A Fixed Parameter Algorithm for the Minimum Number Convex Partition Problem

2005 ◽  
pp. 83-94 ◽  
Author(s):  
Magdalene Grantson ◽  
Christos Levcopoulos
Keyword(s):  
Partition Problem ◽  
Fixed Parameter ◽  
Minimum Number ◽  
Convex Partition ◽  
Fixed Parameter Algorithm

scholarly journals Minimum Common String Partition Problem: Hardness and Approximations

10.37236/1947 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Avraham Goldstein ◽  
Petr Kolman ◽  
Jie Zheng
Keyword(s):  
Genome Rearrangement ◽  
Linear Time ◽  
Fundamental Problem ◽  
Text Processing ◽  
Partition Problem ◽  
Sorting By Reversals ◽  
String Comparison ◽  
Minimum Number ◽  
Tight Connection ◽  
Minimum Common String Partition

String comparison is a fundamental problem in computer science, with applications in areas such as computational biology, text processing and compression. In this paper we address the minimum common string partition problem, a string comparison problem with tight connection to the problem of sorting by reversals with duplicates, a key problem in genome rearrangement. A partition of a string $A$ is a sequence ${\cal P} = (P_1,P_2,\dots,P_m)$ of strings, called the blocks, whose concatenation is equal to $A$. Given a partition ${\cal P}$ of a string $A$ and a partition ${\cal Q}$ of a string $B$, we say that the pair $\langle{{\cal P},{\cal Q}}\rangle$ is a common partition of $A$ and $B$ if ${\cal Q}$ is a permutation of ${\cal P}$. The minimum common string partition problem (MCSP) is to find a common partition of two strings $A$ and $B$ with the minimum number of blocks. The restricted version of MCSP where each letter occurs at most $k$ times in each input string, is denoted by $k$-MCSP. In this paper, we show that $2$-MCSP (and therefore MCSP) is NP-hard and, moreover, even APX-hard. We describe a $1.1037$-approximation for $2$-MCSP and a linear time $4$-approximation algorithm for $3$-MCSP. We are not aware of any better approximations.

Multiplicative Parameterization Above a Guarantee

2021 ◽  
Vol 13 (3) ◽  
pp. 1-16
Author(s):  
Fedor V. Fomin ◽  
Petr A. Golovach ◽  
Daniel Lokshtanov ◽  
Fahad Panolan ◽  
Saket Saurabh ◽  
...  
Keyword(s):  
Minimum Size ◽  
Input Graph ◽  
Fixed Parameter Tractable ◽  
Long Cycle ◽  
Fixed Parameter ◽  
Parameterized Problem ◽  
Fixed Constant ◽  
Fixed Parameter Algorithm ◽  
Np Hardness ◽  
Additive Form

Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k + g(I) . Here, g(I) is usually a lower bound on the minimum size of a solution. Since its introduction in 1999 for M AX SAT and M AX C UT (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research. We highlight a multiplicative form of parameterization above (or, rather, times) a guarantee: Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k · g(I) . In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, which is the most natural guarantee for this problem, and provide a fixed-parameter algorithm. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ε > 0, multiplicative parameterization above g(I) 1+ε of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of fixed-parameter algorithms as well as kernelization algorithms for additional problems parameterized multiplicatively above girth.

A fixed-parameter algorithm for the maximum agreement forest problem on multifurcating trees

2015 ◽  
Vol 59 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Feng Shi ◽  
Jianxin Wang ◽  
Yufei Yang ◽  
Qilong Feng ◽  
Weilong Li ◽  
...  
Keyword(s):  
Maximum Agreement Forest ◽  
Fixed Parameter ◽  
Fixed Parameter Algorithm

scholarly journals A fixed-parameter algorithm for guarding 1.5D terrains

2015 ◽  
Vol 595 ◽  
pp. 130-142 ◽  
Author(s):  
Farnoosh Khodakarami ◽  
Farzad Didehvar ◽  
Ali Mohades
Keyword(s):  
Fixed Parameter ◽  
Fixed Parameter Algorithm

A fixed-parameter algorithm for the directed feedback vertex set problem

2008 ◽  
Vol 55 (5) ◽  
pp. 1-19 ◽  
Author(s):  
Jianer Chen ◽  
Yang Liu ◽  
Songjian Lu ◽  
Barry O'sullivan ◽  
Igor Razgon
Keyword(s):  
Feedback Vertex Set ◽  
Fixed Parameter ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
Fixed Parameter Algorithm ◽  
Directed Feedback Vertex Set
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