High-Dimensional Bipartite Matching System

• The high-dimensional bipartite matching system is a stochastic model where customers and servers from large class sets are paired using a compatibility graph under discrete time arrivals.
• Stability is achieved when arrival intensities and compatibility constraints meet necessary conditions, generalizing Hall’s marriage theorem and verified via network flow methods.
• The Match the Longest (ML) policy plays a critical role by minimizing a quadratic Lyapunov function, ensuring maximal stability even in complex, high-dimensional regimes.

A high-dimensional bipartite matching system describes a stochastic model in which two finite sets of classes—customers and servers—are connected by a compatibility graph and paired according to incoming arrivals and matching policies. The model’s dimensionality reflects the cardinality of customer and server classes, which may each be large (the "high-dimensional” regime). The system evolves in discrete time: at each step, a new customer and server pair arrive together, governed by a joint probability measure over all pairs of classes. If both can be matched according to the graph, they leave the system; otherwise, unmatched entities are buffered. The system thus behaves as a Markov chain over the (potentially large) space of buffer configurations. The stability of this Markov chain—and hence the operational feasibility of the matching system—depends critically on arrival intensities, the structure of the compatibility graph, and the adopted matching policy.

1. Matching Model Formulation

The model is constructed over finite sets $C$ (customer classes) and $S$ (server classes). Authorized matches are specified by a bipartite compatibility graph $G = (C \cup S, E)$, so that a pair $(c, s) \in E$ indicates compatibility. At each time step, one customer (class in $C$) and one server (class in $S$) arrive, distributed according to a joint probability measure $\mu$ on $C \times S$, independently of the past.

Key definitions:

• $\mu_C$ and $\mu_S$ are the marginals of $\mu$ on $C$ and $S$.
• For $U \subsetneq C$, $S(U)$ denotes all servers compatible (connected) with any class in $U$. Dually, for $V \subsetneq S$, $C(V)$ denotes all compatible customer classes.

The system is described by the number of unmatched (buffered) customers and servers in each class, forming a Markov chain over $\mathbb{Z}_+^{|C| + |S|}$.

A matching policy selects which buffered entities are to be matched when multiple choices are possible. Typical policies include:

• ML (Match the Longest): match incoming items to those classes with the largest buffer count.
• MS (Match the Shortest): match to classes with the smallest buffer count.
• FIFO: match to the oldest items.
• Priority: match according to a predefined preference ordering.

2. Necessary and Sufficient Stability Conditions

The foundational contribution is the derivation of necessary (and in some cases also sufficient) conditions for the positive recurrence (stability) of the underlying Markov chain.

Necessary Conditions: $\mathsf{NCond}$

For the existence of a stable buffer process, arrival rates must allow for sufficient matching over all subsets. The necessary condition is:

$$\mathsf{NCond}: \quad \begin{cases} \mu_C(U) < \mu_S(S(U)) & \forall\, U\subsetneq C, \ \mu_S(V) < \mu_C(C(V)) & \forall\, V\subsetneq S. \end{cases}$$

This requirement generalizes Hall’s marriage theorem: in the fluid limit, the flow of incoming customers (or servers) to any subset $U$ must be less than the total arrival rate of compatible servers (or customers). Violation results in persistent backlog.

Sufficient Conditions: $\mathsf{SCond}$

The sufficient condition is expressed by refined facet-based inequalities:

$$\mathsf{SCond}:\quad \mu_C((F)) + \mu_S((F)) > 1 - \mu(E \cap ((F) \times (F))) \quad \forall\, F \in \mathfrak{F} \setminus\{\emptyset\},$$

where $\mathfrak{F}$ indexes the nonzero facets, i.e., possible configurations of buffer occupancy. If all nonzero facets are “saturated,” then $\mathsf{SCond}$ and $\mathsf{NCond}$ coincide, and the stability region is maximal.

Maximality refers to the case where the set of $\mu$ stabilizing the Markov chain is exactly those satisfying $\mathsf{NCond}$; in some instances, only a subset is achievable, depending on the matching policy.

3. Influence of Matching Policy

The choice of matching policy dramatically influences the stability region:

• ML (Match the Longest): For any compatibility graph, the ML policy is proved to yield a maximal stability region. It achieves this by minimizing a quadratic Lyapunov function

$$L(x, y) = \sum_{c \in C} x_{c}^{2} + \sum_{s \in S} y_{s}^{2},$$

where $x_c$ and $y_s$ are the buffer counts for customer class $c$ and server class $s$. This ensures a negative drift outside compact sets and equilibrium for all $\mu$ satisfying $\mathsf{NCond}$.

• MS (Match the Shortest), Priority: There exist bipartite graphs (e.g., the "NN" structure with two or three classes per side) where the stability region under MS or certain priorities is strictly smaller than predicted by $\mathsf{NCond}$. This results from non-saturation of some facets; matching policies may inadvertently “trap” certain classes in the buffer, failing to dissipate backlog sufficiently even when $\mathsf{NCond}$ holds.
• FIFO and Random: The question of maximality remains open. Numerical evidence suggests that the stability region likely remains maximal under these policies, though no proof is supplied.

4. Scaling and Structural Complexity in High Dimensions

Increasing the number of customer and server classes, i.e., moving to high-dimensional models, introduces exponential complexity in the number of facets ($2^{|C| + |S|}$). Each facet can potentially generate a new stability constraint.

Key implications:

• For graphs where all nonzero facets are saturated, $\mathsf{SCond}$ and $\mathsf{NCond}$ still coincide, and maximality is preserved despite high dimension.
• If certain facets are non-saturated (possible in high-dimensional graphs with intricate compatibility structures), stability constraints become more restrictive, and $\mathsf{SCond}$ is strictly stronger than $\mathsf{NCond}$.
• Checking $\mathsf{NCond}$ by enumeration is computationally infeasible for large $|C|,|S|$, but the conditions correspond to a max-flow problem. Thus, feasibility can be checked in time $O((|C|+|S|)^3)$ by reduction to network flow.

Algorithmic and policy consequences:

• ML’s robustness is especially advantageous in high dimensions, as its quadratic Lyapunov minimization is independent of the underlying graph's combinatorics.
• MS and some priority policies risk drastic stability region shrinkage as the combinatorial complexity increases.

5. Lyapunov Techniques and Maximal Stability

For $\mathsf{ML}$, a quadratic Lyapunov function $L(x,y)$ is constructed, tracking squared buffer sizes. The ML policy minimizes this function over all possible matching decisions at each arrival step, guaranteeing sufficient negative drift for all arrival distributions in $\mathsf{NCond}$. This approach generalizes well as system dimension increases.

For other policies, the Lyapunov function may not always decrease—certain buffering dynamics can persist even when overall arrival intensities are feasible, particularly when the policy fails to prioritize overloaded classes.

The table below summarizes the impact of policy and structure:

Policy	Maximal Region?	Lyapunov Function	Effect in High Dimensions
ML (Longest)	Yes (all graphs)	Quadratic	Robust, balances all classes
MS (Shortest)	Not always	Linear unstable	Region may be much smaller
Priority	Not always	Policy-dependent	Structure-specific failures
FIFO, Random	Unknown	Not explicit	Suggestively maximal

6. Operational Insights and Practical Implications

The analysis yields several practical takeaways for high-dimensional bipartite matching system design and operation:

• Ensuring stability for all feasible arrival rates (i.e., achieving the maximal region) is guaranteed only under ML across all graphs and dimensions. MS, priority, or ad-hoc policies risk instability even when no overload exists by $\mathsf{NCond}$.
• Algorithmic checking of stability—although exponentially complex by direct enumeration—can be performed efficiently via network flow algorithms, making analysis scalable despite extreme system dimension.
• As matching graph complexity grows, the distinction between maximal and non-maximal policies becomes pronounced; policy robustness is crucial in large systems (e.g., multi-class skill-based queues, cloud allocation, or kidney exchanges).

7. Summary

• The stability of a high-dimensional bipartite matching system is characterized by polyhedral necessary conditions ($\mathsf{NCond}$); for many graphs and policies (notably ML), these are also sufficient.
• Matching policies have sharply different stability characteristics: ML provides strong maximality and robustness, while MS and certain priorities may suffer degraded stability, especially as system dimension increases.
• The Lyapunov approach with quadratic functionals generalizes well and is central in guaranteeing maximality for robust policies in high dimensions.
• In high-dimensional operational regimes, dependence on the combinatorial complexity of compatibility graphs is mitigated by flow-based algorithms and policy choice, with ML offering a strong practical safeguard against stability loss.

This theoretical framework underpins the design of robust, scalable resource allocation and matching systems in modern high-dimensional applications (Bušić et al., 2010).