Closeness-Central Shortest Path

• Closeness-central shortest path is defined by evaluating a shortest path based on the maximum distance from any vertex outside the path.
• The optimization minimizes the worst-case vertex-to-path distance, contrasting with traditional vertex closeness measures and emphasizing a covering criterion.
• Its NP-hardness, established via a reduction from SAT, highlights significant computational challenges in both weighted and unweighted graphs.

Closeness-central shortest path is a shortest-path selection problem in which the object being ranked is a path rather than a vertex. In the formulation studied in "Centrality of shortest paths: Algorithms and complexity results" (Phosavanh et al., 2024), the distance from a vertex $u$ to a path $P$ is the minimum shortest-path distance from $u$ to any vertex of $P$, and the path’s closeness score is the maximum such distance over all vertices outside the path. The optimization task is therefore to choose, among all shortest paths in a graph, one that minimizes its farthest-vertex distance. This makes the notion a path-eccentricity objective constrained to the family of geodesics, and it is distinct from the classical vertex-based notions of closeness centrality used in network analysis (Phosavanh et al., 2024).

1. Definition and conceptual scope

In the path-based formulation, the fundamental quantity is the distance from a vertex to a path,

$d(u, P) := \min_{v \in P}\{d_{uv}\},$

where $d_{uv}$ is the shortest-path distance between vertices $u$ and $v$. The closeness score of a path is then defined as

$$C_c^{\max}(P) := \max_{u : u \in V \setminus P}\{d(u, P)\}.$$

A path is thus judged by how far its farthest outside vertex is from it. The optimization problem asks for

$$\min \left\{ C_c^{\max}(P) : P \in \mathcal{SP}(G) \right\},$$

where $P$0 denotes the set of all shortest paths between any pair of nodes in $P$1 (Phosavanh et al., 2024).

This definition differs sharply from classical vertex closeness, where a node is central because its distances to all other nodes are small on average. For a connected simple undirected graph, one standard node-level definition is

$P$2

and another closely related formulation sums inverse distances, sometimes under the name harmonic centrality (Tuzhilin, 2024). In contrast, the closeness-central shortest path problem is not a vertex-ranking problem and does not aggregate distances by summation or averaging; it minimizes a maximum-distance criterion over shortest paths. The problem is therefore best understood as a shortest-path-constrained covering problem rather than as a direct extension of ordinary vertex closeness.

2. Formal optimization and decision formulations

The optimization version seeks a shortest path whose maximum vertex-to-path distance is as small as possible. The associated decision problem, denoted MCCSP-D, is defined as follows: given a graph $P$3 and a positive integer $P$4, determine whether there exists a shortest path between two vertices in $P$5 such that its closeness centrality is less than $P$6 (Phosavanh et al., 2024).

This formulation encodes a global distance objective around a geodesic. The shortest-path constraint is not expressed as a penalty term but as a feasibility restriction: only paths already optimal with respect to endpoint distance are admissible. The closeness criterion then ranks those admissible paths by the worst-case distance from the rest of the graph to the path.

The decision threshold $P$7 has a direct geometric meaning. A positive instance certifies the existence of a shortest path whose radius, measured against all vertices not on the path, is strictly below $P$8. A negative instance says that every shortest path leaves at least one outside vertex at distance at least $P$9. This interpretation is the basis for the later reformulation through fixed-radius coverage.

3. Complexity landscape

The main complexity result is negative: MCCSP-D is NP-complete, and consequently the optimization problem is NP-hard. The paper explicitly states that this hardness holds regardless of whether the graph is weighted or unweighted (Phosavanh et al., 2024).

The NP-hardness proof is by reduction from SAT. The construction uses a backbone graph formed by two linked chains,

$u$0

with cross edges that allow a shortest path to encode a truth assignment by choosing either $u$1 or $u$2 for each variable. Source and sink vertices $u$3 are attached to this backbone, and additional clause-related vertices and paths are introduced so that those vertices remain far from the path unless the encoded assignment satisfies the clauses. The reduction establishes that a satisfiable SAT instance yields a shortest path with closeness centrality at most $u$4, and conversely that any solution path of closeness at most $u$5 can be transformed in polynomial time into a canonical $u$6-to-$u$7 backbone path encoding a satisfying assignment (Phosavanh et al., 2024).

The paper also provides a structural contrast with adjacent shortest-path centrality problems. Finding the most betweenness-central shortest path is solvable in polynomial time in both weighted and unweighted graphs, whereas finding the most closeness-central shortest path remains NP-hard in both settings. The explanation given is that betweenness is essentially a counting or distance-invariant measure over geodesics, while the closeness objective depends on the global distance structure around the path,

$u$8

which makes the optimization substantially harder (Phosavanh et al., 2024).

4. Reformulation through fixed-radius coverage

A later development recasts the problem through a family of fixed-radius objectives. "Designing Efficient and Reachable Routes: The $u$9-Step-Central Shortest Path Problem" (Phosavanh et al., 12 Jun 2026) defines, for an unweighted graph $P$0, the optimization problem

$P$1

where $P$2 measures how many vertices lie within distance $P$3 of the shortest path $P$4. Using

$P$5

the open $P$6-step neighborhood can be written as

$P$7

and the centrality score is

$P$8

The connection to closeness-central shortest path is explicit. The path-closeness objective can be recovered as the smallest radius at which an optimal shortest path covers the entire graph: $P$9 Here $d(u, P) := \min_{v \in P}\{d_{uv}\},$0 is the optimal value of the inclusive $d(u, P) := \min_{v \in P}\{d_{uv}\},$1-step coverage problem. This equivalence shows that closeness-central shortest path is the threshold at which fixed-radius reachability becomes complete coverage (Phosavanh et al., 12 Jun 2026).

The same paper proves that the $d(u, P) := \min_{v \in P}\{d_{uv}\},$2-step problem on an unweighted graph can be solved in

$d(u, P) := \min_{v \in P}\{d_{uv}\},$3

time, where $d(u, P) := \min_{v \in P}\{d_{uv}\},$4 is the graph diameter, by exploiting a pruning rule based on suffix agreement of partial shortest paths. If two shortest paths share the same final $d(u, P) := \min_{v \in P}\{d_{uv}\},$5 vertices, only the one with the larger $d(u, P) := \min_{v \in P}\{d_{uv}\},$6-neighborhood on the differing prefix needs to be kept. From this, the paper derives that closeness-central shortest path on an unweighted graph can be solved in

$d(u, P) := \min_{v \in P}\{d_{uv}\},$7

The same source also notes that this is not a general-purpose efficient algorithm for large-diameter graphs, because the worst-case runtime for closeness via repeated $d(u, P) := \min_{v \in P}\{d_{uv}\},$8-solves is exponential in the graph diameter (Phosavanh et al., 12 Jun 2026).

5. Relation to other shortest-path centrality objectives

Within the same path-centrality framework, shortest paths can be ranked by several different criteria. The paper that defines closeness-central shortest path also studies degree-central and betweenness-central shortest paths, thereby placing the closeness variant within a broader family of geodesic-selection problems (Phosavanh et al., 2024).

A concise comparison is useful.

Problem	Objective on a shortest path $d(u, P) := \min_{v \in P}\{d_{uv}\},$9	Complexity stated in the sources
Degree-central shortest path	maximize degree centrality of a path	polynomial in unweighted graphs; NP-hard in weighted graphs (Phosavanh et al., 2024)
Betweenness-central shortest path	maximize betweenness centrality of a path	polynomial in weighted and unweighted graphs (Phosavanh et al., 2024)
Closeness-central shortest path	minimize $d_{uv}$0	NP-hard in weighted and unweighted graphs (Phosavanh et al., 2024)
$d_{uv}$1-step-central shortest path	maximize $d_{uv}$2 over shortest paths	polynomial on unweighted graphs for fixed $d_{uv}$3; NP-hard with edge weights (Phosavanh et al., 12 Jun 2026)

The closeness-central variant is therefore exceptional not because it is the only path-centrality notion tied to distances, but because its global max-distance dependence produces a much harder optimization problem than either path degree or path betweenness. The $d_{uv}$4-step formulation provides a more tractable intermediate objective on unweighted graphs, but it does so by fixing the radius and maximizing coverage rather than directly minimizing the worst uncovered distance.

6. Distinctions, misconceptions, and neighboring notions of closeness

A common source of confusion is the term closeness. In ordinary network analysis, closeness usually refers to a vertex score derived from all-pairs shortest-path distances. For example, one standard definition is

$d_{uv}$5

and another is the normalized reciprocal-of-average-distance form

$d_{uv}$6

In other work, especially on disconnected graphs, harmonic centrality is used, with scores based on sums of inverse distances (Skibski, 2021). None of these is the same object as the closeness-central shortest path, which evaluates a path by the distance of its farthest outside vertex rather than evaluating a vertex by total or average proximity.

A second distinction concerns current-flow variants. Current-flow closeness replaces shortest-path distance by effective resistance, so that accessibility depends on all paths in the electrical-network sense rather than on geodesic distance alone. For a node $d_{uv}$7,

$d_{uv}$8

where $d_{uv}$9 is effective resistance distance (Bergamini et al., 2016). This current-flow notion is explicitly presented as a measure that depends on all paths, whereas the closeness-central shortest path problem is built from shortest-path distances and restricts attention to paths that are themselves shortest.

A third distinction is methodological. Shortest-path-based centralities can be distorted when the underlying graph representation creates spurious paths, as in naively aggregated multidimensional networks. One paper shows that aggregation may artificially shorten distances and inflate closeness scores, and proposes computing shortest-path-based centralities directly in the MultiAspect Graph formalism to avoid such artifacts (Wehmuth et al., 2020). This suggests that closeness-central shortest path, being defined entirely through shortest-path geometry, is meaningful only to the extent that the admissible shortest paths faithfully represent the original network constraints.

Taken together, these neighboring results situate closeness-central shortest path as a specialized geodesic-selection problem. It is not a vertex centrality, not a current-flow quantity, and not merely a rephrasing of average-distance closeness. Its defining feature is the minimization of worst-case graph distance to a shortest path, which gives it a natural covering interpretation and also explains its computational difficulty.