Shortest Cross-Layer Augmenting Path Search (CLAP-S)

• The paper introduces CLAP-S, a novel algorithm that unifies driver sets in duplex networks by optimizing cross-layer augmenting paths.
• It reformulates the controllability problem by minimizing disagreement between layer-specific driver nodes using difference mass decomposition.
• Empirical and theoretical results guarantee a one-driver improvement per CLAP, achieving global optimality with polynomial-time performance over various network types.

Shortest Cross-Layer Augmenting Path Search (CLAP-S) is an algorithm for duplex-network controllability that minimizes the size of the union of driver nodes required to control two layers on the same node set. In the structural-controllability formulation used in multiplex control, each layer is represented by a matching problem on a bipartite graph, and unmatched vertices correspond to driver nodes. CLAP-S does not solve the two layers independently; instead, it searches for cross-layer augmenting structures that preserve each layer’s driver budget while increasing overlap between the two layer-wise Minimum Driver Sets (MDSs). The method is introduced for the Universal Minimum Union Driver Set (MinUDS) problem and is accompanied by a global optimality proof and polynomial-time guarantees (Zheng et al., 26 Sep 2025).

1. Duplex controllability and the MinUDS formulation

The setting is a duplex network in which each layer $G_\ell=(V,E_\ell)$ is a directed graph on the same node set. Each layer is converted to the standard bipartite representation $\mathcal B_\ell=(V^+\cup V^-,E_{\mathcal B})$, and a matching $M$ encodes structural controllability. For a matching $M$, the driver set is

$D(M)=\{v\in V:\ v^- \notin V_M\},$

so unmatched $V^-$-vertices are the driver nodes. If $M^*$ is a maximum matching, then the layer’s MDS has size

$N_D = N-|M^*|.$

In a duplex, the two layers share the same physical control resource, so taking the union of independently computed layer-wise MDSs can be unnecessarily large (Zheng et al., 26 Sep 2025).

To formalize this, the paper fixes budgets $k_1,k_2$ and defines the feasible matchings

$$\mathcal M_\ell(k_\ell)=\{M_\ell:\ |M_\ell|=N-k_\ell\}.$$

For a feasible pair $\mathcal B_\ell=(V^+\cup V^-,E_{\mathcal B})$0, the union driver set is

$\mathcal B_\ell=(V^+\cup V^-,E_{\mathcal B})$1

The optimization problem, termed the Budget-Preserving UDS Contraction problem, is

$\mathcal B_\ell=(V^+\cup V^-,E_{\mathcal B})$2

Its optimum is the Universal Minimum Union Driver Set, or MinUDS. The key shift is that the objective is not to reduce the number of drivers in either layer separately, but to realign layer-wise matchings so that the two driver sets overlap as much as possible (Zheng et al., 26 Sep 2025).

2. Difference-mass decomposition and the optimization objective

A central analytical step is to rewrite the union objective in terms of agreement and disagreement between the two driver sets. The node set is partitioned into four classes:

Set	Definition	Role
$\mathcal B_\ell=(V^+\cup V^-,E_{\mathcal B})$3	$\mathcal B_\ell=(V^+\cup V^-,E_{\mathcal B})$4	Consistently Driven Set
$\mathcal B_\ell=(V^+\cup V^-,E_{\mathcal B})$5	$\mathcal B_\ell=(V^+\cup V^-,E_{\mathcal B})$6	Consistently Matched Set
$\mathcal B_\ell=(V^+\cup V^-,E_{\mathcal B})$7	$\mathcal B_\ell=(V^+\cup V^-,E_{\mathcal B})$8	Layer-1-only drivers
$\mathcal B_\ell=(V^+\cup V^-,E_{\mathcal B})$9	$M$0	Layer-2-only drivers

The disagreement quantity is

$M$1

The paper proves the exact identity

$M$2

Hence minimizing the union is equivalent to minimizing $M$3 (Zheng et al., 26 Sep 2025).

This reformulation is the conceptual core of CLAP-S. Rather than enumerating feasible matching pairs directly, the algorithm searches for operations that reduce the difference mass by moving nodes between the consistent and inconsistent classes while keeping the layer budgets fixed. A plausible implication is that the MinUDS problem becomes tractable not because the full combinatorial space is small, but because disagreement admits a local certificate in the form of an augmenting structure.

3. Alternating paths, driver exchange, and the definition of CLAP

The single-layer primitive underlying CLAP-S is the alternating-path exchange. If $M$4 is a matching, an $M$5-alternating path alternates between edges in $M$6 and edges not in $M$7. The Driver Exchange Principle states that if there is an $M$8-alternating path from a driver node $M$9 to a non-driver node $M$0, then flipping the matching along that path preserves the matching size but swaps $M$1 out of the driver set and $M$2 into it: $M$3 CLAP-S coordinates such exchanges across the two layers (Zheng et al., 26 Sep 2025).

A Cross-Layer Augmenting Path (CLAP) is defined as a sequence of admissible segments that alternates between layers. An admissible segment $M$4 is a layer-$M$5 driver exchange supported by an $M$6-alternating witness path, with polarity constraints depending on the layer. In layer 1, the segment moves a node from $M$7 to non-driver status; in layer 2, it moves a node from non-driver status into $M$8. A CLAP is

$M$9

with $D(M)=\{v\in V:\ v^- \notin V_M\},$0, $D(M)=\{v\in V:\ v^- \notin V_M\},$1, alternating layer labels $D(M)=\{v\in V:\ v^- \notin V_M\},$2, and all nodes distinct (Zheng et al., 26 Sep 2025).

The internal relay nodes are constrained by type. If a segment arrives via layer 1, the relay must be in $D(M)=\{v\in V:\ v^- \notin V_M\},$3; if it arrives via layer 2, the relay must be in $D(M)=\{v\in V:\ v^- \notin V_M\},$4. This ensures that the relay has the correct driver/non-driver status when the path switches layers. Thus a CLAP is not an arbitrary path in the multiplex; it is a layer-alternating chain of legally composable driver exchanges.

4. The CLAP-S algorithm and its optimality theory

CLAP-S is the shortest-CLAP search procedure. Starting from any pair of feasible matchings, typically maximum matchings in each layer, it computes the driver sets and then iterates: find a shortest CLAP from $D(M)=\{v\in V:\ v^- \notin V_M\},$5 to $D(M)=\{v\in V:\ v^- \notin V_M\},$6, apply it by updating the matchings in both layers, recompute the driver sets, and stop when no CLAP exists (Zheng et al., 26 Sep 2025).

Shortest-CLAP search is implemented by a layer-alternating BFS. The search begins simultaneously from all nodes in $D(M)=\{v\in V:\ v^- \notin V_M\},$7, expands through admissible alternating reachability within the current layer, and switches layers only through valid relay nodes. Because BFS expands by number of segments, the first time it reaches $D(M)=\{v\in V:\ v^- \notin V_M\},$8 it has found a shortest CLAP. The paper further proves that shortest CLAPs are feasible: their witness alternating paths can be chosen edge-disjoint within each layer, so the batched symmetric-difference update is well-defined (Zheng et al., 26 Sep 2025).

The main progress guarantee is the CLAP Gain Theorem. Every feasible CLAP preserves the fixed budgets and decreases the difference mass by exactly 2: $D(M)=\{v\in V:\ v^- \notin V_M\},$9 which decreases the union driver set size by exactly 1: $V^-$0 Thus each successful CLAP is a guaranteed one-driver improvement in the duplex control cost (Zheng et al., 26 Sep 2025).

The global optimality result is the CLAP-or-Optimal theorem. A state is CLAP-stable if there is no CLAP from $V^-$1 to $V^-$2. The theorem states that if no CLAP exists, the current state is globally optimal for the MinUDS problem over the feasible set; equivalently, $V^-$3 is minimal, and equivalently, $V^-$4 is minimal. The proof is explicitly analogous to Berge’s theorem for matchings: comparing the current state to a better feasible state and decomposing symmetric differences layer by layer produces a layer-labeled meta-graph whose path components reveal a CLAP whenever improvement is still possible (Zheng et al., 26 Sep 2025).

5. Complexity and empirical behavior

Initialization uses Hopcroft–Karp on each layer, with cost

$V^-$5

A shortest CLAP search costs

$V^-$6

where $V^-$7 is the number of segments in the shortest CLAP found. Since each successful CLAP reduces $V^-$8 by $V^-$9, there are at most $M^*$0 iterations. The worst-case bound stated in the paper is

$M^*$1

Empirically, however, CLAPs are very short, with average length near 1, so the observed runtime is much closer to linear in the number of edges per successful contraction step (Zheng et al., 26 Sep 2025).

The experiments compare CLAP-S with RSU, CLAP-G, and ILP-Exact. Synthetic duplex networks include ER-ER, BA-BA, and ER-BA hybrids, with $M^*$2, average degree varied from 2 to 10, and layer overlap controlled by Jaccard similarity from 0.1 to 0.9. Real-world duplexes are assembled from curated multilayer datasets spanning genetic, neuronal, social, and human-relationship systems, including Arabidopsis, Celegans, Drosophila, YeastLandscape, connectomes, Twitter-derived networks, and social-fabric networks (Zheng et al., 26 Sep 2025).

The reported empirical pattern is that CLAP-S consistently outperforms RSU in both solution quality and runtime. On BA-BA graphs, optimization peaks in moderately sparse conditions, with CLAP-S saving nearly 70 driver nodes on average in the most favorable regime. The runtime advantage is typically about an order of magnitude; one example given for synthetic ER-ER systems is roughly $M^*$3 s for CLAP-S versus $M^*$4 s for RSU. On real-world duplexes, CLAP-S matches the exact ILP optimum on every instance where ILP was tractable, and the detailed results include reductions of hundreds of driver nodes in large networks such as Drosophila and YeastLandscape. The average CLAP length in real data is typically around $M^*$5–$M^*$6, which accounts for the strong practical efficiency (Zheng et al., 26 Sep 2025).

6. Position within augmenting-path and shortest-path research

CLAP-S belongs to the augmenting-path tradition rather than to generic shortest-path search. In the broader algorithmic literature, the shortest augmenting path technique is described as one of the fundamental ideas used in maximum matching and maximum flow algorithms, and the tree-case analysis of online bipartite matching establishes a tight $M^*$7 bound for the total length of shortest augmenting paths (Bosek et al., 2017). CLAP-S inherits the logic of local alternating-path improvement, but applies it to a duplex objective in which two layer-wise matchings must be coordinated rather than optimized separately. This suggests a shift from single-layer augmentation to cross-layer alignment.

The phrase “shortest path” in CLAP-S can be misleading if read through the lens of routing or graph search. Learning-guided shortest-path frameworks such as LSearch and HLSearch operate on generic undirected weighted graphs, use predicted distances and hop counts for pruning, and do not discuss CLAP-S or augmenting-path search in the matching sense (Liu et al., 4 Aug 2025). Routing-oriented adaptations of Dijkstra in elastic optical networks and VLSI detailed routing likewise address feasibility-constrained pathfinding over topology, spectrum state, or layered grid geometry rather than driver-set contraction in multiplex control (Szcześniak et al., 2019, Ahrens et al., 2021).

The distinctive feature of CLAP-S is therefore not shortest path search over physical edges, but shortest search over admissible cross-layer exchange sequences. Its state space is induced by pairs of feasible matchings and the induced partition into $M^*$8, $M^*$9, $N_D = N-|M^*|.$0, and $N_D = N-|M^*|.$1. Its terminal condition is not reaching a destination vertex, but reaching a CLAP-stable state that certifies global optimality for MinUDS (Zheng et al., 26 Sep 2025).