Weighted Maximum Planning Problem

• Weighted maximum planning is a multi-objective method that minimizes the highest weighted cost among objectives, ensuring Pareto-complete recovery even in non-convex trade-offs.
• Exact graph search techniques and heuristic methods like WM-LNS are used to navigate the inherent NP-hardness, balancing exploration with computational efficiency.
• Hybrid approaches leveraging weighted-sum subproblems and adaptive weight strategies offer practical solutions to scaling complex planning problems.

The weighted maximum planning problem is a multi-objective planning problem in which a feasible plan is chosen by minimizing a weighted maximum of objective values rather than a weighted sum. In the graph formulation, a feasible path is $x\in \mathcal X\equiv \mathcal P_{s,g}$, the set of simple paths from start $s$ to goal $g$, with non-negative objective functions $f_i\colon \mathcal X\to\mathbb R_{\ge0}$ and a weight vector $$w\in\mathcal W=\{w\in\mathbb R_{\ge0}^m\mid \sum_{i=1}^m w_i=1\}$$, leading to

$$x^*=\arg\min_{x\in\mathcal X}\;\max_{i=1,\dots,m}\;w_i\,f_i(x).$$

In continuous or discrete trajectory spaces, the same scalarization is written as

$$c^{max}(T;w)=\max_{i=1,\dots,n}\{w_i f_i(T)\}+\rho\sum_{i=1}^n f_i(T),$$

where $\rho>0$ is a small tie-breaking constant (Kalavadia et al., 6 Apr 2026, Wilde et al., 2023). The main rationale for this scalarization is expressiveness: unlike weighted-sum scalarization, the weighted maximum is Pareto-complete and can recover Pareto-optimal solutions in non-convex regions of the trade-off space (Kalavadia et al., 6 Apr 2026, Wilde et al., 2023).

1. Formal optimization model

Two closely related formulations appear in the recent planning literature. In the discrete graph setting, the environment is a graph $G=(V,E)$, and the feasible set is the collection of simple $s\to g$ paths. The objective is to minimize the weighted maximum $s$0 over that path set, optionally with a small tie-breaker $s$1 to avoid weakly Pareto-optimal ties (Kalavadia et al., 6 Apr 2026).

In the broader motion-planning formulation of Wilde et al., one assumes a continuous or discrete state-action space $s$2, a start state $s$3, a goal set $s$4, and a set of feasible trajectories $s$5 satisfying dynamics and obstacle constraints. The scalarized cost is again

$s$6

and it can be rewritten with an auxiliary variable $s$7 as

$s$8

subject to

$s$9

where $g$0 encodes kinodynamic and obstacle constraints (Wilde et al., 2023).

For graph search, Wilde et al. also allow each edge $g$1 to carry non-negative costs $g$2, while the path-level objective is assembled through a monotone composition

$g$3

with examples including summation and maximum over edges (Wilde et al., 2023). This is technically significant because the correctness of dominance pruning depends on monotonicity of the $g$4.

2. Expressiveness and relation to weighted sums

The weighted maximum and weighted sum scalarizations differ fundamentally. The weighted sum is

$g$5

whereas the weighted maximum is

$g$6

Geometrically, weighted-sum optimization identifies a point of contact between a supporting hyperplane with normal $g$7 and the convex hull of the feasible objective-space image, so it can recover only Pareto-optimal solutions on the convex portion of the front. By contrast, the weighted-maximum scalarization is Pareto-complete: for every Pareto-optimal plan there exists some $g$8 such that the plan minimizes the weighted maximum, including solutions in non-convex regions of the Pareto front (Kalavadia et al., 6 Apr 2026).

Wilde et al. state the same distinction for motion planning more generally: the weighted sum only recovers Pareto-optimal solutions on the convex hull of the front, whereas every minimizer of $g$9 is Pareto-optimal, and for every Pareto-optimal trajectory $f_i\colon \mathcal X\to\mathbb R_{\ge0}$0 there exists some weight vector $f_i\colon \mathcal X\to\mathbb R_{\ge0}$1 such that $f_i\colon \mathcal X\to\mathbb R_{\ge0}$2 (Wilde et al., 2023).

Several theoretical consequences follow. A convex-hull characterization is given in the form that the set $f_i\colon \mathcal X\to\mathbb R_{\ge0}$3 of all weighted-sum-optimal plans coincides with those Pareto-optimal cost vectors lying on the convex hull boundary of the feasible objective-space (Kalavadia et al., 6 Apr 2026). For the same weight vector $f_i\colon \mathcal X\to\mathbb R_{\ge0}$4, an approximation bound relates the two scalarizations: $f_i\colon \mathcal X\to\mathbb R_{\ge0}$5 where $f_i\colon \mathcal X\to\mathbb R_{\ge0}$6 and $f_i\colon \mathcal X\to\mathbb R_{\ge0}$7 are the weighted-sum and weighted-maximum optima, respectively (Kalavadia et al., 6 Apr 2026). At the same time, selecting a proxy weight vector $f_i\colon \mathcal X\to\mathbb R_{\ge0}$8 so that a weighted-sum solution tightly approximates the weighted-maximum solution is itself NP-hard (Kalavadia et al., 6 Apr 2026). This establishes that weighted maximum is not merely a differently scaled weighted sum; it defines a different recovery map over the Pareto set.

3. Exact graph-search methods

For finite graphs, Wilde et al. propose an exact discrete planning algorithm that combines Dijkstra-style best-first expansion with Pareto dominance checks on partial paths (Wilde et al., 2023). The algorithm maintains an open list of partial paths $f_i\colon \mathcal X\to\mathbb R_{\ge0}$9 ending at vertex $$w\in\mathcal W=\{w\in\mathbb R_{\ge0}^m\mid \sum_{i=1}^m w_i=1\}$$0, ordered by the current weighted-maximum cost. At each iteration it removes the minimum-cost entry, returns when the goal is popped, and otherwise extends the path to neighboring vertices.

Three pruning rules are central. First, cyclic extensions are discarded. Second, if an existing partial path $$w\in\mathcal W=\{w\in\mathbb R_{\ge0}^m\mid \sum_{i=1}^m w_i=1\}$$1 to the same vertex $$w\in\mathcal W=\{w\in\mathbb R_{\ge0}^m\mid \sum_{i=1}^m w_i=1\}$$2 satisfies $$w\in\mathcal W=\{w\in\mathbb R_{\ge0}^m\mid \sum_{i=1}^m w_i=1\}$$3 for all $$w\in\mathcal W=\{w\in\mathbb R_{\ge0}^m\mid \sum_{i=1}^m w_i=1\}$$4, then the new partial path $$w\in\mathcal W=\{w\in\mathbb R_{\ge0}^m\mid \sum_{i=1}^m w_i=1\}$$5 is rejected. Third, if the new path dominates existing partial paths to $$w\in\mathcal W=\{w\in\mathbb R_{\ge0}^m\mid \sum_{i=1}^m w_i=1\}$$6, those entries are removed. The key lemma is a subpath-elimination property: if two partial paths end at the same vertex and one dominates the other entry-wise in the objectives, then any extension of the dominated path cannot improve over an extension of the dominating path. Because the path compositions $$w\in\mathcal W=\{w\in\mathbb R_{\ge0}^m\mid \sum_{i=1}^m w_i=1\}$$7 are monotone, this lemma transfers from prefixes to complete paths (Wilde et al., 2023).

Under these conditions, the algorithm is complete and returns an optimal weighted-maximum path whenever one exists. Wilde et al. also establish NP-hardness of the discrete weighted-maximum planning problem, even when the path objectives are simple sums, by reduction from the Multi-Objective Shortest Path problem (Wilde et al., 2023). Worst-case complexity is exponential because the method may have to explore all simple paths; the paper states a worst-case bound of $$w\in\mathcal W=\{w\in\mathbb R_{\ge0}^m\mid \sum_{i=1}^m w_i=1\}$$8 for the unbudgeted algorithm (Wilde et al., 2023).

Two practical adaptations are introduced. An admissible cost-to-go heuristic adds a virtual edge to the goal with objective-wise underestimates $$w\in\mathcal W=\{w\in\mathbb R_{\ge0}^m\mid \sum_{i=1}^m w_i=1\}$$9, yielding an A$$x^*=\arg\min_{x\in\mathcal X}\;\max_{i=1,\dots,m}\;w_i\,f_i(x).$$0-style acceleration while preserving optimality. A second adaptation imposes a per-vertex budget $$x^*=\arg\min_{x\in\mathcal X}\;\max_{i=1,\dots,m}\;w_i\,f_i(x).$$1 on stored Pareto-nondominated partial paths. This creates an anytime algorithm: as $$x^*=\arg\min_{x\in\mathcal X}\;\max_{i=1,\dots,m}\;w_i\,f_i(x).$$2 increases, solution quality improves, and $$x^*=\arg\min_{x\in\mathcal X}\;\max_{i=1,\dots,m}\;w_i\,f_i(x).$$3 recovers optimality (Wilde et al., 2023).

4. Large Neighbourhood Search for weighted-maximum planning

A more recent development is WM-LNS, which embeds weighted-maximum planning in an Adaptive Large Neighbourhood Search framework (Kalavadia et al., 6 Apr 2026). The method begins from an initial solution, repeatedly destroys a contiguous segment of the current plan, repairs the gap by solving a weighted-sum subproblem, and evaluates the repaired path under the weighted-maximum objective.

The repair phase is the main hybridization mechanism. After destruction, a repair weight vector $$x^*=\arg\min_{x\in\mathcal X}\;\max_{i=1,\dots,m}\;w_i\,f_i(x).$$4 is chosen either by random sampling or, for $$x^*=\arg\min_{x\in\mathcal X}\;\max_{i=1,\dots,m}\;w_i\,f_i(x).$$5, by a short Generalized Pattern Search in the weight simplex $$x^*=\arg\min_{x\in\mathcal X}\;\max_{i=1,\dots,m}\;w_i\,f_i(x).$$6. Repair then solves a weighted-sum shortest-path problem between cut points via A$$x^*=\arg\min_{x\in\mathcal X}\;\max_{i=1,\dots,m}\;w_i\,f_i(x).$$7 on the scalar cost $$x^*=\arg\min_{x\in\mathcal X}\;\max_{i=1,\dots,m}\;w_i\,f_i(x).$$8 (Kalavadia et al., 6 Apr 2026). This design exploits polynomial weighted-sum subproblems while searching for weighted-maximum optima.

WM-LNS uses five destroy operators:

• Worst removal: remove the contiguous segment of length $$x^*=\arg\min_{x\in\mathcal X}\;\max_{i=1,\dots,m}\;w_i\,f_i(x).$$9 with largest WM cost.
• Best removal: remove the segment of length $$c^{max}(T;w)=\max_{i=1,\dots,n}\{w_i f_i(T)\}+\rho\sum_{i=1}^n f_i(T),$$0 with smallest WM cost.
• Unbalanced-objective removal: remove the segment with largest mean absolute deviation among $$c^{max}(T;w)=\max_{i=1,\dots,n}\{w_i f_i(T)\}+\rho\sum_{i=1}^n f_i(T),$$1.
• Balanced-objective removal: remove the segment with smallest deviation.
• Random removal: remove a randomly chosen segment.

Acceptance is controlled by a simulated-annealing rule on

$$c^{max}(T;w)=\max_{i=1,\dots,n}\{w_i f_i(T)\}+\rho\sum_{i=1}^n f_i(T),$$2

with improving moves always accepted, identical moves skipped, and deteriorating moves accepted with probability $$c^{max}(T;w)=\max_{i=1,\dots,n}\{w_i f_i(T)\}+\rho\sum_{i=1}^n f_i(T),$$3. Destroy heuristics are selected adaptively using roulette-wheel sampling from operator scores updated from past performance (Kalavadia et al., 6 Apr 2026).

The method is heuristic rather than exact. The paper gives no formal convergence proof, although it notes that each repair step solves a polynomial weighted-sum subproblem and that the overall search explores exponentially many combinations in practice (Kalavadia et al., 6 Apr 2026).

5. Empirical behavior

The empirical record reported so far separates into exact or near-exact graph search studies and large-neighbourhood heuristic studies.

For Wilde et al., the reported metrics are dispersion, coverage, and the number of unique solutions. On four benchmark settings, the paper gives the following weighted-sum and weighted-maximum results (Wilde et al., 2023):

Problem	WS: Dispersion / Coverage / #Solutions	WM: Dispersion / Coverage / #Solutions
Obstacles	0.49 / 0.49 / 9	0.15 / 0.65 / 17
Driver	0.52 / 0.95 / 22	0.28 / 0.97 / 29
Graph-1	0.26 / 0.71 / 11	0.19 / 0.75 / 40
Graph-2	0.42 / 0.73 / 6	0.20 / 0.80 / 10

The same study reports that, on a PRM with $$c^{max}(T;w)=\max_{i=1,\dots,n}\{w_i f_i(T)\}+\rho\sum_{i=1}^n f_i(T),$$4 nodes, achieving $$c^{max}(T;w)=\max_{i=1,\dots,n}\{w_i f_i(T)\}+\rho\sum_{i=1}^n f_i(T),$$5 suboptimality without the heuristic requires budget $$c^{max}(T;w)=\max_{i=1,\dots,n}\{w_i f_i(T)\}+\rho\sum_{i=1}^n f_i(T),$$6 and runs about $$c^{max}(T;w)=\max_{i=1,\dots,n}\{w_i f_i(T)\}+\rho\sum_{i=1}^n f_i(T),$$7 slower than Dijkstra, at approximately $$c^{max}(T;w)=\max_{i=1,\dots,n}\{w_i f_i(T)\}+\rho\sum_{i=1}^n f_i(T),$$8 s. With the admissible cost-to-go heuristic, the same quality is obtained in about $$c^{max}(T;w)=\max_{i=1,\dots,n}\{w_i f_i(T)\}+\rho\sum_{i=1}^n f_i(T),$$9 s, approximately $\rho>0$0 slower than Dijkstra; even $\rho>0$1 yields $\rho>0$2 suboptimality and about a $\rho>0$3 slowdown (Wilde et al., 2023).

The WM-LNS study evaluates indoor maps and a 7-DOF manipulator with up to $\rho>0$4 objectives and PRMs of up to $\rho>0$5 nodes. The reported metrics are coverage, number of Pareto-optimal solutions found over $\rho>0$6 weight samples, runtime ratio relative to weighted sum, and percentage error relative to exact WM (Kalavadia et al., 6 Apr 2026). The paper reports that WM-LNS matches the coverage of exact WM, outperforms weighted sum by up to $\rho>0$7 in the $\rho>0$8-objective setting and $\rho>0$9 in the $G=(V,E)$0-objective setting, and finds homotopy classes that weighted sum misses. The reported runtime improvement is one to two orders of magnitude: on large PRMs, WM-LNS is $G=(V,E)$1 faster in the $G=(V,E)$2-objective maze and up to $G=(V,E)$3 faster in the $G=(V,E)$4-objective cluttered-boxes setting while preserving less than $G=(V,E)$5 error to the true WM optimum; on the $G=(V,E)$6-DOF manipulator, WM-LNS is approximately $G=(V,E)$7 faster than exact WM with approximately $G=(V,E)$8 error (Kalavadia et al., 6 Apr 2026).

Taken together, these results indicate that weighted maximum can broaden Pareto-front recovery relative to weighted sum, but exact recovery remains computationally expensive and motivates hybrid or budgeted methods.

6. Scope, neighboring formulations, and open issues

The weighted maximum planning problem should be distinguished from several nearby planning formulations that use weights or maximize weighted quantities but optimize different objects. In weighted coverage path planning, one or multiple agents move in a continuous search space and collect rewards from a spatial distribution, with rewards collectible only once; the paper formulates this as a nonlinear MPC problem with Coverage Constraints and a TSP-based initialization heuristic (Schweppe et al., 2024). In persistent monitoring on metric graphs, the objective is instead to minimize the maximum weighted latency between observations, where weighted latency is the maximum time between visits to a vertex multiplied by its importance weight (Alamdari et al., 2012). These are planning problems with weighted objectives, but they are not the weighted-maximum scalarization used in multi-objective robot planning.

Within weighted-maximum planning itself, the principal limitations are explicit. In the discrete case, the problem is NP-hard (Wilde et al., 2023, Kalavadia et al., 6 Apr 2026). Exact graph search may require exponential exploration of simple paths in the worst case (Wilde et al., 2023). WM-LNS has no formal convergence proof, and its performance depends on hyperparameters (Kalavadia et al., 6 Apr 2026). Weight selection becomes more challenging as the number of objectives grows; the current proposals use random logarithmic sampling in $G=(V,E)$9D and short Generalized Pattern Search for $s\to g$0 (Kalavadia et al., 6 Apr 2026).

The stated future directions are correspondingly technical. Suggested next steps include more informed or deterministic weight-generation strategies, formal approximation guarantees for WM-LNS beyond empirical error bounds, extension to dynamic or stochastic environments, and integration of deterministic proxy-problem bounds (Kalavadia et al., 6 Apr 2026). In the exact-search line, the runtime-quality trade-off exposed by admissible heuristics and per-vertex budgets suggests that scalable weighted-maximum planning will continue to rely on structured pruning and hybridization with weighted-sum subproblems (Wilde et al., 2023).