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Exploration of temporal graphs with bounded degree Thomas Erlebach - - PowerPoint PPT Presentation

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Exploration of temporal graphs with bounded degree Thomas Erlebach and Jakob Spooner University of Leicester { te17 | jts21 } @leicester.ac.uk ICALP Workshop Algorithmic Aspects of Temporal Graphs 9 July 2018 Thomas Erlebach and Jakob

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Exploration of temporal graphs with bounded degree

Thomas Erlebach and Jakob Spooner University of Leicester {te17|jts21}@leicester.ac.uk

ICALP Workshop “Algorithmic Aspects of Temporal Graphs” 9 July 2018

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018

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Outline

1 Temporal graphs 2 Temporal graph exploration problem (TEXP) 3 Known results

Instances that require Ω(n2) steps

4 Faster exploration of degree-bounded graphs 5 Conclusions Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 1 / 20

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Temporal Graphs

Temporal graph (Dynamic, time-varying graph) A graph in which the edge set can change in every (time) step. Step 0:

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 2.1 / 20

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Temporal Graphs

Temporal graph (Dynamic, time-varying graph) A graph in which the edge set can change in every (time) step. Step 1:

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 2.2 / 20

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Temporal Graphs

Temporal graph (Dynamic, time-varying graph) A graph in which the edge set can change in every (time) step. Step 2:

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 2.3 / 20

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Temporal Graphs

Temporal graph (Dynamic, time-varying graph) A graph in which the edge set can change in every (time) step. Underlying graph The graph with all edges that are present in at least one step.

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 2.4 / 20

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Temporal (Time-Respecting) Path

Time edge A pair (e, t) where e is an edge of the underlying graph and t is a time step when e is present. Temporal path (journey) A sequence of time edges (e1, t1), . . . , (ek, tk) such that (e1, e2, . . . , ek) is a path in the underlying graph and t1 < t2 < · · · < tk. Example:

3 4 7 9

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 3.1 / 20

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Temporal (Time-Respecting) Path

Time edge A pair (e, t) where e is an edge of the underlying graph and t is a time step when e is present. Temporal path (journey) A sequence of time edges (e1, t1), . . . , (ek, tk) such that (e1, e2, . . . , ek) is a path in the underlying graph and t1 < t2 < · · · < tk. Example:

3 4 7 9

Temporal walk: temporal path where vertices may repeat

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 3.2 / 20

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Temporal Graph Exploration

Temporal graph exploration problem (TEXP) Starting at a given vertex s at time 0, find a fastest temporal walk that visits all vertices. Equivalently: Schedule an agent: In each time step, traverse an edge or wait. Minimize time when last vertex is visited.

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 4.1 / 20

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Temporal Graph Exploration

Temporal graph exploration problem (TEXP) Starting at a given vertex s at time 0, find a fastest temporal walk that visits all vertices. Equivalently: Schedule an agent: In each time step, traverse an edge or wait. Minimize time when last vertex is visited. We assume: The whole temporal graph is known in advance.

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 4.2 / 20

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Temporal Graph Exploration

Temporal graph exploration problem (TEXP) Starting at a given vertex s at time 0, find a fastest temporal walk that visits all vertices. Equivalently: Schedule an agent: In each time step, traverse an edge or wait. Minimize time when last vertex is visited. We assume: The whole temporal graph is known in advance. Michail and Spirakis [MFCS’14] It is NP-complete to decide if a temporal graph can be explored if it need not be connected in each step.

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 4.3 / 20

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Temporal Graph Exploration

Temporal graph exploration problem (TEXP) Starting at a given vertex s at time 0, find a fastest temporal walk that visits all vertices. Equivalently: Schedule an agent: In each time step, traverse an edge or wait. Minimize time when last vertex is visited. We assume: The whole temporal graph is known in advance. Michail and Spirakis [MFCS’14] It is NP-complete to decide if a temporal graph can be explored if it need not be connected in each step. ⇒ Like Michail and Spirakis, we consider temporal graphs that are connected in each step and have lifetime ≥ n2. (Note: We consider undirected graphs only.)

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 4.4 / 20

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Temporal Graph Exploration

Temporal graph exploration problem (TEXP) Starting at a given vertex s at time 0, find a fastest temporal walk that visits all vertices. Equivalently: Schedule an agent: In each time step, traverse an edge or wait. Minimize time when last vertex is visited. We assume: The whole temporal graph is known in advance. Reachability lemma: Let G be a temporal graph with n vertices. Agent can reach any vertex v from vertex u in n time steps. Proof. Since G always has a u-v path, the set of vertices reachable from u increases in each step until v is reached.

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 4.5 / 20

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Temporal Graph Exploration

Temporal graph exploration problem (TEXP) Starting at a given vertex s at time 0, find a fastest temporal walk that visits all vertices. Equivalently: Schedule an agent: In each time step, traverse an edge or wait. Minimize time when last vertex is visited. We assume: The whole temporal graph is known in advance. Reachability lemma: Let G be a temporal graph with n vertices. Agent can reach any vertex v from vertex u in n time steps. Corollary Any temporal graph can be explored in n2 time steps.

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 4.6 / 20

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Example

Instance of Temporal Graph Exploration problem:

Step 0 Step 1 Step 2 Step 3 Step 4 Step 5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 5.1 / 20

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Example

Instance of Temporal Graph Exploration problem:

Step 0 Step 1 Step 2 Step 3 Step 4 Step 5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 5.2 / 20

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Example

Instance of Temporal Graph Exploration problem:

Step 0 Step 1 Step 2 Step 3 Step 4 Step 5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 5.3 / 20

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Example

Instance of Temporal Graph Exploration problem:

Step 0 Step 1 Step 2 Step 3 Step 4 Step 5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 5.4 / 20

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Example

Instance of Temporal Graph Exploration problem:

Step 0 Step 1 Step 2 Step 3 Step 4 Step 5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 5.5 / 20

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Example

Temporal exploration completed in Step 5.

Step 0 Step 1 Step 2 Step 3 Step 4 Step 5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 5.6 / 20

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Previous Work on TEXP

Avin, Kouck´ y, Lotker, ICALP’08: Analyze cover time of random walk in temporal graph (with self-loops) Star construction shows that simple random walk may take Ω(2n) steps Lazy random walk that leaves v only with probability deg(v)/(∆ + 1) has cover time O(∆2n3 log2 n) Michail and Spirakis, MFCS’14: D-approximation algorithm for temporal graph exploration, where D is the dynamic diameter Note: 1 ≤ D ≤ n − 1, can be equal to n − 1 No (2 − ε)-approximation algorithm unless P = NP (1.7 + ε)-approximation algorithm for temporal TSP with dynamic edge weights in {1, 2}

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 6 / 20

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Previous Work on TEXP

E, Hoffmann, Kammer, ICALP’15: Instances of TEXP that require Ω(n2) steps No O(n1−ε)-approximation algorithm unless P = NP Results for restricted underlying graphs:

treewidth k: O(n1.5k1.5 log n) steps planar: O(n1.8 log n) steps cycle, cycle with chord: O(n) steps 2 × n grid: O(n log3 n) steps Instances of TEXP where underlying graph is planar with ∆ = 4 that require Ω(n log n) steps

Further results on temporal graphs with randomly present edges or regularly present edges.

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 7 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let c0 be the center of a star in step 0. c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.1 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let c1 be the center of a star in step 1. c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.2 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let c2 be the center of a star in step 2. c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.3 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let c3 be the center of a star in step 3. c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.4 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let c4 be the center of a star in step 4. c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.5 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let c5 be the center of a star in step 5. c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.6 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let c0 be the center of a star in step 6. c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.7 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let ci be the center of a star in step i, n

2 + i, n + i, . . .

c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.8 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let ci be the center of a star in step i, n

2 + i, n + i, . . .

Agent starts in c0. Let us only focus on exploring . c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.9 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let ci be the center of a star in step i, n

2 + i, n + i, . . .

Agent starts in c0. Let us only focus on exploring . c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.10 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let ci be the center of a star in step i, n

2 + i, n + i, . . .

Agent starts in c0. Let us only focus on exploring . c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.11 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let ci be the center of a star in step i, n

2 + i, n + i, . . .

Agent starts in c0. Let us only focus on exploring . c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.12 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let ci be the center of a star in step i, n

2 + i, n + i, . . .

Agent starts in c0. Let us only focus on exploring . c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.13 / 20

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SLIDE 36

TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let ci be the center of a star in step i, n

2 + i, n + i, . . .

Agent starts in c0. Let us only focus on exploring . c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.14 / 20

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TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let ci be the center of a star in step i, n

2 + i, n + i, . . .

After returning to ci, wait until ci is center again. c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.15 / 20

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SLIDE 38

TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let ci be the center of a star in step i, n

2 + i, n + i, . . .

After returning to ci, wait until ci is center again. c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.16 / 20

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SLIDE 39

TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let ci be the center of a star in step i, n

2 + i, n + i, . . .

After returning to ci, wait until ci is center again. Each move from x to y with x = y: Ω(n) time steps. c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.17 / 20

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SLIDE 40

TEXP instances that require Ω(n2) steps

Consider the temporal graph below that is a star in each step. Let ci be the center of a star in step i, n

2 + i, n + i, . . .

After returning to ci, wait until ci is center again. Each move from x to y with x = y: Ω(n) time steps. In total, Ω(n2) time steps. c2 c0 c1 c3 c4 c5 v0 v1 v2 v3 v4 v5

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 8.18 / 20

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Observations

The TEXP instances requiring Ω(n2) steps have these properties: The underlying graph is very dense (Ω(n2) edges). The graph in each step has a high-degree vertex (the center of the star has degree n − 1). The graph changes in every step. Questions What if we place a restriction on one of these?

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 9.1 / 20

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Observations

The TEXP instances requiring Ω(n2) steps have these properties: The underlying graph is very dense (Ω(n2) edges). The graph in each step has a high-degree vertex (the center of the star has degree n − 1). The graph changes in every step. Questions What if we place a restriction on one of these? Today: What if the graph in each step has bounded degree?

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 9.2 / 20

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Bounded Degree Graph Exploration

Temporal graph of bounded degree A temporal graph G has degree bounded by ∆ if the graph in each step has maximum degree at most ∆. Question: What is the worst-case exploration time for temporal graphs of bounded degree? We know: Upper bound O(n2) holds for arbitrary graphs. Lower bound Ω(n log n) for underlying planar graphs with maximum degree ∆ = 4.

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 10 / 20

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Main Result

Theorem A temporal graph G with degree bounded by ∆ can always be explored in O

  • log ∆ ·

n2 log n

  • steps.

Remarks: For log ∆ = o(log n), the exploration time is o(n2). For ∆ = O(1), the exploration time is O( n2

log n).

There is still a huge gap to the lower bound of Ω(n log n).

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 11 / 20

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Proof Overview

Theorem A temporal graph G with degree bounded by ∆ can be explored in O

  • log ∆ ·

n2 log n

  • steps.

Proof. While there are Ω(

n log∆ n) unexplored vertices, visit

O(log∆ n) unexplored vertices in O(n) steps. ⇒ O

  • n

log∆ n · n

  • = O
  • log ∆ ·

n2 log n

  • steps

Visit the last O(

n log∆ n) unexplored vertices in O(n) steps

per vertex. ⇒ O

  • n

log∆ n · n

  • = O
  • log ∆ ·

n2 log n

  • steps

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 12 / 20

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Visiting many vertices quickly

Lemma (Main Lemma) While there are Ω(

n log∆ n) unexplored vertices, we can visit

O(log∆ n) unexplored vertices in O(n) steps. Proof idea. Assume current vertex is v, current step is t. Let U be the current set of unexplored vertices. Claim: There exists a walk W starting at some u ∈ U at time t + n that visits O(log∆ n) unexplored vertices in O(n) steps. ⇒ Move from v to u during time t to t + n, then follow W .

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 13 / 20

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Auxiliary Lemma

Lemma (Auxiliary Lemma) Let T be a set of k = |T| unexplored vertices. There are Ω( k

∆)

disjoint pairs (u, v) ∈ T 2 s.t. u can reach v in O( ∆n

k ) steps.

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 14 / 20

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SLIDE 48

Proof of Claim

Claim There is a walk W starting at some u ∈ U at time t + n that visits O(log∆ n) unexplored vertices in O(n) steps. Proof sketch.

O(∆n

k )

O(∆2n

k )

O(∆3n

k )

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 15.1 / 20

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SLIDE 49

Proof of Claim

Claim There is a walk W starting at some u ∈ U at time t + n that visits O(log∆ n) unexplored vertices in O(n) steps. Proof sketch.

O( k

∆) pairs

O(∆n

k )

O(∆2n

k )

O(∆3n

k )

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 15.2 / 20

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SLIDE 50

Proof of Claim

Claim There is a walk W starting at some u ∈ U at time t + n that visits O(log∆ n) unexplored vertices in O(n) steps. Proof sketch.

O( k

∆) pairs

O(∆3n

k )

O(∆2n

k )

O(∆n

k )

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 15.3 / 20

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SLIDE 51

Proof of Claim

Claim There is a walk W starting at some u ∈ U at time t + n that visits O(log∆ n) unexplored vertices in O(n) steps. Proof sketch.

O( k

∆) pairs

O( k

∆2) pairs

O(∆n

k )

O(∆2n

k )

O(∆3n

k )

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 15.4 / 20

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SLIDE 52

Proof of Claim

Claim There is a walk W starting at some u ∈ U at time t + n that visits O(log∆ n) unexplored vertices in O(n) steps. Proof sketch.

O( k

∆) pairs

O( k

∆2) pairs

O(∆n

k )

O(∆2n

k )

O(∆3n

k )

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 15.5 / 20

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SLIDE 53

Proof of Claim

Claim There is a walk W starting at some u ∈ U at time t + n that visits O(log∆ n) unexplored vertices in O(n) steps. Proof sketch.

O( k

∆) pairs

O( k

∆2) pairs

O( k

∆3) pairs

O(∆n

k )

O(∆2n

k )

O(∆3n

k )

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 15.6 / 20

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SLIDE 54

Proof of Claim

Claim There is a walk W starting at some u ∈ U at time t + n that visits O(log∆ n) unexplored vertices in O(n) steps. Proof sketch.

O( k

∆) pairs

O( k

∆2) pairs

O( k

∆3) pairs

O(∆n

k )

O(∆2n

k )

O(∆3n

k )

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 15.7 / 20

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SLIDE 55

Proof of Auxiliary Lemma

Lemma (Auxiliary Lemma) Let T be a set of k = |T| unexplored vertices. There are Ω( k

∆)

disjoint pairs (u, v) ∈ T 2 s.t. u can reach v in O( ∆n

k ) steps.

Proof. Maintain a home set Hv ⊆ T of each v ∈ L = V \ T:

0 ≤ |Hv| ≤ 2 Each u ∈ Hv can reach v by the current time step.

If a vertex w ∈ T is adjacent to a vertex v ∈ L with u ∈ Hv for some u = w, a pair (u, w) is formed.

w { } u u v

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 16 / 20

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SLIDE 56

Potential function

Potential function Φ =

v∈L(|Hv| + 1) ≤ 3n.

We can show:

Φ increases by ≈

k 2∆ in each step.

Formation of a pair decreases potential by at most 20∆n

k

.

If fewer than

k 20∆ pairs were formed in 10∆n k

steps, we would have Φ > 10∆n k · k 2∆ − k 20∆ · 20∆n k = 5n − n > 3n , a contradiction.

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 17 / 20

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SLIDE 57

Obtaining the potential increase

Consider spanning tree T of current graph. Find Ω( k

∆) disjoint paths Pu,w between vertices u, w ∈ T.

On path Pu,w, increase potential of one vertex v ∈ L by adding u or w to its home set Hv (and possibly adjusting

  • ther home sets).

Example:

u w {u} {u, a} {u} {a, b} {w}

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 18.1 / 20

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SLIDE 58

Obtaining the potential increase

Consider spanning tree T of current graph. Find Ω( k

∆) disjoint paths Pu,w between vertices u, w ∈ T.

On path Pu,w, increase potential of one vertex v ∈ L by adding u or w to its home set Hv (and possibly adjusting

  • ther home sets).

Example:

u w {u} {a, b} {u} {u, a} a {w}

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 18.2 / 20

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SLIDE 59

Obtaining the potential increase

Consider spanning tree T of current graph. Find Ω( k

∆) disjoint paths Pu,w between vertices u, w ∈ T.

On path Pu,w, increase potential of one vertex v ∈ L by adding u or w to its home set Hv (and possibly adjusting

  • ther home sets).

Example:

u w {u} {a, b} {u} {u, a} {w}

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 18.3 / 20

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SLIDE 60

Obtaining the potential increase

Consider spanning tree T of current graph. Find Ω( k

∆) disjoint paths Pu,w between vertices u, w ∈ T.

On path Pu,w, increase potential of one vertex v ∈ L by adding u or w to its home set Hv (and possibly adjusting

  • ther home sets).

Example:

b u w {u} {a, b} {u} {u, a} {w}

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 18.4 / 20

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SLIDE 61

Obtaining the potential increase

Consider spanning tree T of current graph. Find Ω( k

∆) disjoint paths Pu,w between vertices u, w ∈ T.

On path Pu,w, increase potential of one vertex v ∈ L by adding u or w to its home set Hv (and possibly adjusting

  • ther home sets).

Example:

u w {u} {u, a} {u, b} {a} {w}

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 18.5 / 20

slide-62
SLIDE 62

Obtaining the potential increase

Consider spanning tree T of current graph. Find Ω( k

∆) disjoint paths Pu,w between vertices u, w ∈ T.

On path Pu,w, increase potential of one vertex v ∈ L by adding u or w to its home set Hv (and possibly adjusting

  • ther home sets).

Example:

add u u w {u} {u, a} {u, b} {a} {w}

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 18.6 / 20

slide-63
SLIDE 63

Obtaining the potential increase

Consider spanning tree T of current graph. Find Ω( k

∆) disjoint paths Pu,w between vertices u, w ∈ T.

On path Pu,w, increase potential of one vertex v ∈ L by adding u or w to its home set Hv (and possibly adjusting

  • ther home sets).

Example:

+1 u w {u} {u, a} {a, u} {u, b} {w}

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 18.7 / 20

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SLIDE 64

Conclusions

We have shown that temporal graphs whose degree is bounded by ∆ in each step can be explored in O(log ∆ ·

n2 log n) steps.

The best known lower bound for small ∆ is only Ω(n log n) steps, so a large gap remains. We are still only at the beginning of understanding how restrictions on the underlying graph or on the graph in each step affect the worst-case exploration time.

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 19 / 20

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SLIDE 65

Open Problems

Close the gap for temporal graphs of bounded degree in each step. Exploration of temporal graphs whose underlying graph is planar:

What is the largest number of steps required?

Upper bound: O(n1.8 log n) steps Lower bound: Ω(n log n) steps

Approximation algorithms?

Underlying graphs from other graph classes:

n × n grids Planar graphs of bounded degree Arbitrary graphs of bounded degree

Instance-dependent lower bounds on exploration time Graphs that change only every c > 1 steps

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018 20 / 20

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SLIDE 66

Thank you!

Thomas Erlebach and Jakob Spooner ICALP Workshop, Prague, 9 July 2018