, the bipartite graph $K_{n,n}$ is weighted by i.i.d. $\mathrm{Exp}(1)$ edge costs. A matching is stable (or cheapest-feasible) if it contains no blocking-pair: an unmatched edge whose cost is less than the minimum cost of either endpoint’s current match (Ahlberg et al., 7 Jun 2024).

2. CFM: Mathematical Rule and Algorithm

CFM proceeds greedily, selecting for each arriving job the cheapest available provider capable of fulfilling the required service, resorting to a maximal-availability fallback if no eligible provider remains. Formally, with providers $\S=\{s_1,\ldots,s_m\}$ and jobs $\D=\{d_1,\ldots,d_n\}$ each with demand $w_j$, the matching $\M\subseteq \S\times\D$ is built by iterating over jobs:

1. For job $d_j$, define the feasible provider set $\F_j = \{s\in \S\setminus\operatorname{dom}(\M):\tau_s\geq w_j\}$.
2. If $\F_j\neq\emptyset$, pick $s_j^* \in \arg\min_{s\in\F_j} \hat c_s$ (cheapest feasible provider).
3. Otherwise, pick $s_j^*$ as the remaining provider with $\max \tau_s$.
4. Update the matching: $\M\leftarrow \M\cup\{(s_j^*,d_j)\}$, remove $s_j^*$ from the set of eligible providers.

Each tie is broken by a fixed ordering over $\S$ to guarantee uniqueness (Zang et al., 20 Nov 2025).

In the exponential edge-weighted setting, this rule is equivalent to sequentially pairing the two unmatched vertices connected by the minimal remaining edge, removing all incident edges after each step—yielding the unique stable (cheapest-feasible) matching (Ahlberg et al., 7 Jun 2024).

3. Theoretical Guarantees and Limit Laws

Existence and Uniqueness

For arbitrary arrival sequences and availability/cost vectors, CFM always produces a well-defined matching, never gets stuck, and is unique once tie-breaking is fixed (Zang et al., 20 Nov 2025). In the mean-field random graph setting, the greedy-edge algorithm gives the solitary stable matching (Ahlberg et al., 7 Jun 2024).

Cost and Regret Bounds

Define two offline optima: GCM (greedy-cheapest-ever, ignoring feasibility) and GSM (greedy-shortest-ever, maximizing completed jobs). Supply-regret and demand-regret are given by

• $R_S(\mathrm{CFM}) = \sum_{(s,d)\in\M_\mathrm{CFM}} \hat c_s - \min_\M \sum_{(s,d)\in \M} \hat c_s$,
• $R_D(\mathrm{CFM}) = (\max_\M \#\{\text{feasible matches}\}) - \#\{\text{feasible matches in } \M_\mathrm{CFM}\}$.

CFM is $1/2$-competitive in size:

• For $n\leq m$ jobs and providers, $R_D(\mathrm{CFM})\leq\lfloor n/2\rfloor$.
• When $\hat c_s \geq P_f$ ($P_f$ a floor price), supply regret satisfies $R_S(\mathrm{CFM}) \leq \lfloor m/2\rfloor (P-P_f)$ (Zang et al., 20 Nov 2025).

Distributional Results in Random Setting

In $K_{n,n}$ with i.i.d. $\mathrm{Exp}(1)$ edge costs, the total CFM matching cost $C_{n,n}$ satisfies

• $\mathbb{E}[C_{n,n}] = H_n = \ln n + \gamma + o(1)$ (harmonic sum; $\gamma$ Euler–Mascheroni constant),
• $\operatorname{Var}(C_{n,n}) = \pi^2/6 + o(1)$,
• $C_{n,n} - \ln n$ converges to Gumbel$(0,1)$,
• Typical edge costs are $O(1/n)$ with explicit density $f_W(x)=(1+x)^{-2}$ for $x\geq 0$ (Ahlberg et al., 7 Jun 2024).

The rank of a typical edge (its position among candidate edge costs) exhibits a power-law tail with infinite mean: $\mathbb{P}(R\geq r)\sim 1/r$ as $r\to\infty$.

4. Computational Aspects

A practical implementation of CFM maintains a balanced data structure (segment tree or BST) keyed by remaining availability $\tau_s$ and a min-heap at each node by $\hat c_s$, supporting:

• Range-minimum queries in $O(\log m)$ time for providers with sufficient availability,
• Popping minimum-cost feasible providers and updating availability in $O(\log m)$,
• Per-period CFM execution in $O(n\log m)$ time with $O(m)$ space for $n$ jobs and $m$ providers (Zang et al., 20 Nov 2025).

In random exponential cost settings, the entire matching process can be represented as a sum of independent exponential increments, enabling exact analysis of matching cost distributions (Ahlberg et al., 7 Jun 2024).

5. Incentive Properties and Market Mechanism Design

CFM alone prioritizes lower-cost providers, but truthful cost reporting and early availability staking are only achieved when paired with a suitable surplus-sharing payment mechanism. The premium-sharing pool aggregates the surplus (market price minus reported costs) and distributes it equally among contemporaneously matched providers whose jobs started at or after a reference time:

• Providers' payoff per hour is $$\hat c_s + \frac{1}{|\S^t_{\geq h}|}\sum_{s’\in\S^t_{\geq h}}(P^t-\hat c_{s’})$$.
• Under mild conditions, this structure ensures (i) earlier staking increases total pool participation and (ii) reporting lower $\hat c_s$ increases matching probability, with exactly offsetting marginal incentives, yielding truthful cost revelation (Zang et al., 20 Nov 2025).

By combining CFM with the premium-sharing pool, the mechanism is incentive compatible: providers optimally report true costs and stake full available capacity immediately.

6. Robustness and Sensitivity

Overlap Robustness

CFM exhibits structural robustness to small perturbations in underlying cost data. In $K_{n,n}$ with i.i.d. exponential edge costs, resampling an $\epsilon$ fraction of costs preserves the majority of the matching: the expected fraction of unchanged edges converges to $1-O(\epsilon/\log(1/\epsilon))$ as $\epsilon\to 0$ and $n\to\infty$ (Ahlberg et al., 7 Jun 2024). This is attributed to the local nature of greedy matching, which depends on descending chains of edge weights that, for bounded-length sequences, remain typically unaffected by small random perturbations.

Cost and Tail Noise Sensitivity

Despite this overlap robustness, the total CFM cost is highly sensitive to noise: resampling an $\epsilon$-fraction of costs causes the most expensive $m$ edge assignments (for $m\ll \epsilon \ln n$) to be completely reshuffled with high probability. Consequently, nearly all variance in total cost derives from these tail edges, and their independence under resampling drives the cost correlation $\operatorname{Corr}(C_{n,n},C_{n,n}')\to 0$ for $\epsilon \ln n\to\infty$ (Ahlberg et al., 7 Jun 2024).

Multi-period Regret

In multi-period regimes with a finite number of providers, CFM demonstrates vanishing long-run regret compared to the feasible optimum as total service capacities increase, reflecting robustness in dynamic, adversarially structured demand scenarios (Zang et al., 20 Nov 2025).

7. Applications and Broader Connections

CFM serves as (i) the basis for large-scale, realistically implementable market platforms for perishable utility resources such as compute (where it enables near real-time matching, robust incentive alignment, and provable regret bounds) (Zang et al., 20 Nov 2025), and (ii) as an analytically tractable model of stable matching in mean-field random graph settings, yielding new limit theorems and robust matching algorithms (Ahlberg et al., 7 Jun 2024). The explicit links between CFM and classic stable matching concepts, their uniquely greedy realization, and the connection to Gumbel fluctuation limits and power-law edge ranks offer a rich foundation for both theoretical investigation and market infrastructure design.