TY - GEN

T1 - On Hardness of Approximation of Parameterized Set Cover and Label Cover

T2 - 4th Symposium on Simplicity in Algorithms, SOSA 2021, co-located with SODA 2021

AU - Karthik, C. S.

AU - Livni-Navon, Inbal

N1 - Publisher Copyright:
Copyright © 2021 by SIAM.

PY - 2021/1/1

Y1 - 2021/1/1

N2 - In the (k, h)-SetCover problem, we are given a collection S of sets over a universe U, and the goal is to distinguish between the case that S contains k sets which cover U, from the case that at least h sets in S are needed to cover U. Lin (ICALP’19) recently showed a gap creating reduction from the (k, k + 1)-SetCover problem on universe of size Ok(log |S|) to the (k, qklogloglog|S||S| · k ) -SetCover problem on universe of size |S|. In this paper, we prove a more scalable version of his result: given any error correcting code C over alphabet [q], rate ρ, and relative distance δ, we use C to create a reduction from the (k, k + 1)-SetCover problem on universe U to the (k, 2qk1−2δ) -SetCover problem on universe of size logρ|S| |U|qk. Lin established his result by composing the input SetCover instance (that has no gap) with a special threshold graph constructed from extremal combinatorial object called universal sets, resulting in a final SetCover instance with gap. Our reduction follows along the exact same lines, except that we generate the threshold graphs specified by Lin simply using the basic properties of the error correcting code C. We further show that one can recover the precise result of Lin by using a code which also achieves optimal parameters as a perfect hash function.

AB - In the (k, h)-SetCover problem, we are given a collection S of sets over a universe U, and the goal is to distinguish between the case that S contains k sets which cover U, from the case that at least h sets in S are needed to cover U. Lin (ICALP’19) recently showed a gap creating reduction from the (k, k + 1)-SetCover problem on universe of size Ok(log |S|) to the (k, qklogloglog|S||S| · k ) -SetCover problem on universe of size |S|. In this paper, we prove a more scalable version of his result: given any error correcting code C over alphabet [q], rate ρ, and relative distance δ, we use C to create a reduction from the (k, k + 1)-SetCover problem on universe U to the (k, 2qk1−2δ) -SetCover problem on universe of size logρ|S| |U|qk. Lin established his result by composing the input SetCover instance (that has no gap) with a special threshold graph constructed from extremal combinatorial object called universal sets, resulting in a final SetCover instance with gap. Our reduction follows along the exact same lines, except that we generate the threshold graphs specified by Lin simply using the basic properties of the error correcting code C. We further show that one can recover the precise result of Lin by using a code which also achieves optimal parameters as a perfect hash function.

UR - http://www.scopus.com/inward/record.url?scp=85101952273&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:85101952273

T3 - 4th Symposium on Simplicity in Algorithms, SOSA 2021

SP - 210

EP - 223

BT - 4th Symposium on Simplicity in Algorithms, SOSA 2021

A2 - King, Valerie

A2 - Le, Hung Viet

PB - Society for Industrial and Applied Mathematics Publications

Y2 - 11 January 2021 through 12 January 2021

ER -