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A game of cops and robbers on graphs with periodic edge-connectivity - - PowerPoint PPT Presentation

Dec 20, 2022 365 likes •969 views

A game of cops and robbers on graphs with periodic edge-connectivity Thomas Erlebach Jakob Spooner Department of Informatics Algorithmic Aspects of Temporal Graphs II ICALP 2019 Satellite Workshop Patras, Greece, 8 July 2019 Thomas Erlebach

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A game of cops and robbers on graphs with periodic edge-connectivity

Thomas Erlebach Jakob Spooner

Department of Informatics

Algorithmic Aspects of Temporal Graphs II ICALP 2019 Satellite Workshop Patras, Greece, 8 July 2019

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 1

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Cops and Robbers

One or several cops chase a robber in a graph Also known as pursuit-evasion games Many variations:

Move along edges or arbitrarily Knowledge about position of opponent Turn-based or simultaneous moves . . .

Some variants relate to graph parameters such as treewidth

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 2

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Discrete Cop and Robber Game

Undirected graph G = (V , E). One cop, one robber.

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 3

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Discrete Cop and Robber Game

Undirected graph G = (V , E). One cop, one robber. Cop chooses its start vertex.

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Discrete Cop and Robber Game

Undirected graph G = (V , E). One cop, one robber. Cop chooses its start vertex. Then robber chooses its start vertex.

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 3

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Discrete Cop and Robber Game

Undirected graph G = (V , E). One cop, one robber. Cop chooses its start vertex. Then robber chooses its start vertex. Then cop and robber take turns. In each turn, can stay at a vertex or move over an edge to an adjacent vertex.

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 3

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Discrete Cop and Robber Game

Undirected graph G = (V , E). One cop, one robber. Cop chooses its start vertex. Then robber chooses its start vertex. Then cop and robber take turns. In each turn, can stay at a vertex or move over an edge to an adjacent vertex. Full knowledge about other player’s position.

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 3

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Discrete Cop and Robber Game

Undirected graph G = (V , E). One cop, one robber. Cop chooses its start vertex. Then robber chooses its start vertex. Then cop and robber take turns. In each turn, can stay at a vertex or move over an edge to an adjacent vertex. Full knowledge about other player’s position. Game ends when cop and robber are on the same vertex (cop wins), or continues indefinitely (robber wins).

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Discrete Cop and Robber Game

Undirected graph G = (V , E). One cop, one robber. Cop chooses its start vertex. Then robber chooses its start vertex. Then cop and robber take turns. In each turn, can stay at a vertex or move over an edge to an adjacent vertex. Full knowledge about other player’s position. Game ends when cop and robber are on the same vertex (cop wins), or continues indefinitely (robber wins). G is cop-win if the cop can guarantee to be at the same vertex as the robber eventually, otherwise robber-win.

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Discrete Cop and Robber Game - Examples

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Discrete Cop and Robber Game - Examples

robber-win

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Discrete Cop and Robber Game - Examples

robber-win cop-win

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Discrete Cop and Robber Game - Examples

robber-win cop-win Studied by Quiliot (1978) and Nowakowski and Winkler (1983). G is cop-win if and only if it can be dismantled.

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Cop and Robber in Temporal Graphs?

We want to consider graphs with infinite lifetime whose edge set can change in every step.

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Cop and Robber in Temporal Graphs?

We want to consider graphs with infinite lifetime whose edge set can change in every step. Assume each edge e has a periodic appearance schedule be of length ℓe, meaning the edge appears in step t if: be[t mod ℓe] = 1

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 5

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Cop and Robber in Temporal Graphs?

We want to consider graphs with infinite lifetime whose edge set can change in every step. Assume each edge e has a periodic appearance schedule be of length ℓe, meaning the edge appears in step t if: be[t mod ℓe] = 1 Example: be = 01001 means ℓe = 5 and the edge appears in steps 1, 4, 6, 9, 11, 14, . . .

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Cop and Robber in Temporal Graphs?

We want to consider graphs with infinite lifetime whose edge set can change in every step. Assume each edge e has a periodic appearance schedule be of length ℓe, meaning the edge appears in step t if: be[t mod ℓe] = 1 Example: be = 01001 means ℓe = 5 and the edge appears in steps 1, 4, 6, 9, 11, 14, . . . Such graphs are called edge-periodic graphs (Casteigts et al., 2011). An edge-periodic graph G is given by a graph G = (V , E) together with be and ℓe for each e ∈ E. Define LCM as the least common multiple of the edge periods ℓe.

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 5

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Edge-Periodic Cop and Robber Game (EPCR)

Edge-periodic graph G. One cop, one robber.

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 6

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Edge-Periodic Cop and Robber Game (EPCR)

Edge-periodic graph G. One cop, one robber. Cop chooses its start vertex.

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 6

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Edge-Periodic Cop and Robber Game (EPCR)

Edge-periodic graph G. One cop, one robber. Cop chooses its start vertex. Then robber chooses its start vertex.

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Edge-Periodic Cop and Robber Game (EPCR)

Edge-periodic graph G. One cop, one robber. Cop chooses its start vertex. Then robber chooses its start vertex. Then cop and robber take turns. In each turn, can stay at a vertex or move over an edge to a vertex that is adjacent in the current time step.

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 6

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Edge-Periodic Cop and Robber Game (EPCR)

Edge-periodic graph G. One cop, one robber. Cop chooses its start vertex. Then robber chooses its start vertex. Then cop and robber take turns. In each turn, can stay at a vertex or move over an edge to a vertex that is adjacent in the current time step. In each time step, the cop moves first, then the robber.

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 6

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Edge-Periodic Cop and Robber Game (EPCR)

Edge-periodic graph G. One cop, one robber. Cop chooses its start vertex. Then robber chooses its start vertex. Then cop and robber take turns. In each turn, can stay at a vertex or move over an edge to a vertex that is adjacent in the current time step. In each time step, the cop moves first, then the robber. Full knowledge about other player’s position and edge appearance schedule.

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 6

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Edge-Periodic Cop and Robber Game (EPCR)

Edge-periodic graph G. One cop, one robber. Cop chooses its start vertex. Then robber chooses its start vertex. Then cop and robber take turns. In each turn, can stay at a vertex or move over an edge to a vertex that is adjacent in the current time step. In each time step, the cop moves first, then the robber. Full knowledge about other player’s position and edge appearance schedule. Game ends when cop and robber are on the same vertex (cop wins), or continues indefinitely (robber wins).

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 6

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Edge-Periodic Cop and Robber Game (EPCR)

Edge-periodic graph G. One cop, one robber. Cop chooses its start vertex. Then robber chooses its start vertex. Then cop and robber take turns. In each turn, can stay at a vertex or move over an edge to a vertex that is adjacent in the current time step. In each time step, the cop moves first, then the robber. Full knowledge about other player’s position and edge appearance schedule. Game ends when cop and robber are on the same vertex (cop wins), or continues indefinitely (robber wins). G is cop-win if the cop can guarantee to be at the same vertex as the robber eventually, otherwise robber-win.

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Result for Edge-Periodic Cop and Robber Games

Theorem There is an algorithm to decide if an edge-periodic graph G with n vertices is cop-win or robber-win (and to compute a winning strategy for the winning player) in time O(n3 · LCM).

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Result for Edge-Periodic Cop and Robber Games

Theorem There is an algorithm to decide if an edge-periodic graph G with n vertices is cop-win or robber-win (and to compute a winning strategy for the winning player) in time O(n3 · LCM). Proof outline. Edge appearance schedule for G repeats every LCM steps. Translate game into a reachability game on a suitable directed graph with O(n2 · LCM) vertices and O(n3 · LCM) edges. Apply known algorithm for reachability games.

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Main Tool in the Proof: Reachability Games

Directed graph G = (V1 ∪ V2, E) with set F ⊆ V1 ∪ V2 of winning positions for player 1. In state v ∈ Vi, player i chooses an outgoing edge to determine the next state. Player 1 wins if the game reaches a state in F eventually. Otherwise, player 2 wins.

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Main Tool in the Proof: Reachability Games

Directed graph G = (V1 ∪ V2, E) with set F ⊆ V1 ∪ V2 of winning positions for player 1. In state v ∈ Vi, player i chooses an outgoing edge to determine the next state. Player 1 wins if the game reaches a state in F eventually. Otherwise, player 2 wins. Theorem (Berwanger (2011), Gr¨ adel et al. (2002)) For a given reachability game, one can determine in linear time the winning states for each player and corresponding memoryless winning strategies.

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Translating EPCR into a Reachability Game

Define a game state (c, r, p, t) for each possible configuration

  • f the EPCR:

c ∈ V is the cop’s location r ∈ V is the robber’s location p ∈ {cop, robber} is the player making the next move t ∈ {0, 1, 2, . . . LCM − 1} is the current time step modulo LCM

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Translating EPCR into a Reachability Game

Define a game state (c, r, p, t) for each possible configuration

  • f the EPCR:

c ∈ V is the cop’s location r ∈ V is the robber’s location p ∈ {cop, robber} is the player making the next move t ∈ {0, 1, 2, . . . LCM − 1} is the current time step modulo LCM

Define directed edges to represent the possible moves of the player whose turn it is.

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Translating EPCR into a Reachability Game

Define a game state (c, r, p, t) for each possible configuration

  • f the EPCR:

c ∈ V is the cop’s location r ∈ V is the robber’s location p ∈ {cop, robber} is the player making the next move t ∈ {0, 1, 2, . . . LCM − 1} is the current time step modulo LCM

Define directed edges to represent the possible moves of the player whose turn it is. There are 2n2 · LCM game states, and at most n outgoing edges per step.

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Translating EPCR into a Reachability Game

Define a game state (c, r, p, t) for each possible configuration

  • f the EPCR:

c ∈ V is the cop’s location r ∈ V is the robber’s location p ∈ {cop, robber} is the player making the next move t ∈ {0, 1, 2, . . . LCM − 1} is the current time step modulo LCM

Define directed edges to represent the possible moves of the player whose turn it is. There are 2n2 · LCM game states, and at most n outgoing edges per step. An edge-periodic cop and robber game G is cop-win if and

  • nly if there is a vertex c ∈ V such that (c, r, cop, 0) is a

cop-winning state in the reachability game for all choices of r. The winning strategies of the reachability game translate into winning strategies of EPCR.

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Strictly Edge-Periodic Cycles

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Strictly Edge-Periodic Graphs and Cycles

An edge-periodic graph G is strictly edge-periodic if each edge appearance schedule is of the form 000 · · · 01. In other words, each e is present once every ℓe steps, namely in the i-th step (step i − 1) where i is a multiple of ℓe.

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Strictly Edge-Periodic Graphs and Cycles

An edge-periodic graph G is strictly edge-periodic if each edge appearance schedule is of the form 000 · · · 01. In other words, each e is present once every ℓe steps, namely in the i-th step (step i − 1) where i is a multiple of ℓe. A strictly edge-periodic cycle is a strictly edge-periodic graph whose edge set forms a cycle. Example:

4 7 12 13 1 5 16 8 32

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Strictly Edge-Periodic Graphs and Cycles

An edge-periodic graph G is strictly edge-periodic if each edge appearance schedule is of the form 000 · · · 01. In other words, each e is present once every ℓe steps, namely in the i-th step (step i − 1) where i is a multiple of ℓe. A strictly edge-periodic cycle is a strictly edge-periodic graph whose edge set forms a cycle. Example:

4 7 12 13 1 5 16 8 32

cop-win

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Strictly Edge-Periodic Graphs and Cycles

An edge-periodic graph G is strictly edge-periodic if each edge appearance schedule is of the form 000 · · · 01. In other words, each e is present once every ℓe steps, namely in the i-th step (step i − 1) where i is a multiple of ℓe. A strictly edge-periodic cycle is a strictly edge-periodic graph whose edge set forms a cycle. Example:

4 7 12 13 1 5 16 8 32 cop

cop-win

Thomas Erlebach and Jakob Spooner A game of cops and robbers on periodic graphs 11

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Strictly Edge-Periodic Graphs and Cycles

An edge-periodic graph G is strictly edge-periodic if each edge appearance schedule is of the form 000 · · · 01. In other words, each e is present once every ℓe steps, namely in the i-th step (step i − 1) where i is a multiple of ℓe. A strictly edge-periodic cycle is a strictly edge-periodic graph whose edge set forms a cycle. Example:

4 7 12 13 1 5 16 8 32 cop robber

cop-win

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Result for Strictly Edge-Periodic Cycles

Question Can we determine a value B such that if the strictly edge-periodic cycle has at least B edges, then it must be robber-win?

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Result for Strictly Edge-Periodic Cycles

Question Can we determine a value B such that if the strictly edge-periodic cycle has at least B edges, then it must be robber-win? Recall that LCM is the least common multiple of all edge periods le. Let f = 1 if LCM ≥ 2 maxe ℓe, and f = 2 otherwise.

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Result for Strictly Edge-Periodic Cycles

Question Can we determine a value B such that if the strictly edge-periodic cycle has at least B edges, then it must be robber-win? Recall that LCM is the least common multiple of all edge periods le. Let f = 1 if LCM ≥ 2 maxe ℓe, and f = 2 otherwise. Theorem If a strictly edge-periodic cycle has at least 4f · LCM + 4 edges, then it is robber-win.

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Auxiliary problem: Infinite edge-periodic path

Infinite path starting at v0, finite set of edge periods le. Cop initially placed at v0.

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Auxiliary problem: Infinite edge-periodic path

Infinite path starting at v0, finite set of edge periods le. Cop initially placed at v0. Lemma The cop and robber game played on an infinite edge-periodic path is robber-win.

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Auxiliary problem: Infinite edge-periodic path

Infinite path starting at v0, finite set of edge periods le. Cop initially placed at v0. Lemma The cop and robber game played on an infinite edge-periodic path is robber-win. Proof. Partition path into segments, such that the cop passes through one segment in f · LCM steps (assuming it never waits unnecessarily).

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Auxiliary problem: Infinite edge-periodic path

Infinite path starting at v0, finite set of edge periods le. Cop initially placed at v0. Lemma The cop and robber game played on an infinite edge-periodic path is robber-win. Proof. Partition path into segments, such that the cop passes through one segment in f · LCM steps (assuming it never waits unnecessarily). Observe: Each segment consists of at least two edges.

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Auxiliary problem: Infinite edge-periodic path

Infinite path starting at v0, finite set of edge periods le. Cop initially placed at v0. Lemma The cop and robber game played on an infinite edge-periodic path is robber-win. Proof. Partition path into segments, such that the cop passes through one segment in f · LCM steps (assuming it never waits unnecessarily). Observe: Each segment consists of at least two edges. Place robber at start of second segment, and always move right

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Robber Strategy for Cycles with ≥ 4f · LCM + 4 Edges

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Robber Strategy for Cycles with ≥ 4f · LCM + 4 Edges

Robber chooses vertex opposite the cop’s start vertex.

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Robber Strategy for Cycles with ≥ 4f · LCM + 4 Edges

Robber chooses vertex opposite the cop’s start vertex. Dodge mode: While robber is in segment of 2–3 edges

  • pposite cop’s current vertex, move towards cop’s antipodal

vertex if possible.

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Robber Strategy for Cycles with ≥ 4f · LCM + 4 Edges

d Robber chooses vertex opposite the cop’s start vertex. Dodge mode: While robber is in segment of 2–3 edges

  • pposite cop’s current vertex, move towards cop’s antipodal

vertex if possible. Escape mode: If cop moves so that robber leaves the segment, use modified infinite path strategy. Note: Initial distance is d ≥ 2f · LCM edges.

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Escape Mode

Consider infinite path starting at cop’s current position following the ring towards the robber.

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Escape Mode

Consider infinite path starting at cop’s current position following the ring towards the robber. Partition path into segments, such that the cop passes through the last edge of each segment at i-th step such that i is a multiple of f · LCM.

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Escape Mode

Consider infinite path starting at cop’s current position following the ring towards the robber. Partition path into segments, such that the cop passes through the last edge of each segment at i-th step such that i is a multiple of f · LCM. Each segment is at most f · LCM edges long.

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Escape Mode

Consider infinite path starting at cop’s current position following the ring towards the robber. Partition path into segments, such that the cop passes through the last edge of each segment at i-th step such that i is a multiple of f · LCM. Each segment is at most f · LCM edges long. As the initial distance between robber and cop is at least 2f · LCM edges, the robber is located in the ≥3rd segment.

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Escape Mode

Consider infinite path starting at cop’s current position following the ring towards the robber. Partition path into segments, such that the cop passes through the last edge of each segment at i-th step such that i is a multiple of f · LCM. Each segment is at most f · LCM edges long. As the initial distance between robber and cop is at least 2f · LCM edges, the robber is located in the ≥3rd segment. Robber strategy: Wait until cop reaches end of first segment, then walk through segments in parallel with the (non-waiting) cop.

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Escape Mode

Consider infinite path starting at cop’s current position following the ring towards the robber. Partition path into segments, such that the cop passes through the last edge of each segment at i-th step such that i is a multiple of f · LCM. Each segment is at most f · LCM edges long. As the initial distance between robber and cop is at least 2f · LCM edges, the robber is located in the ≥3rd segment. Robber strategy: Wait until cop reaches end of first segment, then walk through segments in parallel with the (non-waiting) cop. If the robber is ever located in the segment opposite the cop again, revert to Dodge mode.

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Conclusion

Results for cop and robber games on edge-periodic graphs: O(n3 · LCM) algorithm to determine cop-win or robber-win

Can be extended to k cops using O(n2k+1 · LCM) time or O(nk+1k · LCM) time.

Strictly edge-periodic cycles with at least 4f · LCM + 4 edges are robber-win. Future work: Can we check if an edge-periodic graph is cop-win or robber-win in polynomial time? (The input size is O(|V | + |E| +

e∈E ℓe), but LCM can be

exponential in maxe∈E ℓe.) Are there cop-and-robber games on temporal graphs that lead to useful graph parameters (similar to treewidth for static graphs)?

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Thank you! Questions?