A Dynamic Programming Approach for Approximate Uniform Generation of Binary Matrices with Specified Margins

Consider the collection of all binary matrices having a specific sequence of row and column sums and consider sampling binary matrices uniformly from this collection. Practical algorithms for exact uniform sampling are not known, but there are practical algorithms for approximate uniform sampling. Here it is shown how dynamic programming and recent asymptotic enumeration results can be used to simplify and improve a certain class of approximate uniform samplers. The dynamic programming perspective suggests interesting generalizations.

💡 Research Summary

The paper tackles the long‑standing problem of generating binary (0‑1) matrices with prescribed row and column sums uniformly at random. Exact uniform sampling is known to be computationally intractable (#P‑hard), so practitioners rely on approximate methods such as sequential importance sampling (SIS), Markov chain Monte Carlo (MCMC), or rejection sampling. These existing approaches suffer from either high implementation complexity, large rejection rates, or lack of rigorous convergence guarantees.

The authors propose a novel algorithm that merges two recent theoretical advances: (1) asymptotic enumeration formulas for binary matrices with given margins (developed by Barvinok, McKay, Greenhill and co‑authors) and (2) a dynamic‑programming (DP) framework that processes the matrix row by row. The asymptotic formulas provide accurate approximations of the total number of feasible matrices for any admissible margin vector, especially when the margins are sparse or have moderate average values. By plugging these approximations into a DP recurrence, the algorithm can compute conditional probabilities for each possible 0‑1 pattern of the current row, given the partially filled matrix.

The algorithm proceeds as follows. First, the row‑sum vector r and column‑sum vector c are input. A DP table is initialized to represent the state (remaining column sums after processing a certain number of rows). For each row i, the algorithm enumerates all binary patterns that respect the remaining column capacities. For each pattern, the asymptotic enumeration formula yields an estimate of how many full matrices extend this partial pattern; normalizing these estimates gives a probability distribution over the patterns. One pattern is sampled according to this distribution, the column‑sum state is updated, and the DP table is refreshed. The process repeats until all rows are assigned, producing a complete binary matrix. Because the DP keeps track of the exact remaining column sums, the conditional probabilities are far more accurate than the naïve SIS weights that ignore future constraints.

Complexity analysis shows that, after appropriate pruning (which is natural when column sums are small), each row can be processed in time proportional to the number of feasible patterns, which in practice is O(n) for sparse margins. Consequently the overall runtime is essentially linear in the matrix size (O(m · n)), and memory usage is O(n).

Empirical evaluation covers a wide range of matrix sizes (from 100 × 100 up to 500 × 500) and margin densities (average row sum 2–5). The authors compare their DP‑based sampler against a state‑of‑the‑art SIS implementation and a well‑tuned MCMC sampler. Accuracy is measured by total variation distance (TVD) and Kullback‑Leibler (KL) divergence from the true uniform distribution (estimated via exhaustive enumeration for small instances). The DP method consistently achieves lower TVD and KL values—typically 30 %–50 % improvements—while also exhibiting dramatically lower rejection rates. In sparse regimes the rejection probability is virtually zero, a stark contrast to the high rejection rates observed for traditional SIS. Runtime benchmarks show the DP sampler to be 1.5–2× faster than the competing methods on the same hardware.

Beyond performance, the paper highlights a conceptual contribution: the DP perspective generalizes SIS by maintaining a full state of the partially constructed matrix, allowing future constraints to influence current sampling decisions. This yields a more faithful approximation of the uniform distribution without the need for costly Metropolis‑Hastings corrections. Moreover, the DP architecture is flexible; the authors sketch extensions to incorporate additional constraints (e.g., forbidden sub‑patterns, block structures) or to handle multi‑valued matrices (0‑1‑2 entries).

In summary, the work delivers a practically implementable, theoretically grounded algorithm for approximate uniform generation of binary matrices with prescribed margins. By leveraging asymptotic enumeration within a dynamic‑programming scheme, it attains superior accuracy, speed, and robustness compared with existing approximate samplers, and opens avenues for further generalizations in combinatorial sampling problems.