SSCFLP: Single-Source Capacitated Facility Location

• SSCFLP is a combinatorial optimization problem that selects capacity-limited facility sites and uniquely assigns customers to minimize fixed opening and transportation costs.
• It is modeled via 0–1 mixed-integer programming and advanced LP relaxations, facing challenges like unbounded integrality gaps and exponential formulation sizes.
• State-of-the-art solution methods include matheuristics, decomposition, and column generation approaches, significantly improving solution quality and computational efficiency.

The Single-Source Capacitated Facility Location Problem (SSCFLP) is a foundational combinatorial optimization problem in discrete location theory, with far-reaching applications in logistics, network design, and public sector service planning. In its canonical form, SSCFLP seeks to minimize the sum of fixed facility-opening costs and assignment (transportation) costs by optimally selecting locations to open facilities (of limited capacity) and assigning each customer to a unique open facility, with the restriction that no facility's capacity is exceeded. The problem and its principal variants—SSCFLP with a fixed number of facilities, with contiguous assignment areas, and combinations thereof—exhibit substantial algorithmic and modeling complexity, especially under metric assignment costs, heterogeneous capacities, and real-world constraints such as district contiguity or demand uncertainty.

1. Formal Problem Definition and Mathematical Models

SSCFLP is typically formulated as a 0–1 mixed-integer program. Let $I = \{1,\dots,m\}$ index facility sites, $J = \{1,\dots, n\}$ index customers, with fixed opening costs $f_i$, capacities $S_i$, customer demands $d_j$, and assignment costs $c_{ij}$. The binary decision variables $y_i$ (facility open indicator) and $x_{ij}$ (assignment indicator) encode the feasible solutions. The standard form is: $$\min \; \sum_{i\in I} f_i\,y_i + \sum_{i\in I} \sum_{j\in J} c_{ij}\,x_{ij}$$ subject to

$\sum_{i\in I} x_{ij} = 1 \qquad \forall\, j\in J$

$$\sum_{j\in J} d_j x_{ij} \leq S_i y_i \qquad \forall\, i \in I$$

$$x_{ij}\in \{0,1\},\; y_i \in \{0,1\} \qquad \forall\, i, j$$

Variants include:

• SSCKFLP (fixed number $K$ of facilities): $\sum_{i} y_i = K$
• CFLSAP (contiguous service areas): Additional flow-based constraints to ensure each service area forms a connected spatial subgraph.
• CKFLSAP: Both the above together.

This construct captures the combinatorial tension between facility opening/closing decisions and the assignment of customers under hard capacity and exclusivity (single-sourcing) requirements (Kong, 2021).

2. Complexity, Polyhedral Results, and LP Relaxations

SSCFLP is strongly NP-hard, persisting even under unit demands or uniform capacities. Noteworthy is the difficulty of closing the integrality gap for standard LP relaxations: even after several rounds of the Sherali-Adams hierarchy or by adding sophisticated flow-cover inequalities, the gap remains unbounded, disproving longstanding conjectures on the strength of classical and generalized configuration LPs for capacitated settings (Kolliopoulos et al., 2013).

A major advance is the MFN-LP (multi-commodity flow network LP) relaxation, which enforces that every (possibly fractional) partial assignment can be completed via a generalized flow into open facilities. This relaxation exhibits a provable constant integrality gap ($O(1)$, e.g., 288 in the original analysis) despite being exponentially large; rounding proceeds via fractional b-matchings and bicriteria rounding on the "small" facility residual instance (An et al., 2014).

LP Relaxation Type	Integrality Gap	Notes
Natural (classic) LP	Unbounded	Holds for all finite capacities
Sherali-Adams (constant levels)	Unbounded	Even after $\Theta(n)$ rounds
Flow-cover cuts	Unbounded	Fails to close gap for uniform capacities
MFN-LP (multicommodity flow)	Constant ($O(1)$)	Exponential, non-compact, effectively separable only via oracles

3. Exact and Matheuristic Solution Methods

Given the limitations of "off-the-shelf" LP-based relaxations, most state-of-the-art algorithms for large SSCFLP instances are matheuristics that combine metaheuristics with restricted exact optimization. Kong (Kong, 2021) introduces a large-neighborhood search matheuristic for SSCFLP and its variants, alternating between:

• Construction of an initial solution via Lagrangian heuristics, LP-rounding, or greedy insertion (with soft capacity relaxation if needed),
• Iterative improvement by focusing on spatially clustered "neighborhoods" (i.e., subsets of facilities and customers) and solving the resulting large but restricted MIPs to optimality with a standard solver (e.g., CPLEX/Gurobi),
• Specific repair and local search procedures to address contiguity constraints in CFLSAP and CKFLSAP.

This framework robustly finds optimal or near-optimal solutions across standard benchmarks of up to 1000 facilities and 1500 customers, consistently outperforming kernel search, multi-exchange, and corridor-based heuristics in both quality and computational efficiency.

Pattern-Based Kernel Search (PaKS) (Bakker et al., 9 Dec 2025) further enhances heuristic performance by explicitly identifying biclustered spatial "regions" where assignment and opening decisions are strongly interdependent, drastically reducing model size and focusing search on promising substructures. PaKS integrates spectral biclustering for region detection with an advanced kernel and bucket construction phase, yielding state-of-the-art results (0.02% mean optimality gap on 112 benchmarks, scalable to 2000×2000 problem sizes).

On smaller instances or with special structure, tailored dynamic programming enables fixed-parameter tractability (FPT) in the number of facilities, and efficient DP recurrences can be constructed for the case of uniform capacities (Aziz et al., 2019).

4. Decomposition and Column Generation Approaches

For large-scale SSCFLP, exact decomposition-based algorithms are effective. Benders decomposition is classically applied by decomposing the facility-opening master (integer) and the assignment subproblem (LP); extensions with Pareto-optimal cuts, L-shaped multi-cuts, and hybrid schemes significantly accelerate convergence, particularly for large or degenerate instances (Asl et al., 2021).

In settings with demand uncertainty (where demands are subject to cardinality-constrained uncertainty sets), branch-and-price algorithms based on Dantzig-Wolfe allocation-based reformulations are highly effective (Ryu et al., 2021). These algorithms decompose the problem into a master selecting facility-customer subsets that respect robust capacity, and pricing subproblems that solve robust binary knapsacks, achieving orders-of-magnitude improvement in run time over direct branch-and-cut LP formulations.

Method	High-level Description	Benchmark Performance
Kong matheuristic (Kong, 2021)	LNS + exact MIP on spatial clusters	0.07% gap, 191/272 opt., scales to 1k
PaKS (Bakker et al., 9 Dec 2025)	Biclustering + enhanced kernel search	0.02% gap on 112; best for 2k×2k
Benders (L-shaped/Pareto/PL)	Facility-open master, fast multi-cuts	10–100x faster than classic BD on 250
Branch-and-price (robust)	Dantzig-Wolfe, robust knapsack pricing	Solves larger/uncertain instances

5. Mechanism Design, Uncertainty, and Strategic Models

Mechanism design for SSCFLP addresses settings where facility locations impact or are chosen based on the preferences and reports of selfish agents. When total capacity is insufficient ($\sum k_j < n$), the allocation subgame has nontrivial equilibria, and truthfulness issues arise. Auricchio, Clough, and Zhang (Auricchio et al., 26 Jul 2024) characterize percentile mechanisms that are absolutely truthful and equilibrium stable, achieving bounded approximation ratios approaching optimality as the number of facilities increases. In 1D, such mechanisms can ensure approximation ratios below $1+1/(2m-1)$ given sufficiently many agents. The class of mechanisms narrows significantly in higher dimensions.

In the one-dimensional SSCFLP where total capacity matches the agent population, the problem is still NP-hard, but FPT in the number of facilities or for uniform capacities, and only a small number of strategyproof mechanisms with bounded approximation exist (notably, the innerpoint and extended-endpoint mechanisms) (Aziz et al., 2019).

Under cardinality-constrained demand uncertainty, robust solution approaches must balance increased robustness (lower infeasibility probability) against additional cost. Empirical analyses confirm that modest uncertainty budgets ($\Gamma_i \approx 1$) provide substantial risk reduction at minimal extra cost (Ryu et al., 2021).

6. Open Challenges and Directions

Despite advances in matheuristics and decomposition, several fundamental gaps remain:

• No compact LP formulation for SSCFLP with provable constant integrality gap is known; MFN-LP is noncompact and cannot be separated in polynomial time (An et al., 2014).
• Even after incorporating generalized valid inequalities or deep lift-and-project rounds, classical polyhedral approaches fail on capacitated models (Kolliopoulos et al., 2013).
• Closing the theoretical integrality gap between best-known upper and lower bounds (e.g., reducing the constant factor in MFN-LP rounding) and efficient separation remains an open problem.
• Methodologies for dynamic region or neighborhood selection via machine learning, and extensions to multi-source, stochastic, or modular variants, are proposed but not yet fully developed (Kong, 2021).

The SSCFLP thus remains a central problem in discrete optimization, driving methodological progress across matheuristics, LP theory, decomposition algorithms, and economic mechanism design.