Optimal Distributed Covering Algorithms Ran Ben-Basat 1 , Guy Even 2 , Ken-ichi Kawarabayashi 3 , Gregory Schwartzman 3 1 Harvard 2 Tel-Aviv U. 3 NII 1 / 8
Lower Bound on Number of Communication Rounds Theorem ([KMW16]) Any distributed constant-factor approximation algorithm requires Ω (log ∆ / log log ∆ ) rounds to terminate. lower bound holds for: every constant approximation ratio unweighted graphs, and even if the message lengths are not bounded. 6 / 8
Weighted Graph Vertex Cover Results det. weighted approximation time algorithm yes no 3 O ( ∆ ) [PS09] O ( ∆ 2 ) yes no 2 [Ast+09] yes yes 2 O (1) for ∆ ≤ 3 [Ast+09] O ( ∆ + log ∗ n ) yes yes 2 [PR01] O ( ∆ + log ∗ W ) yes yes 2 [AS10] O (log 2 n ) yes yes 2 [KVY94] O (log n log ∆ / log 2 log ∆ ) yes yes 2 [Ben+18] no yes 2 O (log n ) [GKP08; KY11] yes yes 2 O ( log n ) This work O ( ✏ − 4 log( W · ∆ )) yes yes 2 + ✏ [Hoc82; KMW06] O (log ✏ − 1 log n ) yes yes 2 + ✏ [KVY94] O ( ✏ − 1 log ∆ / log log ∆ ) yes yes 2 + ✏ [BCS17; EGM18] log log ∆ + log ✏ − 1 log ∆ ⇣ ⌘ log ∆ yes yes 2 + ✏ O [Ben+18] log 2 log ∆ ⇣ log log ∆ + log ✏ − 1 · ( log ∆ ) 0 . 001 ⌘ log ∆ yes yes 2 + ✏ O This work 2 + log log ∆ yes yes O (log ∆ / log log ∆ ) [BCS17], ∀ c = O (1) c · log ∆ 2 + (log ∆ ) − c yes yes O (log ∆ / log log ∆ ) [Ben+18], ∀ c = O (1) 2 + 2 − c · (log ∆ ) 0 . 99 yes yes O ( log ∆ / log log ∆ ) This work, ∀ c = O ( 1 ) 7 / 8
Weighted Hypergraph Vertex Cover Results weighted approximation time algorithm f 2 ∆ 2 + f ∆ log ∗ W yes � � [AS10] f O f log 2 n � � yes f O [KVY94] yes f O ( f log n ) This work ⇣ ⌘ log( f ∆ ) ✏ − 1 · f · [EGM18] 1 no f + ✏ O log log( f ∆ ) yes f + ✏ O ( f · log( f / ✏ ) · log n ) [KVY94] ✏ − 4 · f 4 · log f · log( W · ∆ ) yes f + ✏ � � [KMW06] O ⇣ f · log ( f / ✏ ) · ( log ∆ ) 0 . 001 + ⌘ log ∆ yes f + ✏ O This work log log ∆ no f + 1 / c O (log ∆ / log log ∆ ) [EGM18], ∀ f , c = O (1) f + 2 − c · (log ∆ ) 0 . 99 yes O ( log ∆ / log log ∆ ) This work , ∀ f , c = O ( 1 ) 8 / 8
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