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Temporal Vertex Cover with a Sliding Time Window George B. Mertzios Durham University These results have been presented in ICALP 2018 Joint work with: Eleni C. Akrida, University of Liverpool, Paul G. Spirakis, University of Liverpool, Viktor

  1. Basic definitions: Sliding Window Temporal Vertex Cover Definition (Time Windows) 1 For every time slot t ∈ [1 , T − ∆ + 1] : the time window W t = [ t, t + ∆ − 1] is the sequence of the ∆ consecutive time slots t, t + 1 , . . . , t + ∆ − 1 . 2 E [ W t ] = � i ∈ W t E i is the union of all edges appearing at least once in the time window W t . 3 S [ W t ] = { ( w, t ) ∈ S : t ∈ W t } is the restriction of the temporal vertex subset S to the window W t . George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 9 / 40

  2. Basic definitions: Sliding Window Temporal Vertex Cover Definition (Sliding ∆ -Window Temporal Vertex Cover) A sliding ∆ -window temporal vertex cover of ( G, λ ) is a temporal vertex subset S of ( G, λ ) such that: for every time window W t and for every edge e ∈ E [ W t ] , e is temporally covered by at least one vertex appearance ( w, t ) ∈ S [ W t ] . George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 10 / 40

  3. Basic definitions: Sliding Window Temporal Vertex Cover Example ( ∆ = 4 ) u v w u v w u v w u v w u v w u v w u v w u v w c c c c c c c c 1 2 3 4 5 6 7 8 – { ( c, 2) , ( c, 3) , ( c, 6) , ( c, 8) } is not a sliding ∆ -window temporal vertex cover, as edges cv, cw ∈ E [ W 4 ] are not temporally covered in window W 4 . George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 11 / 40

  4. Basic definitions: Sliding Window Temporal Vertex Cover Example ( ∆ = 4 ) u v w u v w u v w u v w u v w u v w u v w u v w c c c c c c c c 1 2 3 4 5 6 7 8 – { ( c, 2) , ( c, 3) , ( c, 6) , ( c, 8) } is not a sliding ∆ -window temporal vertex cover, as edges cv, cw ∈ E [ W 4 ] are not temporally covered in window W 4 . u v w u v w u v w u v w u v w u v w u v w u v w c c c c c c c c 1 2 3 4 5 6 7 8 – { ( c, 1) , ( c, 5) } is a sliding ∆ -window temporal vertex cover. George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 11 / 40

  5. Basic definitions: Sliding Window Temporal Vertex Cover Sliding Window Temporal Vertex Cover ( SW-TVC ) Input: A temporal graph ( G, λ ) with lifetime T , and an integer ∆ ≤ T . Output: A sliding ∆ -window temporal vertex cover S of ( G, λ ) with the minimum |S| . George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 12 / 40

  6. Overview Basic definitions Temporal vertex cover Temporal vertex cover with a sliding time window Open problems George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 13 / 40

  7. Temporal Vertex Cover: the star temporal case Lemma TVC on star temporal graphs is equivalent to Set Cover . leafs of the underlying star ↔ ground set of the Set Cover instance each snapshot graph ↔ a set in the Set Cover instance Goal: Choose sets (snapshots) to cover all elements (leafs’ edges) Example: u v w u v w u v w u v w u v w u v w u v w u v w c c c c c c c c 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 14 / 40

  8. Temporal Vertex Cover: the star temporal case Lemma TVC on star temporal graphs is equivalent to Set Cover . leafs of the underlying star ↔ ground set of the Set Cover instance each snapshot graph ↔ a set in the Set Cover instance Goal: Choose sets (snapshots) to cover all elements (leafs’ edges) Example: u v w u v w u v w u v w u v w u v w u v w u v w c c c c c c c c 1 2 3 4 5 6 7 8 1 Universe: { u, v, w } 2 Sets: S 1 = { u, v, w } , S 2 = { u } , S 3 = { v } , S 4 = { w } , . . . George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 14 / 40

  9. Temporal Vertex Cover: the star temporal case Lemma TVC on star temporal graphs is equivalent to Set Cover . Lemma TVC on star temporal graphs is equivalent to Hitting Set . George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 15 / 40

  10. Temporal Vertex Cover: the star temporal case Lemma TVC on star temporal graphs is equivalent to Set Cover . Lemma TVC on star temporal graphs is equivalent to Hitting Set . Consequences: 1 TVC is NP-complete even on star temporal graphs. 2 For any ε < 1 , TVC on star temporal graphs cannot be optimally solved in O (2 εT ) time, unless SETH fails (due to Hitting Set). George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 15 / 40

  11. Temporal Vertex Cover: the star temporal case Lemma TVC on star temporal graphs is equivalent to Set Cover . Lemma TVC on star temporal graphs is equivalent to Hitting Set . Consequences: 1 TVC is NP-complete even on star temporal graphs. 2 For any ε < 1 , TVC on star temporal graphs cannot be optimally solved in O (2 εT ) time, unless SETH fails (due to Hitting Set). 3 TVC does not admit a polynomial-time (1 − ε ) ln n -approximation algorithm, unless NP has n O (log log n ) -time deterministic algorithms. George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 15 / 40

  12. Temporal Vertex Cover: the star temporal case Lemma TVC on star temporal graphs is equivalent to Set Cover . Lemma TVC on star temporal graphs is equivalent to Hitting Set . Consequences: 1 TVC is NP-complete even on star temporal graphs. 2 For any ε < 1 , TVC on star temporal graphs cannot be optimally solved in O (2 εT ) time, unless SETH fails (due to Hitting Set). 3 TVC does not admit a polynomial-time (1 − ε ) ln n -approximation algorithm, unless NP has n O (log log n ) -time deterministic algorithms. 4 TVC on star temporal graphs can be ln n -approximated in polynomial time. 5 For general graphs: 2 ln n -approximation algorithm by a similar reduction from TVC to Set Cover George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 15 / 40

  13. Overview Basic definitions Alternative models Temporal vertex cover Temporal vertex cover with a sliding time window Open problems George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 16 / 40

  14. SW-TVC : always star temporal graphs 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 17 / 40

  15. SW-TVC : always star temporal graphs 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 17 / 40

  16. SW-TVC : always star temporal graphs 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 17 / 40

  17. SW-TVC : always star temporal graphs 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 17 / 40

  18. SW-TVC : always star temporal graphs 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 17 / 40

  19. SW-TVC : always star temporal graphs 1 2 3 4 5 6 7 8 On always star temporal graphs, a minimum size SW-TVC contains at most one vertex (the star center) in each snapshot ⇒ we assign a Boolean variable x i ∈ { 0 , 1 } for the snapshot at time i George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 17 / 40

  20. SW-TVC : always star temporal graphs 1 2 3 4 5 6 7 8 On always star temporal graphs, a minimum size SW-TVC contains at most one vertex (the star center) in each snapshot ⇒ we assign a Boolean variable x i ∈ { 0 , 1 } for the snapshot at time i For variables x 1 , x 2 , . . . , x ∆ we define f ( t ; x 1 , x 2 , . . . , x ∆ ) to be the smallest cardinality of a sliding ∆ -window temporal vertex cover S of ( G, λ ) | [1 ,t +∆ − 1] , such that the solution at times t, t + 1 , . . . , t + ∆ − 1 is defined by the variables x 1 , x 2 , . . . , x ∆ . George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 17 / 40

  21. SW-TVC : always star temporal graphs f ( 6; 1, 0, 1 ) 1 2 3 4 5 6 7 8 On always star temporal graphs, a minimum size SW-TVC contains at most one vertex (the star center) in each snapshot ⇒ we assign a Boolean variable x i ∈ { 0 , 1 } for the snapshot at time i For variables x 1 , x 2 , . . . , x ∆ we define f ( t ; x 1 , x 2 , . . . , x ∆ ) to be the smallest cardinality of a sliding ∆ -window temporal vertex cover S of ( G, λ ) | [1 ,t +∆ − 1] , such that the solution at times t, t + 1 , . . . , t + ∆ − 1 is defined by the variables x 1 , x 2 , . . . , x ∆ . George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 17 / 40

  22. SW-TVC : always star temporal graphs f ( 6; 1, 0, 1 ) 1 2 3 4 5 6 7 8 On always star temporal graphs, a minimum size SW-TVC contains at most one vertex (the star center) in each snapshot ⇒ we assign a Boolean variable x i ∈ { 0 , 1 } for the snapshot at time i For variables x 1 , x 2 , . . . , x ∆ we define f ( t ; x 1 , x 2 , . . . , x ∆ ) to be the smallest cardinality of a sliding ∆ -window temporal vertex cover S of ( G, λ ) | [1 ,t +∆ − 1] , such that the solution at times t, t + 1 , . . . , t + ∆ − 1 is defined by the variables x 1 , x 2 , . . . , x ∆ . Lemma (dynamic programming) f ( t ; x 1 , x 2 , . . . , x ∆ ) = x ∆ + min y ∈{ 0 , 1 } { f ( t − 1; y, x 1 , x 2 , . . . , x ∆ − 1 ) } George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 17 / 40

  23. SW-TVC : always star temporal graphs Theorem SW-TVC on always star temporal graphs can be solved in O ( T ∆( n + m ) · 2 ∆ ) time. George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 18 / 40

  24. SW-TVC : always star temporal graphs Theorem SW-TVC on always star temporal graphs can be solved in O ( T ∆( n + m ) · 2 ∆ ) time. Algorithm SW-TVC on always star temporal graphs Input: An always star temporal graph ( G, λ ) with lifetime T and a natural ∆ ≤ T . Output: The cardinality of a minimum sliding ∆ -window temporal vertex cover in ( G, λ ) . for t = 1 to T − ∆ + 1 do for all x 1 , x 2 , . . . , x ∆ ∈ { 0 , 1 } do if { ( c t + i − 1 , t + i − 1) | x i = 1 } is a TVC of ( G, λ ) | [ t,t +∆ − 1] then if t = 1 then f ( t ; x 1 , . . . , x ∆ ) ← � ∆ i =1 x i else f ( t ; x 1 , . . . , x ∆ ) ← x ∆ + min y ∈{ 0 , 1 } { f ( t − 1; y, x 1 , . . . , x ∆ − 1 ) } else f ( t ; x 1 , . . . , x ∆ ) ← ∞ return min x 1 ,...,x ∆ ∈{ 0 , 1 } { f ( T − ∆ + 1; x 1 , . . . , x ∆ ) } George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 18 / 40

  25. SW-TVC : the general case Theorem SW-TVC on general temporal graphs can be solved in O ( T ∆( n + m ) · 2 n (∆+1) ) time. George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 19 / 40

  26. SW-TVC : the general case Theorem SW-TVC on general temporal graphs can be solved in O ( T ∆( n + m ) · 2 n (∆+1) ) time. Main idea: for each of the ∆ snapshots in the (currently) last ∆ -window, we enumerate all 2 n vertex subsets, instead of just enumerating over the truth values of ∆ Boolean variables (“always star” case) George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 19 / 40

  27. SW-TVC : the general case Theorem SW-TVC on general temporal graphs can be solved in O ( T ∆( n + m ) · 2 n (∆+1) ) time. Algorithm SW-TVC on general temporal graphs Input: A temporal graph ( G, λ ) with lifetime T and a natural ∆ ≤ T . Output: The smallest cardinality of a sliding ∆ -window temporal vertex cover in ( G, λ ) . 1: for t = 1 to T − ∆ + 1 do for all A 1 , A 2 , . . . , A ∆ ⊆ V do 2: if � ∆ 3: i =1 ( A i , t + i − 1) is a TVC of ( G, λ ) | [ t,t +∆ − 1] then 4: if t = 1 then f ( t ; A 1 , . . . , A ∆ ) ← � ∆ 5: i =1 | A i | 6: else 7: f ( t ; A 1 , . . . , A ∆ ) ← | A ∆ | + min X ⊆ V { f ( t − 1; X, A 1 , . . . , A ∆ − 1 ) } 8: else 9: f ( t ; A 1 , . . . , A ∆ ) ← ∞ return min A 1 ,...,A ∆ ⊆ V { f ( T − ∆ + 1; A 1 , . . . , A ∆ ) } George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 19 / 40

  28. SW-TVC : Optimality under ETH Theorem For any two (arbitrarily growing) functions f : N → N and g : N → N , there exists a constant ε ∈ (0 , 1) such that SW-TVC cannot be solved in f ( T ) · 2 εn · g (∆) time assuming ETH. George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 20 / 40

  29. SW-TVC : Optimality under ETH Theorem For any two (arbitrarily growing) functions f : N → N and g : N → N , there exists a constant ε ∈ (0 , 1) such that SW-TVC cannot be solved in f ( T ) · 2 εn · g (∆) time assuming ETH. Proof (idea): reduction from Vertex Cover T = ∆ = 2 G 1 = G ; G 2 is an independent set given f and g , choose appropriate ε George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 20 / 40

  30. SW-TVC : Optimality under ETH Theorem For any two (arbitrarily growing) functions f : N → N and g : N → N , there exists a constant ε ∈ (0 , 1) such that SW-TVC cannot be solved in f ( T ) · 2 εn · g (∆) time assuming ETH. Proof (idea): reduction from Vertex Cover T = ∆ = 2 G 1 = G ; G 2 is an independent set given f and g , choose appropriate ε Corollary Our O ( T ∆( n + m ) · 2 n (∆+1) ) -time algorithm is asymptotically almost optimal. George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 20 / 40

  31. SW-TVC: always bounded vertex cover number temporal graphs Let C k be the class of graphs with the vertex cover number at most k . Theorem SW-TVC on always C k temporal graphs can be solved in O ( T ∆( n + m ) · n k (∆+1) ) time. George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 21 / 40

  32. SW-TVC: always bounded vertex cover number temporal graphs Let C k be the class of graphs with the vertex cover number at most k . Theorem SW-TVC on always C k temporal graphs can be solved in O ( T ∆( n + m ) · n k (∆+1) ) time. Main idea: in the optimal solution, the choice at step i is a subset of a minimum vertex cover at this snapshot ⇒ for each of the ∆ last snapshots, enumerate all n k vertex subsets (candidates for vertex cover at snapshot i ) George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 21 / 40

  33. ∆ - TVC If the parameter ∆ (the size of a sliding window) is fixed, we refer to SW-TVC as ∆ - TVC (i.e. ∆ is a part of the problem name). George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 22 / 40

  34. ∆ - TVC If the parameter ∆ (the size of a sliding window) is fixed, we refer to SW-TVC as ∆ - TVC (i.e. ∆ is a part of the problem name). Observation (∆ + 1) - TVC is at least as hard as ∆ - TVC . George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 22 / 40

  35. ∆ - TVC If the parameter ∆ (the size of a sliding window) is fixed, we refer to SW-TVC as ∆ - TVC (i.e. ∆ is a part of the problem name). Observation (∆ + 1) - TVC is at least as hard as ∆ - TVC . . . . . . . . . . . . . . . . . . . G 1 G 2 G ∆ ∅ G ∆+1 G 2∆ ∅ t = T + ⌊ T ∆ ⌋ t = 1 t = 2 t = ∆ t = ∆ + 2 t = 2∆ + 1 t = ∆ + 1 t = 2∆ + 2 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 22 / 40

  36. 2 - TVC is hard to approximate Let X be the class of graphs whose connected components are induced subgraphs of graph Ψ . Ψ George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 23 / 40

  37. 2 - TVC is hard to approximate Let X be the class of graphs whose connected components are induced subgraphs of graph Ψ . Ψ Clearly, Vertex Cover is linearly solvable on graphs from X . George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 23 / 40

  38. 2 - TVC is hard to approximate Let X be the class of graphs whose connected components are induced subgraphs of graph Ψ . Ψ Clearly, Vertex Cover is linearly solvable on graphs from X . Theorem There is no PTAS for 2 - TVC on always X temporal graphs. George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 23 / 40

  39. 2 - TVC is hard to approximate Theorem There is no PTAS for 2 - TVC on always X temporal graphs. Proof (sketch): 1 Let Y be the class of graphs which can be obtained from cubic graphs by subdividing every edge exactly 4 times. 2 There is no PTAS for Vertex Cover on Y . 3 Reduce Vertex Cover on Y to 2 - TVC on always X temporal graphs such that optimal solutions of both problems have same size. K 4 The 4-subdivision of K 4 Snapshot G 1 Snapshot G 2 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 24 / 40

  40. SW-TVC : approximation algorithms I Reduction from SW-TVC to Set Cover . George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 25 / 40

  41. SW-TVC : approximation algorithms I Reduction from SW-TVC to Set Cover . 1 The universe: the set of all pairs ( e, t ) ∈ E × [ T − ∆ + 1] such that e appears (and so must be temporally covered) within window W t . George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 25 / 40

  42. SW-TVC : approximation algorithms I Reduction from SW-TVC to Set Cover . 1 The universe: the set of all pairs ( e, t ) ∈ E × [ T − ∆ + 1] such that e appears (and so must be temporally covered) within window W t . 2 The sets: for every vertex appearance ( v, s ) we define C v,s to be the set of elements ( e, t ) in the universe, such that ( v, s ) temporally covers e in window W t . George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 25 / 40

  43. SW-TVC : approximation algorithms I Reduction from SW-TVC to Set Cover . 1 The universe: the set of all pairs ( e, t ) ∈ E × [ T − ∆ + 1] such that e appears (and so must be temporally covered) within window W t . 2 The sets: for every vertex appearance ( v, s ) we define C v,s to be the set of elements ( e, t ) in the universe, such that ( v, s ) temporally covers e in window W t . Consequences: 1 O (ln n + ln ∆) -approximation (every set C v,s has at most n ∆ elements ⇒ approximation factor H n ∆ − 1 2 ≈ ln n + ln ∆ ) George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 25 / 40

  44. SW-TVC : approximation algorithms I Reduction from SW-TVC to Set Cover . 1 The universe: the set of all pairs ( e, t ) ∈ E × [ T − ∆ + 1] such that e appears (and so must be temporally covered) within window W t . 2 The sets: for every vertex appearance ( v, s ) we define C v,s to be the set of elements ( e, t ) in the universe, such that ( v, s ) temporally covers e in window W t . Consequences: 1 O (ln n + ln ∆) -approximation (every set C v,s has at most n ∆ elements ⇒ approximation factor H n ∆ − 1 2 ≈ ln n + ln ∆ ) 2 2 k -approximation, where k is the maximum edge ∆ -frequency (just take both vertex appearances for every appearance of an edge) George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 25 / 40

  45. SW-TVC : approximation algorithms I Reduction from SW-TVC to Set Cover . 1 The universe: the set of all pairs ( e, t ) ∈ E × [ T − ∆ + 1] such that e appears (and so must be temporally covered) within window W t . 2 The sets: for every vertex appearance ( v, s ) we define C v,s to be the set of elements ( e, t ) in the universe, such that ( v, s ) temporally covers e in window W t . Consequences: 1 O (ln n + ln ∆) -approximation (every set C v,s has at most n ∆ elements ⇒ approximation factor H n ∆ − 1 2 ≈ ln n + ln ∆ ) 2 2 k -approximation, where k is the maximum edge ∆ -frequency (just take both vertex appearances for every appearance of an edge) ⇒ 2∆ -approximation George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 25 / 40

  46. SW-TVC : approximation algorithms II Single-edge temporal graph: exact algorithm 1 2 3 4 5 6 7 8

  47. SW-TVC : approximation algorithms II Single-edge temporal graph: exact algorithm 1 2 3 4 5 6 7 8

  48. SW-TVC : approximation algorithms II Single-edge temporal graph: exact algorithm 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 26 / 40

  49. SW-TVC : approximation algorithms II Single-edge temporal graph: exact algorithm 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 27 / 40

  50. SW-TVC : approximation algorithms II Single-edge temporal graph: exact algorithm 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 28 / 40

  51. SW-TVC : approximation algorithms II Single-edge temporal graph: exact algorithm 1 2 3 4 5 6 7 8

  52. SW-TVC : approximation algorithms II Single-edge temporal graph: exact algorithm 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 29 / 40

  53. SW-TVC : approximation algorithms II Single-edge temporal graph: exact algorithm 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 30 / 40

  54. SW-TVC : approximation algorithms II Single-edge temporal graph: exact algorithm 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 31 / 40

  55. SW-TVC : approximation algorithms II Single-edge temporal graph: exact algorithm Algorithm SW-TVC on single-edge temporal graphs Input: A temporal graph ( G, λ ) of lifetime T with V ( G ) = { u, v } ; and ∆ ≤ T . Output: A minimum-cardinality sliding ∆ -window temporal vertex cover S of ( G, λ ) . 1: S ← ∅ 2: t = 1 3: while t ≤ T − ∆ + 1 do if ∃ r ∈ [ t, t + ∆ − 1] such that uv ∈ E r then 4: 5: choose maximum such r and add ( u, r ) to S 6: t ← r + 1 7: else 8: t ← t + 1 return S greedy algorithm linear time George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 32 / 40

  56. SW-TVC : approximation algorithms II Always degree at most d temporal graphs: d -approx. algorithm Main idea: solve independently each single-edge subgraph of G take the union of the solutions George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 33 / 40

  57. SW-TVC : approximation algorithms II Always degree at most d temporal graphs: d -approx. algorithm 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 34 / 40

  58. SW-TVC : approximation algorithms II Always degree at most d temporal graphs: d -approx. algorithm 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 34 / 40

  59. SW-TVC : approximation algorithms II Always degree at most d temporal graphs: d -approx. algorithm 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 34 / 40

  60. SW-TVC : approximation algorithms II Always degree at most d temporal graphs: d -approx. algorithm 1 2 3 4 5 6 7 8 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 34 / 40

  61. SW-TVC : approximation algorithms II Always degree at most d temporal graphs: d -approx. algorithm 1 2 3 4 5 6 7 8 · · · George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 34 / 40

  62. SW-TVC : approximation algorithms II Always degree at most d temporal graphs: d -approx. algorithm 1 2 3 4 5 6 7 8 · · · George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 35 / 40

  63. SW-TVC : approximation algorithms II Always degree at most d temporal graphs: d -approx. algorithm Algorithm d -approximation of SW-TVC on always degree at most d temporal graphs Input: An always degree at most d temporal graph ( G, λ ) of lifetime T , and ∆ ≤ T . Output: A sliding ∆ -window temporal vertex cover S of ( G, λ ) . 1: for every edge uv ∈ E ( G ) do 2: Compute an optimal solution S uv of the problem for ( G [ { u, v } ] , λ ) ] 3: S ← S ∪ S uv return S George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 36 / 40

  64. SW-TVC : approximation algorithms II Always degree at most d temporal graphs: d -approx. algorithm Algorithm d -approximation of SW-TVC on always degree at most d temporal graphs Input: An always degree at most d temporal graph ( G, λ ) of lifetime T , and ∆ ≤ T . Output: A sliding ∆ -window temporal vertex cover S of ( G, λ ) . 1: for every edge uv ∈ E ( G ) do 2: Compute an optimal solution S uv of the problem for ( G [ { u, v } ] , λ ) ] 3: S ← S ∪ S uv return S Lemma The above algorithm is a O ( mT ) -time d -approximation algorithm for SW-TVC on always degree at most d temporal graphs. George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 36 / 40

  65. SW-TVC : approximation algorithms II Always degree at most d temporal graphs: d -approx. algorithm Algorithm d -approximation of SW-TVC on always degree at most d temporal graphs Input: An always degree at most d temporal graph ( G, λ ) of lifetime T , and ∆ ≤ T . Output: A sliding ∆ -window temporal vertex cover S of ( G, λ ) . 1: for every edge uv ∈ E ( G ) do 2: Compute an optimal solution S uv of the problem for ( G [ { u, v } ] , λ ) ] 3: S ← S ∪ S uv return S Lemma The above algorithm is a O ( mT ) -time d -approximation algorithm for SW-TVC on always degree at most d temporal graphs. Corollary SW-TVC can be optimally solved in O ( mT ) time on the class of always degree at most 1 (matching) temporal graphs. George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 36 / 40

  66. Overview Basic definitions Alternative models Temporal vertex cover Temporal vertex cover with a sliding time window Open problems George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 37 / 40

  67. Open Problems Problem 1 Determine the complexity status of ∆ -TVC on degree at most 2 temporal graphs. 1 ∆ -TVC on always degree at most 1 can be solved in linear time. George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 38 / 40

  68. Open Problems Problem 1 Determine the complexity status of ∆ -TVC on degree at most 2 temporal graphs. 1 ∆ -TVC on always degree at most 1 can be solved in linear time. 2 ∆ -TVC on always degree at most 3: no PTAS George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 38 / 40

  69. Open Problems Problem 1 Determine the complexity status of ∆ -TVC on degree at most 2 temporal graphs. 1 ∆ -TVC on always degree at most 1 can be solved in linear time. 2 ∆ -TVC on always degree at most 3: no PTAS, even when: the underlying graph has degree at most 3; and 1 connected components of snapshots have at most 7 vertices. 2 George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 38 / 40

  70. Open Problems Problem 1 Determine the complexity status of ∆ -TVC on degree at most 2 temporal graphs. 1 ∆ -TVC on always degree at most 1 can be solved in linear time. 2 ∆ -TVC on always degree at most 3: no PTAS, even when: the underlying graph has degree at most 3; and 1 connected components of snapshots have at most 7 vertices. 2 Problem 2 Can ∆ -TVC on general graphs be approximated within a factor better than 2∆ ? George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 38 / 40

  71. Open Problems Problem 1 Determine the complexity status of ∆ -TVC on degree at most 2 temporal graphs. 1 ∆ -TVC on always degree at most 1 can be solved in linear time. 2 ∆ -TVC on always degree at most 3: no PTAS, even when: the underlying graph has degree at most 3; and 1 connected components of snapshots have at most 7 vertices. 2 Problem 2 Can ∆ -TVC on general graphs be approximated within a factor better than 2∆ ? Problem 3 Can ∆ -TVC on always degree at most d temporal graphs be approximated within a factor better than d ? George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 38 / 40

  72. Open Problems Problem 4 FPT results for ∆ -TVC with respect to ∆ on general temporal graphs? George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 39 / 40

  73. Open Problems Problem 4 FPT results for ∆ -TVC with respect to ∆ on general temporal graphs? 1 SW-TVC on general temporal graphs can be solved in O ( T ∆( n + m ) · 2 n (∆+1) ) time. George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 39 / 40

  74. Open Problems Problem 4 FPT results for ∆ -TVC with respect to ∆ on general temporal graphs? 1 SW-TVC on general temporal graphs can be solved in O ( T ∆( n + m ) · 2 n (∆+1) ) time. Problem 5 FPT results for SW-TVC with respect to the maximum vertex cover number k of snapshots? George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 39 / 40

  75. Open Problems Problem 4 FPT results for ∆ -TVC with respect to ∆ on general temporal graphs? 1 SW-TVC on general temporal graphs can be solved in O ( T ∆( n + m ) · 2 n (∆+1) ) time. Problem 5 FPT results for SW-TVC with respect to the maximum vertex cover number k of snapshots? 1 SW-TVC on always vertex cover number at most k temporal graphs can be solved in O ( T ∆( n + m ) · n k (∆+1) ) time. George B. Mertzios Temporal Vertex Cover AlgoUK Liverpool, Sept. 2018 39 / 40

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