COLOGR: Risk-Aware Constrained Graph Reduction

• The paper introduces COLOGR as an algorithmic framework that transforms the resource-constrained RCDP into a risk-aware Weight-Constrained Shortest Path Problem using Lagrangian relaxation and vertex elimination.
• It systematically prunes infeasible and dominated graph vertices through a two-phase process—COGR and LOGR—thereby preserving the optimal surrogate path under budget constraints.
• Empirical evaluations show that COLOGR outperforms greedy baselines in traversal cost, variance reduction, and computational efficiency in complex uncertain obstacle environments.

Searching arXiv for COLOGR and closely related papers to ground the article. COLOGR is an algorithmic framework for solving the resource-constrained extension of the Random Disambiguation Path problem, a planning problem in uncertain obstacle fields where an agent may disambiguate ambiguous obstacles at heterogeneous cost subject to a global budget. In the formulation introduced on arXiv, COLOGR combines Lagrangian relaxation with a two-phase vertex elimination procedure after reformulating the planning task as a risk-aware Weight-Constrained Shortest Path Problem (WCSPP). The framework is designed to prune infeasible and dominated graph structure while preserving the optimal surrogate solution, and its analysis emphasizes feasibility guarantees, surrogate optimality under mild assumptions, and frequent empirical zero duality gap (Zhou et al., 8 Jul 2025).

1. Nomenclature and conceptual scope

The paper does not provide a single monolithic expansion of the letters in “COLOGR.” Instead, it presents COLOGR as the combination of two components: COGR, “Cost- and Obstacle-based Graph Reduction,” and LOGR, “Lagrangian Optimization with Graph Reduction.” It also states that COLOGR and Two-Phase Vertex Elimination (TPVE) are used interchangeably (Zhou et al., 8 Jul 2025).

COLOGR is introduced in the context of the Random Constrained Disambiguation Path problem, denoted RCDP, which extends the Random Disambiguation Path problem and, more distantly, the classical Stochastic Obstacle Scene problem. The distinctive addition is a global disambiguation budget together with potentially heterogeneous obstacle-specific disambiguation costs. The motivating setting is an agent moving from a source $s$ to a target $t$ in a bounded planar region $\Omega \subset \mathbb{R}^2$ populated with ambiguous disk-shaped obstacles whose statuses are assumed independent and static during traversal (Zhou et al., 8 Jul 2025).

The term should be distinguished from two unrelated arXiv usages. The text-generation paper “Modeling Comparative Logical Relation with Contrastive Learning for Text Generation” introduces CoLo for Comparative Logic in Comparative Logical Relation Generation, not COLOGR (Dan et al., 2024). Separately, the arXiv record “Proceedings of the First Workshop on Coalgebra, Horn Clause Logic Programming and Types” does not provide proceedings content in the supplied material from which a COLOGR-specific technical definition could be verified (Komendantskaya et al., 2017).

2. Mathematical formulation of constrained disambiguation

RCDP is discretized on an 8-adjacency integer lattice, yielding an undirected graph $G=(V,E)$ in which each vertex is a lattice point and each edge has Euclidean length $\ell_e$. If a path $p$ intersects ambiguous obstacles, its cost is written as

$\mathcal{C}_p = \ell_p + \delta_p,$

where $\ell_p$ is the total traversal length and

$$\delta_p = \sum_{x: p \cap D(x) \neq \emptyset} \delta_x$$

is the total disambiguation cost. The original objective is

$$\min_{p \in \mathcal{P}(s,t)} \mathbf{E}[\mathcal{C}_p].$$

Because the exact expected-cost problem is stated to be generally intractable, the paper introduces a deterministic surrogate with risk-adjusted edge penalties:

$t$0

and solves

$t$1

This is the WCSPP used by COLOGR (Zhou et al., 8 Jul 2025).

The edge-level quantities are derived from the set of obstacles intersecting an edge,

$t$2

with symmetric allocation

$t$3

and adjusted edge cost

$t$4

The obstacle-level risk penalty $t$5 is instantiated through three families of risk functions (Zhou et al., 8 Jul 2025):

Risk model	Formula	Emphasis
RD	$t$6	strong penalty as $t$7
DT	$t$8	proximity to goal and blockage probability
LU	$t$9	linear undesirability with tunable $\Omega \subset \mathbb{R}^2$0

These constructions encode three coupled quantities on each path: geometric length, ambiguity risk, and budget consumption. A plausible implication is that COLOGR is best viewed not as a purely geometric planner but as a constrained risk-allocation mechanism on a graph.

3. Algorithmic organization: graph initialization, COGR, and LOGR

COLOGR has three stated stages: Graph Initialization (GI), Phase 1: COGR, and Phase 2: LOGR (Zhou et al., 8 Jul 2025).

At initialization, each edge is assigned $\Omega \subset \mathbb{R}^2$1 and $\Omega \subset \mathbb{R}^2$2, converting the uncertain obstacle-planning problem into a deterministic WCSPP. The algorithm then computes two baseline paths: $\Omega \subset \mathbb{R}^2$3, the minimum-cost path with $\Omega \subset \mathbb{R}^2$4, and $\Omega \subset \mathbb{R}^2$5, the minimum-weight path, interpreted in practice as the shortest obstacle-free path. The decision logic is explicit. If $\Omega \subset \mathbb{R}^2$6 already satisfies $\Omega \subset \mathbb{R}^2$7, then it is optimal and the algorithm terminates. If $\Omega \subset \mathbb{R}^2$8 violates $\Omega \subset \mathbb{R}^2$9, the instance is infeasible. Otherwise the lower- and upper-bound paths are initialized as $G=(V,E)$0 and $G=(V,E)$1 (Zhou et al., 8 Jul 2025).

Phase 1, COGR, performs pre-optimization pruning. For each vertex $G=(V,E)$2, the algorithm computes a best path through $G=(V,E)$3 under the weight criterion, $G=(V,E)$4, and a best path through $G=(V,E)$5 under the cost criterion, $G=(V,E)$6. Two elimination rules are then applied. If

$G=(V,E)$7

the vertex is removed as infeasible. If

$G=(V,E)$8

the vertex is removed because it cannot improve the incumbent feasible solution. The phase repeats until no more eliminations occur, and feasible improvements to the incumbent update $G=(V,E)$9 (Zhou et al., 8 Jul 2025).

Phase 2, LOGR, embeds pruning inside iterative Lagrangian optimization. For each multiplier $\ell_e$0, the algorithm computes shortest modified-cost paths through vertices and removes a vertex $\ell_e$1 when its best attainable dual lower bound already exceeds the current feasible upper bound:

$\ell_e$2

This lower-bound pruning is the central operational meaning of TPVE: vertices are discarded either because no feasible path through them exists or because even their best relaxed path is already too expensive (Zhou et al., 8 Jul 2025).

4. Lagrangian relaxation and multiplier search

The budget constraint is dualized through the Lagrangian function

$\ell_e$3

For fixed $\ell_e$4, the relaxed subproblem is an ordinary shortest-path problem with modified edge cost

$\ell_e$5

If $\ell_e$6 minimizes the penalized problem, then

$\ell_e$7

is a valid dual lower bound (Zhou et al., 8 Jul 2025).

The multiplier update is based on two bracketing paths satisfying $\ell_e$8. The next multiplier is

$\ell_e$9

This is obtained by intersecting two affine pieces of the dual function. The paper states that $p$0 is optimal if either the new path exactly exhausts the budget or if three adjacent paths have identical penalized value at $p$1. This reflects the stated piecewise-linear concavity of $p$2 (Zhou et al., 8 Jul 2025).

The paper also gives a cut-based property. For a minimum $p$3-vertex cut $p$4,

$p$5

This bound motivates vertex-local reasoning around cut-critical regions. This suggests that the graph-reduction strategy is not merely heuristic; it is organized around explicit lower and upper bounds at the path level (Zhou et al., 8 Jul 2025).

5. Theoretical properties, optimality conditions, and complexity

A central proposition states Preservation of Optimal Path under TPVE: if $p$6 is the optimal constrained path with $p$7, then $p$8 is retained after both COGR and LOGR. The rationale is explicit. In Phase 1, every vertex on $p$9 lies on a feasible path of cost at most the optimum; in Phase 2, every such vertex has a lower bound not exceeding the optimal surrogate cost, so it never satisfies an elimination condition (Zhou et al., 8 Jul 2025).

The main theorem is framed as Feasibility and Surrogate Optimality of COLOGR. Under the assumption that there exists a path $\mathcal{C}_p = \ell_p + \delta_p,$0 in the reduced graph that minimizes the Lagrangian-relaxed objective at the optimal multiplier $\mathcal{C}_p = \ell_p + \delta_p,$1 and also satisfies the budget constraint, the framework guarantees both feasibility and optimality for the surrogate problem:

$\mathcal{C}_p = \ell_p + \delta_p,$2

The paper is explicit that this is surrogate optimality, not exact optimality for the original stochastic expected-cost problem (Zhou et al., 8 Jul 2025).

Several recovery propositions refine this statement. One concerns identification via a unique vertex signature, where vertex-conditioned shortest-path evaluation can isolate the true constrained optimum even when multiple paths minimize the Lagrangian objective. Another states that if $\mathcal{C}_p = \ell_p + \delta_p,$3 is feasible and also minimizes the Lagrangian objective for some $\mathcal{C}_p = \ell_p + \delta_p,$4, then COLOGR correctly identifies $\mathcal{C}_p = \ell_p + \delta_p,$5, and $\mathcal{C}_p = \ell_p + \delta_p,$6 is an optimal multiplier. The paper also states a property that spurious dual optima vanish under high RBG, where RBG denotes the risk blockage gradient; stronger risk differentiation is said to reduce the persistence of infeasible or suboptimal dual minimizers (Zhou et al., 8 Jul 2025).

The reported worst-case time complexity is

$\mathcal{C}_p = \ell_p + \delta_p,$7

for $\mathcal{C}_p = \ell_p + \delta_p,$8 and $\mathcal{C}_p = \ell_p + \delta_p,$9. Phase 1 contributes roughly

$\ell_p$0

whereas Phase 2 is dominated by a number of Lagrangian breakpoints bounded by

$\ell_p$1

The comparison baseline SNE is assigned higher complexity,

$\ell_p$2

The stated interpretation is that COLOGR becomes especially effective when initial graph reduction is substantial, obstacle density is large enough that many vertices become infeasible or dominated, and risk penalties strongly separate candidate paths (Zhou et al., 8 Jul 2025).

6. Empirical behavior, application domains, and limitations

The simulation setting is a COBRA-style minefield navigation problem on

$\ell_p$3

with start $\ell_p$4, target $\ell_p$5, and 8-adjacency grid discretization. Obstacle counts are $\ell_p$6, with $\ell_p$7 true obstacles, disk radius $\ell_p$8, and a Strauss spatial process with inhibition distance $\ell_p$9 and interaction parameter $$\delta_p = \sum_{x: p \cap D(x) \neq \emptyset} \delta_x$$0. Sensor probabilities are drawn from $$\delta_p = \sum_{x: p \cap D(x) \neq \emptyset} \delta_x$$1 for true obstacles and $$\delta_p = \sum_{x: p \cap D(x) \neq \emptyset} \delta_x$$2 for false obstacles. Budget scenarios include a uniform-cost simplification with $$\delta_p = \sum_{x: p \cap D(x) \neq \emptyset} \delta_x$$3 and $$\delta_p = \sum_{x: p \cap D(x) \neq \emptyset} \delta_x$$4, and a heterogeneous-cost case with $$\delta_p = \sum_{x: p \cap D(x) \neq \emptyset} \delta_x$$5 and $$\delta_p = \sum_{x: p \cap D(x) \neq \emptyset} \delta_x$$6 (Zhou et al., 8 Jul 2025).

The empirical findings reported for COLOGR-based RCDP policies are consistent across several dimensions. They consistently outperform greedy baselines in average traversal cost, especially at high obstacle density, under tight budgets, and under heterogeneous risk/cost settings. They show lower variance, narrower confidence intervals, and better 25th/75th percentile behavior. Greedy baselines often violate resource constraints in dense environments, whereas COLOGR-based policies respect the budget by construction. Among the risk models, LU-based policies are strongest overall, $$\delta_p = \sum_{x: p \cap D(x) \neq \emptyset} \delta_x$$7 often gives a strong tradeoff, $$\delta_p = \sum_{x: p \cap D(x) \neq \emptyset} \delta_x$$8 can do well in sparse to moderate densities, and $$\delta_p = \sum_{x: p \cap D(x) \neq \emptyset} \delta_x$$9 performs better with generous budgets but degrades under tight constraints (Zhou et al., 8 Jul 2025).

The appendix comparison against SNE gives the clearest direct evidence about TPVE:

Statistic	TPVE	SNE
Selected cases: average graph size	142	3948
Selected cases: average duality gap	0	0.026
Selected cases: average path cost	57.304	61.015
Over 1000 cases: average graph size	2423	3745
Over 1000 cases: average duality gap	0	0.001
Over 1000 cases: average path cost	58.444	58.667

These results are presented as support for the claim that COLOGR’s pruning is stronger while still preserving the optimal surrogate path (Zhou et al., 8 Jul 2025).

The paper also includes a toy example with six obstacles and $$\min_{p \in \mathcal{P}(s,t)} \mathbf{E}[\mathcal{C}_p].$$0, where the COLOGR/RCDP policy has expected cost

$$\min_{p \in \mathcal{P}(s,t)} \mathbf{E}[\mathcal{C}_p].$$1

compared with $$\min_{p \in \mathcal{P}(s,t)} \mathbf{E}[\mathcal{C}_p].$$2 for constrained RD. This example is used to illustrate that globally budget-aware planning can outperform local greedy disambiguation (Zhou et al., 8 Jul 2025).

The framework’s stated application domains include stochastic network design, mobility planning, and constrained decision-making under uncertainty, together with minefield navigation, disaster response, logistics, autonomous systems, and GNSS-denied navigation. Its limitations are equally explicit: there is no exact guarantee for the original stochastic expected-cost problem, only for the surrogate problem under assumptions; obstacle statuses are assumed independent; obstacles are disks with binary status; dynamic or adversarial environments are not handled; and the formulation addresses a single-budget constraint rather than multiple simultaneous resource constraints (Zhou et al., 8 Jul 2025).

In that sense, COLOGR occupies a specific methodological niche. It is not a generic stochastic planner, nor merely a shortest-path heuristic. It is a constrained dual-optimization framework whose core contribution lies in combining risk-aware surrogate modeling, Lagrangian relaxation, and safe graph reduction to make budget-feasible disambiguation planning computationally tractable in static uncertain environments (Zhou et al., 8 Jul 2025).