Coordination-Exhaustive Search (CES)

• CES is a family of exhaustive search methods that integrates coordination in multi-agent planning, defining support pair ordering and agent matching.
• It decomposes the problem into ordering, pairing, and movement planning to balance the tradeoff between coordination overhead and search efficiency.
• Empirical analyses reveal that while CES offers optimal solutions, its super-exponential scaling in support pairs confines its use to smaller instances or as a baseline.

Searching arXiv for the cited CES-related papers to ground the article in current metadata. I’ll look up the arXiv records for the main CES sources by identifier. Coordination-Exhaustive Search (CES) denotes, in the literature considered here, both a concrete exact algorithm for coordinated multi-agent planning and a broader systems perspective on how exhaustive search behaves under different coordination regimes. In its explicit algorithmic form, CES was introduced for Team Coordination on Graphs with Risky Edges (TCGRE), where agents may support one another at designated support nodes to reduce traversal costs on risky edges; in the broader distributed-search sense, CES concerns the allocation of search regions, the exploitation of heterogeneous agent speeds, and the tradeoff between coordination overhead and search efficacy (Zhou et al., 8 Sep 2025, Stojanovski et al., 2012).

1. Scope and nomenclature

The acronym “CES” is not used uniformly across the search literature. In TCGRE, CES means Coordination-Exhaustive Search: an exhaustive optimizer over coordination assignments, support pairs, and their temporal ordering. In distributed exhaustive search, the same acronym is also used in the phrase “CES systems,” referring to coordinated exhaustive-search systems whose performance depends on subregion allocation and cooperation level. In a distinct line of work on parallel Bayesian and ordered search, “CES” appears as Completely Even Search, a classical non-coordinating search style in which agents act independently and identically (Zhou et al., 8 Sep 2025, Stojanovski et al., 2012, Korman et al., 2017).

This terminological overlap matters because the underlying design questions differ. The TCGRE version of CES is an exact combinatorial solver; the distributed-systems usage studies how coordination changes the average time to find a target; and the non-coordinating-search usage analyzes the constant-factor penalty incurred by abandoning coordination. This suggests that CES is best understood not as a single method, but as a family of coordination-sensitive exhaustive-search viewpoints whose common concern is how much structure or communication should be imposed on parallel search.

2. CES as an exact solver for coordinated multi-agent path planning

In the TCGRE formulation, the environment is an undirected graph $G=(V,E)$ with a subset of risky edges $E' \subset E$. Each of $N$ agents has a start vertex $v_n^o$ and a goal vertex $v_n^g$. At each discrete time step $t$, the planner must choose movement decisions and, when feasible, coordination decisions specifying whether an agent provides or receives support. The objective is to minimize the total path cost while accounting for cost reductions enabled by support:

$\min \sum_{t=0}^{T-1} \sum_{n=1}^N C_{n,t}(M,S).$

The problem is reformulated as a 3D matching problem over robot pairs, support pairs, and time steps, and its NP-hardness is proved via reduction from Minimum 3D Matching. CES addresses this by exhaustive enumeration. Its operational structure has three layers: first, it checks every possible order of support pairs; second, for each order, it assigns at most one robot pair to each support pair; third, it computes the corresponding movement and coordination plans and retains the feasible plan with lowest total cost. In the terminology used in the paper, CES first matches time orders and support pairs, and then robot pairs (Zhou et al., 8 Sep 2025).

This decomposition is significant because it makes explicit where the combinatorial burden lies. Rather than enumerating the entire joint configuration space directly, CES enumerates coordination structure: which support opportunities occur, in what order, and which agent pairs realize them. The algorithm is therefore exhaustive with respect to coordination assignments, not merely path prefixes.

3. Guarantees, scaling behavior, and relation to alternative exact methods

CES is both optimal and complete. It is optimal because it explores all feasible coordination assignments and returns the lowest-cost solution; it is complete because it finds a solution whenever one exists. Its principal limitation is combinatorial growth. For $S$ support pairs and $R$ robot pairs, there are up to $S!$ ways to order the support events and up to $E' \subset E$0 ways to assign robot pairs. The paper further notes that CES “was already proven polynomial to # agents in the conference version, but it has worse than exponential runtime growth wrt. # support pairs,” which makes the number of risky-edge/support-node combinations the decisive bottleneck (Zhou et al., 8 Sep 2025).

The TCGRE paper positions CES against three alternatives: JSG, RHOCA*, and Dynamic-HJSG. Their distinctions are structural rather than cosmetic.

Method	Optimality	Principal bottleneck or mechanism
JSG	Yes	Exponential in number of agents
CES	Yes	Polynomial in number of agents, but super-exponential in support pairs
RHOCA*	No	Receding-horizon approximation for scalability
Dynamic-HJSG	Yes	Dynamic graph construction with agent-homogeneity pruning

The empirical comparison reported in the paper is consistent with this decomposition. Dynamic-HJSG had Completion % 98%, Avg. Runtime (Success) 120ms, Timeout Cases 2%, and Effective Runtime 132ms; HCES had 85%, 450ms, 15%, and 9.45s; JSG had 65%, 1.2s, 35%, and 42.2s; CES had 40%, 3.8s, 60%, and 63.8s. These figures do not undermine CES’s exactness; rather, they localize its practical role as a correctness baseline and a small-instance optimal solver (Zhou et al., 8 Sep 2025).

A common misconception is that “optimal and exhaustive” implies uniformly poor scaling in all problem dimensions. The reported analysis is more specific: CES is presented as polynomial in the number of agents but worse than exponential in the number of support pairs. The distinction is central, because it separates coordination combinatorics from agent-count combinatorics.

4. Coordination regimes in exhaustive search systems

A broader CES literature studies exhaustive search conducted by multiple agents over a search space of total size $E' \subset E$1. The core performance metric is the average time to find a unique solution, denoted $E' \subset E$2. In the general model,

$E' \subset E$3

and, when speed $E' \subset E$4 and subregion length $E' \subset E$5 are mutually independent,

$E' \subset E$6

where $E' \subset E$7 is the number of agents. This formalism makes region allocation the decisive object: performance depends not only on how many agents search, but on how the search space is partitioned among them (Stojanovski et al., 2012).

For homogeneous agents, equal subregions are the best coordinated strategy. If each agent has common speed $E' \subset E$8 and receives a region of length $E' \subset E$9, then

$N$0

Semi-equal subregions approximate this when agents join dynamically, but random subregions are substantially worse because overlap and waste are possible. The paper gives a concrete comparison: with 10 agents in equal subregions, the same performance as 19 agents in random subregion allocation is achieved.

For heterogeneous agents, the strongest result is speed-proportional allocation. If agent $N$1 has speed $N$2, then the optimal subregion length is

$N$3

so that all agents finish at the same time,

$N$4

and the average search time becomes

$N$5

The paper’s general conclusion is direct: performance improves as the level of cooperation increases, and homogeneous and heterogeneous agents can achieve the same search performance provided that subregion lengths follow differences in agent speeds (Stojanovski et al., 2012).

5. Non-coordinating search and the price of abandoning coordination

A separate but closely related literature asks what happens when exhaustive parallel search is performed without coordination. The model is often the treasure-hunt problem: boxes are ordered, a treasure is hidden in one of them, and $N$6 searchers can inspect one box per time step. Under full coordination, the optimum is immediate. In the Bayesian formulation, the coordinated algorithm has expected running time

$N$7

Non-coordinating algorithms are intentionally symmetric and communication-free; each searcher acts independently, differing only by probabilistic choices (Korman et al., 2017).

The key message is that the penalty for non-coordination is bounded by a constant factor rather than an asymptotic collapse. For arbitrary distributions, the universal algorithm satisfies

$N$8

and the memory-efficient variant satisfies

$N$9

When the distribution is uniform over $v_n^o$0 boxes, the optimal non-coordinated expected running time is approximately

$v_n^o$1

whereas the coordinated optimum is approximately $v_n^o$2, so non-coordination is essentially twice as slow. For Pareto distributions $v_n^o$3 with $v_n^o$4, the suggested algorithm selects uniformly among unchecked boxes in $v_n^o$5 with $v_n^o$6; it is asymptotically optimal, and the price of non-coordination is about $v_n^o$7 (Korman et al., 2017).

The ordered-set version yields a matching asymptotic bound on speed-up. The best achievable speed-up for $v_n^o$8 non-coordinating searchers is

$v_n^o$9

giving $v_n^g$0 for $v_n^g$1, $v_n^g$2 for $v_n^g$3, and asymptotically $v_n^g$4. These bounds are tight. The same work emphasizes why non-coordination remains attractive: such algorithms are naturally robust to searcher crashes, require no IDs or shared state, and are suitable when communication is costly, unreliable, privacy-invasive, or operationally difficult (Fraigniaud et al., 2015).

A common misconception is that abandoning coordination renders exhaustive parallel search nearly useless. The cited bounds show a subtler picture: the loss can be as small as a factor of $v_n^g$5 in uniform settings and at worst about a factor of $v_n^g$6 in the asymptotic general bound.

6. Boundary cases, related coordination methods, and disambiguations

CES is also invoked as a principle of necessity in complexity-theoretic work on CSP and SAT. Using Model RB, a random CSP model with large domains, the paper “SAT Requires Exhaustive Search” proves that Model RB cannot be solved in $v_n^g$7 time for any $v_n^g$8, and, via log-encoding, that SAT with $v_n^g$9 Boolean variables and unrestricted clause length cannot be solved in $t$0 for any $t$1. In that setting, the “CES principle” means that exhaustive search is not merely a design choice but the only viable option on the constructed instances, because satisfiable and unsatisfiable twins are indistinguishable under local reductions (Xu et al., 2023).

Robotic search under restricted communication supplies another related coordination perspective. A strategy based on cellular decomposition and soft obstacles assigns bounded, non-overlapping cells to robots, treats cells assigned to other robots as obstacles, and updates interference regions after rendezvous. The reported simulations on $t$2 grids with 2 to 10 robots indicate that the Soft Obstacle Strategy (SOS) yields the best coverage, especially in time-bounded settings, at approximately 87% of maximum possible coverage, while avoiding the interruptibility associated with periodic rendezvous. This suggests that high-quality coordination can emerge from sparse, event-driven information exchange rather than continuous communication (Masakuna et al., 2019).

Finally, CES should be distinguished from coordinate exchange (CEXCH) in optimal experimental design. The paper “Revisiting the effect of greediness on the efficacy of exchange algorithms for generating exact optimal experimental designs” studied 21 exact response-surface design scenarios and found essentially no difference between single-element, most-greedy CEXCH and design-row, medium-greedy CEXCH, while PSO exhibited better efficacy for generating $t$3-optimal designs, and for most $t$4-optimal designs than CEXCH, but not to a strong degree under the reported parametrization. The relevance here is terminological and conceptual: acronymic proximity should not obscure that CEXCH concerns exchange-based design optimization rather than coordination-exhaustive search proper, and that greater greediness does not automatically imply better efficacy (Gullion et al., 2023).

Taken together, these literatures place CES at the intersection of combinatorial optimization, distributed search, and coordination theory. In one form, CES is an exact solver over coordination assignments; in another, it is a systems framework for distributing exhaustive search among agents; and in a stronger lower-bound sense, it names situations in which exhaustive search cannot be algorithmically circumvented.