Title Text

On Making Graphs Temporally (In)Efficient

Pierluigi Crescenzi

Gran Sasso Science Institute

Algorithmic Aspects of Temporal Graphs V
Paris, July 4, 2022

Joint work with Filippo Brunelli and Laurent Viennot

Based on F. Brunelli, P. Crescenzi, L. Viennot, Maximizing reachability in a temporal graph obtained by assigning starting times to a collection of walks, Networks. (2022), 1– 27.

Pierluigi Crescenzi

July 4, 2022

Introduction

• Find graph (bounded) modification which  maximizes (minimizes) specific optimization criterion
  • Well-known example
    • Maximum Diameter Edge Addition (Schoone et al, 1987)
      • Input: Connected graph G and integers k and D
      • Question: Can we obtain super-graph G' of G by adding k edges to G, such that G' has diameter at most D?
    • Lot of research on this problem and many variations
  • Other graph operations and optimization criteria
    • E.g. node/edge deletion or edge contraction
    • E.g. reachability, information diffusion, centrality measures

Network optimization

Temporal Graphs@ICALP

Introduction

Temporal graphs

• Temporal graph \(\mathbb{G}=(V,\mathbb{E})\)
  • \(V\): set of nodes (\(n=|V|\))
  • \(\mathbb{E}\): set of (temporal) edges \((u,v,t,\lambda)\) (\(m=|\mathbb{E}|\))
    • Starting time: \(t\)
    • Travel time: \(\lambda>0\)
    • Arrival time: \(t+\lambda\)
• Path \(\mathbb{P}\) from \(u\) to \(v\) (\(v\) temporally reachable from \(u\))
  • Sequence of edges \[(u=w_{1},w_{2},t_{1},\lambda_{1}), (w_{2},w_{3},t_{2},\lambda_{2}),\ldots,(w_{k},w_{k+1}=v,t_{k},\lambda_{k})\] such that, for each \(i\) with \(1< i\leq k\), \(t_i\geq t_{i-1}+\lambda_{i-1}\)
  • Departure time: \(t_1\)
  • Arrival time: \(t_{k}+\lambda_{k}\)
• Reachability of \(u\): number of \(v\) temporally reachable from \(u\)
• Reachability of \(\mathbb{G}\): sum of reachability of all nodes (number of pairs \((u,v)\) such that \(v\) temporally reachable from \(u\))

Pierluigi Crescenzi

July 4, 2022

Temporal Graphs@ICALP

Introduction

Temporal graph

• Node \(x\) can reach \(x,y,w\)
• Node \(y,w,z\) can reach \(y,w\)
• Nodes \(w,z\) can reach all nodes
• Reachability is 13

Pierluigi Crescenzi

July 4, 2022

Temporal Graphs@ICALP

Introduction

Temporal graph

• Node \(x\) can reach \(x,y,w\)
• Nodes \(y,w,z\) can reach all nodes
• Reachability is 15

Pierluigi Crescenzi

July 4, 2022

Temporal Graphs@ICALP

Introduction

Edge temporalization

Pierluigi Crescenzi

• Network optimization problem on weighted multidigraph \(D\)
  • Graph operation: edge temporalization
    • Assign to each edge \((u,v,\lambda)\) of \(D\) a starting time \(t\) (making it temporal edge \((u,v,t,\lambda)\))
  • Optimization criterion: maximizing reachability of resulting temporal graph
• Example

July 4, 2022

Temporal Graphs@ICALP

Introduction

Minimizing maximum reachability

Pierluigi Crescenzi

• Network optimization problem on unweighted (di)graph \(D\)
  • Graph operation: edge temporalization
  • Optimization criterion: minimizing maximum reachability of all nodes in resulting temporal graph
• Example

July 4, 2022

Temporal Graphs@ICALP

Introduction

Minimizing reachability through edge delaying

Pierluigi Crescenzi

• Network optimization problem on graph \(G\)
  • Graph operation: edge temporalization
  • Optimization criterion: minimizing number of time-labels so that the resulting temporal is temporally connected

July 4, 2022

Temporal Graphs@ICALP

Introduction

Minimizing reachability through edge delaying

Pierluigi Crescenzi

• Network optimization problem on temporal graph \(\mathbb{G}\)
  • Graph operation: edge delaying
    • \(\delta\)-delaying of \((u,v,t,\lambda)\): \((u,v,t+\delta,\lambda)\)
  • Optimization criterion: minimizing reachability of resulting temporal graph

July 4, 2022

Temporal Graphs@ICALP

Introduction

Minimizing reachability through edge delaying

Pierluigi Crescenzi

• Network optimization problem on temporal graph \(\mathbb{G}\)
  • Graph operation: edge shifting
    • \(\delta\)-shifting of \((u,v,t,\lambda)\): \((u,v,t\pm\delta,\lambda)\)
  • Optimization criterion: minimizing maximum time a set of source vertices needs to reach every other vertex of the graph

July 4, 2022

Temporal Graphs@ICALP

Our problem

Trip network

Pierluigi Crescenzi

• Motivation: public transport system
• Given weighted multidigraph \(D\)
  • Trip collection \(\mathbb{T}\) on \(D\): collection of walks
    • Two walks can share same edge
    • Not necessarily pairwise disjoint
• Example

• Trip network: \((D,\mathbb{T})\)
  • Induced multidigraph \(M(D,\mathbb{T})\): disjoint union of all trips

July 4, 2022

Temporal Graphs@ICALP

Our problem

Trip temporalization

Pierluigi Crescenzi

• Trip temporalization of trip network \((D,\mathbb{T})\)
  • Assigns starting time to each trip in \(\mathbb{T}\)
    • Implies edge temporalization of \(M(D,\mathbb{T})\)
      • If \(T=e_1,\ldots,e_k\) with \(e_i=(u_i,v_i,\lambda_i)\) and starting time \(t\) assigned to \(T\), then temporal edges \[(u_i,v_i,t+\sum_{j=1}^{i-1}\lambda_j,\lambda_i)\] in edge temporalization of \(M(D,\mathbb{T})\)
  • Assumption: negligible waiting time at stops

July 4, 2022

Temporal Graphs@ICALP

Our problem

Trip temporalization

Pierluigi Crescenzi

• Assign 1 to blue trip, 6 to green trip, and 10 to red trip
  • Resulting temporal graph

• Reachability: 30

July 4, 2022

Temporal Graphs@ICALP

Our problem

Maximum reachability trip temporalization

Pierluigi Crescenzi

• Maximum Reachability Trip Temporalization (MRTT)
  • Network optimization problem on trip network \((D,\mathbb{T})\)
    • Graph operation: trip temporalization
    • Optimization criterion: maximizing reachability of resulting temporal graph

July 4, 2022

Temporal Graphs@ICALP

Our problem

MRTT variants

Pierluigi Crescenzi

• One-to-One Reachability Trip Temporalization (O2ORTT)
  • Decision problem on trip network \((D,\mathbb{T})\) and nodes \(s\) and \(t\)
    • Solution: trip temporalization
    • Goal: making \(t\) temporally reachable from \(s\) in resulting temporal graph
  • Parameterized version on number of used trips
• Single Source Maximum Reachability Trip Temporalization (SSMRTT)
  • Network optimization problem on trip network \((D,\mathbb{T})\) and node \(s\)
    • Graph operation: trip temporalization
    • Optimization criterion: maximizing number of nodes temporally reachable from \(s\) in resulting temporal graph

July 4, 2022

Temporal Graphs@ICALP

Our problem

Special cases

Pierluigi Crescenzi

• Strongly temporalizable trip network \((D,\mathbb{T})\)
  • For each pair of nodes \(u\) and \(v\), there exists a trip temporalisation of \(\mathbb{T}\) that makes \(v\) temporally reachable from \(u\) in resulting temporal graph
• Symmetric trip network \((D,\mathbb{T})\)
  • For each trip \(T\in\mathbb{T}\), \(\mathbb{T}\) includes also the reverse trip
    • Trip starting from the last node of \(T\), arriving in the first node of \(T\), and passing through all the nodes in \(T\) in reverse order

July 4, 2022

Temporal Graphs@ICALP

Proof sketches

Pierluigi Crescenzi

Reduction from 3-SAT to O2O-RTT

• Variable \(x_i\) gadget

• Clause \(c_j\) gadget

• Reduction example
  \((x_1\vee x_2\vee\neg x_3)\wedge(\neg x_1\vee x_2\vee x_3)\wedge(\neg x_1\vee\neg x_2\vee\neg x_3)\)

• Edge \((l_j^h,t_i^k)\) if \(\neg x_i\) is \(h\)-th literal of \(c_j\) and \(k\)-th occurrence of \(\neg x_i\)

• Edge \((l_j^h,f_i^k)\) if \(x_i\) is \(h\)-th literal of \(c_j\) and \(k\)-th occurrence of \(x_i\)

• Trips \((w_j,l_j^h,t_i^k,s(t_i^k))\) or \((w_j,l_j^h,f_i^k,s(f_i^k))\)

• Edge \((v_{n+1},w_1)\)

July 4, 2022

Temporal Graphs@ICALP

Proof sketches

Pierluigi Crescenzi

\(k\)-O2O-RTT

• Application of color coding technique described in (Alon et al, 2016)
  • Dynamic programming technique to decide whether, given a \(k\)-coloring of the trips, there exists a path from \(s\) to \(t\) which is the concatenation of exactly \(i\) sub-trips of distinct trips in with pairwise distinct colors
    • This implies trip temporalization such that \(t\) is temporally reachable from \(s\)
  • Use of appropriate set of perfect hash functions from \(\mathbb{T}\) to \(\{1,\ldots,k\}\)
    • \(2^{O(k)}\log |\mathbb{T}|\) such functions

July 4, 2022

Temporal Graphs@ICALP

Proof sketches

Pierluigi Crescenzi

MRTT in strongly temporalizable trip networks

• Strongly temporalizable trip network s.t. any trip temporalization cannot connect more than \(n^{3/2}\) pairs of nodes

July 4, 2022

Temporal Graphs@ICALP

Proof sketches

Pierluigi Crescenzi

MRTT in symmetric and strongly temporalizable trip networks

• Still NP-hard
• There exists ordering of trips such that the reachability of the resulting temporal graphs is at least \(2/9\) of all node pairs
  • Construct spanning tree of transfer weighted graph
    • Nodes: \((T,T^r)\)
    • Edges: if they share a node in \(D\)
    • Weight: nodes of \(D\) in \(T\) (and, then, in \(T^r\))
  • Find centroid
    • Node whose removal generates sub-trees of weight at most 2/3 of the total weight
  • Prove that a constant fraction of nodes can "reach" the centroid and a constant fraction of nodes can be reached from the centroid

July 4, 2022

Temporal Graphs@ICALP