The Ordered Covering Problem

The Ordered Covering Problem We study the Ordered Covering (OC) problem. The input is a finite set of n elements X, a color function $$c:X \rightarrow \{0,1\}$$ c : X → { 0 , 1 } and a collection $$\mathcal {S}$$ S of subsets of X. A solution consists of an ordered tuple $$T=(S_1,\dots ,S_{\ell })$$ T = ( S 1 , ⋯ , S ℓ ) of sets from $$\mathcal {S}$$ S which covers X, and a coloring $$g:\{S_i\}_{i=1}^\ell \rightarrow \{0,1\}$$ g : { S i } i = 1 ℓ → { 0 , 1 } such that $$\forall x \in X$$ ∀ x ∈ X , the first set covering x in the tuple, namely $$S_j$$ S j with $$j=\min \{i : x \in S_i\}$$ j = min { i : x ∈ S i } , has color $$g(S_j)=c(x)$$ g ( S j ) = c ( x ) . The minimization version is to find a solution using the minimum number of sets. Variants of OC include OC $$(\alpha _0,\alpha _1)$$ ( α 0 , α 1 ) in which each element of color $$i \in \{0,1\}$$ i ∈ { 0 , 1 } appears in at most $$\alpha _i$$ α i sets of $$\mathcal {S}$$ S , and k–OC in which the first set of the solution $$S_1$$ S 1 is required to have color 0, and there are at most $$k-1$$ k - 1 alternations of colors in the solution. Among other results we show: There is a polynomial time approximation algorithm for Min–OC(2, 2) with approximation ratio 2. (This is best possible unless Vertex Cover can be approximated within a ratio better than 2.) Moreover, Min–OC(2, 2) can be solved optimally in polynomial time if the underlying instance is bipartite. For every $$\alpha _0, \alpha _1 \ge 2$$ α 0 , α 1 ≥ 2 , there is a polynomial time approximation algorithm for Min–3–OC $$(\alpha _0,\alpha _1)$$ ( α 0 , α 1 ) with approximation $$\alpha _1(\alpha _0 - 1)$$ α 1 ( α 0 - 1 ) . Unless the unique games conjecture is false, this is best possible. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Algorithmica Springer Journals

The Ordered Covering Problem

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Publisher
Springer US
Copyright
Copyright © 2017 by Springer Science+Business Media, LLC
Subject
Computer Science; Algorithm Analysis and Problem Complexity; Theory of Computation; Mathematics of Computing; Algorithms; Computer Systems Organization and Communication Networks; Data Structures, Cryptology and Information Theory
ISSN
0178-4617
eISSN
1432-0541
D.O.I.
10.1007/s00453-017-0357-6
Publisher site
See Article on Publisher Site

Abstract

We study the Ordered Covering (OC) problem. The input is a finite set of n elements X, a color function $$c:X \rightarrow \{0,1\}$$ c : X → { 0 , 1 } and a collection $$\mathcal {S}$$ S of subsets of X. A solution consists of an ordered tuple $$T=(S_1,\dots ,S_{\ell })$$ T = ( S 1 , ⋯ , S ℓ ) of sets from $$\mathcal {S}$$ S which covers X, and a coloring $$g:\{S_i\}_{i=1}^\ell \rightarrow \{0,1\}$$ g : { S i } i = 1 ℓ → { 0 , 1 } such that $$\forall x \in X$$ ∀ x ∈ X , the first set covering x in the tuple, namely $$S_j$$ S j with $$j=\min \{i : x \in S_i\}$$ j = min { i : x ∈ S i } , has color $$g(S_j)=c(x)$$ g ( S j ) = c ( x ) . The minimization version is to find a solution using the minimum number of sets. Variants of OC include OC $$(\alpha _0,\alpha _1)$$ ( α 0 , α 1 ) in which each element of color $$i \in \{0,1\}$$ i ∈ { 0 , 1 } appears in at most $$\alpha _i$$ α i sets of $$\mathcal {S}$$ S , and k–OC in which the first set of the solution $$S_1$$ S 1 is required to have color 0, and there are at most $$k-1$$ k - 1 alternations of colors in the solution. Among other results we show: There is a polynomial time approximation algorithm for Min–OC(2, 2) with approximation ratio 2. (This is best possible unless Vertex Cover can be approximated within a ratio better than 2.) Moreover, Min–OC(2, 2) can be solved optimally in polynomial time if the underlying instance is bipartite. For every $$\alpha _0, \alpha _1 \ge 2$$ α 0 , α 1 ≥ 2 , there is a polynomial time approximation algorithm for Min–3–OC $$(\alpha _0,\alpha _1)$$ ( α 0 , α 1 ) with approximation $$\alpha _1(\alpha _0 - 1)$$ α 1 ( α 0 - 1 ) . Unless the unique games conjecture is false, this is best possible.

Journal

AlgorithmicaSpringer Journals

Published: Aug 22, 2017

References

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