Maximum Common Bigraph Problem

• The Maximum Common Bigraph Problem is defined as finding the largest bigraph shared by two given bigraphs under mappings that preserve both place and link structures.
• It generalizes classical maximum common subgraph problems and is essential for applications in bisimulation checking, formal verification, and model reduction in reactive systems.
• Advanced algorithms, including adaptations of McSplit and RRSplit, tackle its NP-hard complexity using branch-and-bound techniques and structural pruning.

The Maximum Common Bigraph Problem (MCB) concerns the computation of the largest bigraph structure shared by two given bigraphs under admissible mappings that preserve both place and link structure. Bigraphs, as introduced by Milner, simultaneously encode spatial (place-graph) and connection (link-graph) relationships, making the MCB problem a strict generalisation of classical maximum common (sub)graph problems, with direct applications to bisimulation checking, formal verification, and model reduction in bigraph reactive systems. Both general bigraph matching and the MCB problem are NP-hard, requiring advanced combinatorial and branch-and-bound techniques, notably adaptations of the McSplit and RRSplit paradigms, to construct practically feasible algorithms for large models (Burns et al., 7 Jan 2026, Yu et al., 17 Feb 2025).

1. Formal Definition of the Maximum Common Bigraph Problem

Given two solid concrete bigraphs,

$$G_1 : \langle m_1,X_1\rangle \to \langle n_1,Y_1\rangle,\quad G_2 : \langle m_2,X_2\rangle \to \langle n_2,Y_2\rangle$$

where a concrete bigraph $G$ consists of

• a place graph $G^P = (V, \mathrm{ctrl}, \mathrm{prnt}): m\to n$, with $V$ nodes, controls $\mathrm{ctrl}:V\to K$, and acyclic parent map $\mathrm{prnt}$;
• a link graph $$G^L = (V, E, \mathrm{ctrl}, \mathrm{link}): X \to Y$$, where $E$ is the edge set and $\mathrm{link}: X \sqcup P \to E \sqcup Y$ connects ports $P$ and names.

A maximum common bigraph (MCB) is a pair $(G_m, M)$ with

1. Existence of contexts $C_1, C_2$ and parameters $D_1, D_2$ such that

$$G_1 = C_1 \circ (\mathrm{id} \otimes G_m) \circ D_1, \quad G_2 = C_2 \circ (\mathrm{id} \otimes G_m) \circ D_2$$

1. Maximality of $G_m$: no strictly larger $G'_m$ exists such that $G'_m$ embeds into both $G_1$ and $G_2$ by support-size, and $G_m$ is RPO-saturated (cannot be extended further by pushout).
2. $M = \{(g_i, a_i, b_i)\}$ is an injective mapping of $G_m$’s entities, ports, and closed links into those of $G_1$ and $G_2$.

In a concise form,

$$\mathrm{MCB}(G_1, G_2) = \max_{\preceq} \{ G\mid G_1 = C_1 \circ (\mathrm{id} \otimes G) \circ D_1,\; G_2 = C_2 \circ (\mathrm{id} \otimes G) \circ D_2 \}$$

where $\preceq$ orders bigraphs by support-size and bigraph composition.

In the bipartite setting, for bigraphs modelled as bipartite graphs $Q$ and $G$, the problem is to find the triple $(q, g, \phi)$ of largest size such that $q$ embeds into $Q$ and $g$ into $G$, with an induced bijection $\phi$ preserving bipartition and adjacency (Yu et al., 17 Feb 2025).

2. Computational Complexity

Bigraph matching is NP-hard, shown to be NP-complete for controls of unbounded arity. Since MCB strictly generalises matching—recovering matching when $G_m$ must embed fully—MCB inherits NP-hardness. The decision variant (“Does there exist a common bigraph of size at least $k$?”) is NP-complete via reduction from the Maximum Common Induced Subgraph (MCIS) problem. There is no closed-form polynomial time bound except special instances; exponential bounds hold in the general case (Burns et al., 7 Jan 2026).

For bipartite bigraphs, worst-case time complexity for branch-and-bound solutions such as RRSplit is

$$O^*((N+1)^n),\quad n = |U_Q| + |V_Q|;\ N = |U_G| + |V_G|$$

where $O^*(\cdot)$ hides polynomial factors (Yu et al., 17 Feb 2025).

3. Algorithmic Methods

Two chief algorithmic paradigms exist: a McSplit-based approach for arbitrary bigraphs, and RRSplit for bipartite bigraphs.

(A) McSplit Adaptation for Bigraphs (Burns et al., 7 Jan 2026):

• Encoding: Place-graph ($\varphi_p$): nodes (entities) become vertices; parent-child links become edges; vertices labelled by control.
• Flattened link-graph ($\varphi_f$): Ports and closures become additional vertices; introduced edges encode membership and attachment to closures.
• Combined encoding: $\varphi(G) = \varphi_f(\varphi_p(G^P), G^L)$, producing a single labelled ordinary graph.
• Search tree state: Maintains a partial mapping $M_{\mathrm{current}}$, label classes grouping unmatched vertices, and upper bounds on extendable mapping size (ignoring port-to-port classes).
• Branching and pruning: Enforces bigraph-specific composition constraints (tensor-disjointness, closure visibility only after all parent ports matched).
• Pseudocode principle: Recursive search with score, bound pruning, class refinement, and visibility update.

Computational cost per node is $O(|V_1|+|V_2|)$ with depth at most $\min(|V_1|,|V_2|)$.

(B) RRSplit for Bipartite Bigraphs (Yu et al., 17 Feb 2025):

• Search node: Triplet $(S, C, D)$ of current partial match, candidate pairs, and exclusion set.
• Vertex-equivalence reduction: Prunes search branches by leveraging structural equivalence classes within each partition.
• Maximality-based reduction: Branches only on pairs guaranteed to be present in some maximum solution, thus reducing redundancy.
• Refined upper bound: For each Cartesian cell $X \times Y$ of potential pairs, computes an upper bound via representative-based splitting.
• Partition-wise degeneracy ordering: Favors enumeration that rapidly reduces candidate set size.
• Implementation: Precomputes static equivalence classes, employs bitset adjacency for fast neighborhood queries.

Table: Main Algorithmic Paradigms

Paradigm	Applicable to	Key Features
McSplit	General bigraphs	Graph encoding, label class refinement, composition constraints, exponential search tree
RRSplit	Bipartite bigraphs	Structural equivalence pruning, cell-based maximality, refined upper bounds, empirical speedup

4. Illustrative Example

For two bigraphs $G_1$ and $G_2$:

• $G_1$ is a flat place graph with two A-nodes ($A_1$, $A_2$), each with one port, and one closed link connecting them.
• $G_2$ is a longer chain with three A-nodes ($A'_1$–$A'_2$–$A'_3$) and two closures: $q_1\leftrightarrow q_2$ (closure $f$) and $q_2\leftrightarrow q_3$ (closure $g$).

1. Encode place graphs: $\varphi_p(G_1)$ has two nodes, $\varphi_p(G_2)$ has three nodes and two edges.
2. Flatten link graphs: Vertices added for each port and closure, edges encode connections.
3. Label classes: Entities (control-A), ports, and closures grouped into respective label classes.
4. Search tree: Starts with an empty mapping and upper bound from label classes.
5. Branching: Map $A_1\to A'_1$, refine label classes. Map $A_2\to A'_2$, refine further. Ports and closures mapped as visible; maximal mapping is two A-nodes linked by a single closure (from $G_1$).

No larger entity-only mapping exists, confirming optimality.

5. Role in Bisimulation Checking and Minimal Context Formation

MCB is pivotal in constructing the Minimal Contextual Transition System (MCTS) for bigraph agents. For an agent bigraph $G$ and a rewrite rule $R$, discovering their maximum common bigraph $G_m$ (the largest overlap) is required to:

• Remove $G_m$ from both $G$ and $R$.
• Define the minimal residual context $C$ necessary for the rule application.
• Systematically enumerate all minimal transitions $G \stackrel{[C]}{\longrightarrow} G'$ by iterating over all $(G, R)$ pairs and their MCBs.

Bisimulation—equivalence up to contextual behavior—relies on comparing isomorphism types of agents' MCTSs, which MCB supports directly. This is stricter than LTS-based equivalence, as LTS ignores unknown context and is therefore incomplete for full bisimulation analysis (Burns et al., 7 Jan 2026).

6. Extensions, Heuristics, and Practical Performance

Proposed optimisations for MCB search include:

• Bit-parallel McSplit implementations for inclusion in bigraph toolkits (e.g., BigraphER).
• Parallelisation of independent MCB computations for different rule/context pairs.
• Heuristic improvements such as PageRank seeding and look-ahead to further prune the search tree.
• Tightening interface compatibility constraints earlier in search to improve pruning efficacy.

Empirical benchmarks for RRSplit show substantial performance improvements over McSplitDAL, with 7,730 out of 9,180 bipartite graph instances solved within 1,800 seconds compared to 4,696 for McSplitDAL, and average speedups exceeding $3.3 \times 10^4$ on biochemical-reaction datasets (Yu et al., 17 Feb 2025). Branch exploration is reduced to 0.1–10% of previous approaches, making very large bipartite instances tractable.

7. Connections to Related Problems and Fields

MCB generalises the classical Maximum Common Induced Subgraph (MCIS) problem, as matching is a degenerate MCB where one bigraph is embedded fully into the other. This positions MCB at the intersection of graph isomorphism, model checking, and rule-based system optimization. In the bipartite case, structural results and algorithmic advances from chemical informatics, reaction network analysis, and database theory directly influence both the theoretical and practical development of MCB solvers (Yu et al., 17 Feb 2025).

Advances in practical branch-and-bound search, such as the reduction-complete RRSplit algorithm, rely on accelerating search-space pruning by leveraging structural symmetries and equivalence classes, relevant to general combinatorial optimization beyond the bigraph framework.

For formal verification, process calculi, and agent-based modelling, robust MCB computation is increasingly integral for model minimization and quantifying behavioral similarity in structurally rich, context-dependent models (Burns et al., 7 Jan 2026).