Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs

We investigate the space complexity of certain perfect matching problems over bipartite graphs embedded on surfaces of constant genus (orientable or non-orientable). We show that the problems of deciding whether such graphs have (1) a perfect matching or not and (2) a unique perfect matching or not, are in the logspace complexity class \SPL. Since \SPL\ is contained in the logspace counting classes $\oplus\L$ (in fact in \modk\ for all $k\geq 2$), \CeqL, and \PL, our upper bound places the above-mentioned matching problems in these counting classes as well. We also show that the search version, computing a perfect matching, for this class of graphs is in $\FL^{\SPL}$. Our results extend the same upper bounds for these problems over bipartite planar graphs known earlier. As our main technical result, we design a logspace computable and polynomially bounded weight function which isolates a minimum weight perfect matching in bipartite graphs embedded on surfaces of constant genus. We use results from algebraic topology for proving the correctness of the weight function.

💡 Research Summary

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The paper investigates the space complexity of perfect‑matching related problems on bipartite graphs that are embedded on surfaces of constant genus, both orientable and non‑orientable. The authors prove that (1) deciding whether such a graph has a perfect matching (Decision‑BPM) and (2) deciding whether it has a unique perfect matching (Unique‑BPM) both belong to the log‑space class SPL. Since SPL is contained in several log‑space counting classes—including ⊕L, Modₖ L for every k ≥ 2, C_=L, and PL—the results immediately place these decision problems into all those classes as well. Moreover, the search problem (Search‑BPM), i.e., constructing a perfect matching when one exists, is shown to be in FL^SPL (log‑space with an SPL oracle).

The technical core is a deterministic isolation scheme that works for graphs embedded on constant‑genus surfaces. The authors first transform any combinatorial embedding of a bipartite graph G on a genus‑g surface into a “genus‑g grid graph” G′′ using a sequence of log‑space, matching‑preserving reductions. This transformation handles both orientable and non‑orientable cases; the latter is reduced to the orientable case via a double‑cover construction that does not increase space requirements.

On the resulting grid graph, they construct a family W = {w₁,…,w_{4g+1}} of polynomially bounded weight functions, each computable in log‑space. The family is designed so that for every simple cycle C in G′′ there exists some wᵢ∈W whose circulation (alternating sum of edge weights along C) is non‑zero. The cycles are classified into three topological types: (a) surface‑non‑separating cycles, (b) surface‑separating cycles that stay entirely inside the fundamental polygon, and (c) surface‑separating cycles that cross the polygon boundary. Algebraic topology guarantees that a non‑separating cycle must intersect an odd number of polygon sides, which yields 2g weight functions (W₁) handling type (a). Type (b) cycles behave like planar cycles, so the planar isolation weight from DKR08 (the single weight w) suffices. Type (c) cycles alternate “entering” and “exiting” the polygon; another 2g weight functions (W₂) capture this alternation. Because g is a constant, a linear combination of the weights—e.g., Σ_{i} wᵢ·(n^{c})^{i} for a fixed constant c—produces a single weight function w* whose circulation is non‑zero on every cycle.

A key lemma (Lemma 1, originally from DKR08) states that if all circulations are non‑zero, then the minimum‑weight perfect matching is unique. Hence, under w* the graph has a uniquely isolated minimum‑weight perfect matching. This deterministic isolation enables the authors to reduce Decision‑BPM and Unique‑BPM to the SPL‑complete problem of checking whether a matrix determinant is 0 or 1 under the promise that it is one of those two values. Consequently, both decision problems lie in SPL, and by known inclusions also in ⊕L, Modₖ L, C_=L, PL, and NC².

For the search problem, the authors use the SPL oracle to compute the determinant of appropriately constructed matrices and to extract the unique minimum‑weight perfect matching, thereby placing Search‑BPM in FL^SPL.

The paper also discusses related work: the isolating lemma of Mulmuley‑Vazirani‑Vazirani, previous deterministic isolation for planar bipartite graphs (DKR08), and parallel algorithms (NC²) for matching on low‑genus graphs. The novelty lies in extending deterministic isolation from planar to constant‑genus surfaces, providing a log‑space algorithmic framework that leverages algebraic topology (Z₂‑homology, fundamental polygons) to guarantee non‑zero circulations for all cycle types.

In summary, the authors achieve three main contributions:

1. A log‑space, deterministic isolation weight function for bipartite graphs on constant‑genus surfaces.
2. Upper bounds placing Decision‑BPM, Unique‑BPM in SPL (hence in several counting classes) and Search‑BPM in FL^SPL.
3. A systematic reduction pipeline (embedding normalization → grid embedding → weight construction) that preserves matching structure and operates within logarithmic space.

These results advance our understanding of the space complexity of matching problems beyond planar graphs, showing that even with modest topological complexity the problems remain within low‑space counting classes.