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Temporal Connectivity Augmentation

T. Bellitto, J. Bouton Popper, B. Escoffier

TL;DR

We study the problem of augmenting temporal graphs to achieve connectivity by selecting a minimum subset of candidate temporal edges under strict and non-strict path semantics (TCA, TSA, TPCA). The paper proves NP-hardness for TCA and TSA even with lifespan $T=2$, and presents polynomial-time solvable cases such as $(1+1)$-TCA; it also shows TPCA is fixed-parameter tractable when the number of pairs $p$ is fixed by reductions to TG-Steiner, while edge-by-edge TPCA is NP-hard. A key technique is a temporal expansion that maps temporal augmentation problems to static Steiner-like problems (e.g., OCTO, DG-Steiner), enabling existing static algorithms to be reused. The results delineate the boundary between hard and tractable augmentation problems in temporal networks and suggest future work on structural parameters and restricted underlying graphs.

Abstract

Connectivity in temporal graphs relies on the notion of temporal paths, in which edges follow a chronological order (either strict or non-strict). In this work, we investigate the question of how to make a temporal graph connected. More precisely, we tackle the problem of finding, among a set of proposed temporal edges, the smallest subset such that its addition makes the graph temporally connected (TCA). We study the complexity of this problem and variants, under restricted lifespan of the graph, i.e. the maximum time step in the graph. Our main result on TCA is that for any fixed lifespan at least 2, it is NP-complete in both the strict and non-strict setting. We additionally provide a set of restrictions in the non-strict setting which makes the problem solvable in polynomial time and design an algorithm achieving this complexity. Interestingly, we prove that the source variant (making a given vertex a source in the augmented graph) is as difficult as TCA. On the opposite, we prove that the version where a list of connectivity demands has to be satisfied is solvable in polynomial time, when the size of the list is fixed. Finally, we highlight a variant of the previous case for which even with two pairs the problem is already NP-hard.

Temporal Connectivity Augmentation

TL;DR

We study the problem of augmenting temporal graphs to achieve connectivity by selecting a minimum subset of candidate temporal edges under strict and non-strict path semantics (TCA, TSA, TPCA). The paper proves NP-hardness for TCA and TSA even with lifespan , and presents polynomial-time solvable cases such as -TCA; it also shows TPCA is fixed-parameter tractable when the number of pairs is fixed by reductions to TG-Steiner, while edge-by-edge TPCA is NP-hard. A key technique is a temporal expansion that maps temporal augmentation problems to static Steiner-like problems (e.g., OCTO, DG-Steiner), enabling existing static algorithms to be reused. The results delineate the boundary between hard and tractable augmentation problems in temporal networks and suggest future work on structural parameters and restricted underlying graphs.

Abstract

Connectivity in temporal graphs relies on the notion of temporal paths, in which edges follow a chronological order (either strict or non-strict). In this work, we investigate the question of how to make a temporal graph connected. More precisely, we tackle the problem of finding, among a set of proposed temporal edges, the smallest subset such that its addition makes the graph temporally connected (TCA). We study the complexity of this problem and variants, under restricted lifespan of the graph, i.e. the maximum time step in the graph. Our main result on TCA is that for any fixed lifespan at least 2, it is NP-complete in both the strict and non-strict setting. We additionally provide a set of restrictions in the non-strict setting which makes the problem solvable in polynomial time and design an algorithm achieving this complexity. Interestingly, we prove that the source variant (making a given vertex a source in the augmented graph) is as difficult as TCA. On the opposite, we prove that the version where a list of connectivity demands has to be satisfied is solvable in polynomial time, when the size of the list is fixed. Finally, we highlight a variant of the previous case for which even with two pairs the problem is already NP-hard.

Paper Structure

This paper contains 11 sections, 17 theorems, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Temporal connectivity augmentation
  4. Strict TCA
  5. (Non-strict) TCA
  6. (1+1)-TCA
  7. Temporal pair connectivity augmentation
  8. Strict TPCA for a fixed number of pairs
  9. Non-strict TPCA for a fixed number of pairs
  10. Edge-by-edge TPCA
  11. Conclusion

Key Result

Proposition 5

There is a polynomial-time reduction from the temporal spanner problem to the temporal connectivity augmentation problem.

Figures (2)

  • Figure 3: A TG-Steiner instance and its equivalent DG-Steiner expansion. The backward arcs in gray correspond to the non-strict version and the unlabelled arcs have weight 0.
  • Figure : $(1+1)$-TCA algorithm

Theorems & Definitions (21)

  • Definition 1: temporal graph
  • Definition 2: journey
  • Definition 3: temporal connectivity
  • Definition 4: augmentation
  • Proposition 5
  • Theorem 6
  • Corollary 7
  • Theorem 8
  • Corollary 9
  • Lemma 10
  • ...and 11 more