# Differential approximation of NP-hard problems with equal size feasible solutions

RAIRO - Operations Research - Recherche Opérationnelle (2002)

- Volume: 36, Issue: 4, page 279-297
- ISSN: 0399-0559

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topMonnot, Jérôme. "Differential approximation of NP-hard problems with equal size feasible solutions." RAIRO - Operations Research - Recherche Opérationnelle 36.4 (2002): 279-297. <http://eudml.org/doc/245799>.

@article{Monnot2002,

abstract = {In this paper, we focus on some specific optimization problems from graph theory, those for which all feasible solutions have an equal size that depends on the instance size. Once having provided a formal definition of this class of problems, we try to extract some of its basic properties; most of these are deduced from the equivalence, under differential approximation, between two versions of a problem $\pi $ which only differ on a linear transformation of their objective functions. This is notably the case of maximization and minimization versions of $\pi $, as well as general minimization and minimization with triangular inequality versions of $\pi $. Then, we prove that some well known problems do belong to this class, such as special cases of both spanning tree and vehicles routing problems. In particular, we study the strict rural postman problem (called $SRPP$) and show that both the maximization and the minimization versions can be approximately solved, in polynomial time, within a differential ratio bounded above by $1/2$. From these results, we derive new bounds for standard ratio when restricting edge weights to the interval $[a,ta]$ (the $SRPP[t]$ problem): we respectively provide a $2/(t+1)$- and a $(t+1)/2t$-standard approximation for the minimization and the maximization versions.},

author = {Monnot, Jérôme},

journal = {RAIRO - Operations Research - Recherche Opérationnelle},

keywords = {approximate algorithms; differential ratio; performance ratio; analysis of algorithms; Approximate algorithms},

language = {eng},

number = {4},

pages = {279-297},

publisher = {EDP-Sciences},

title = {Differential approximation of NP-hard problems with equal size feasible solutions},

url = {http://eudml.org/doc/245799},

volume = {36},

year = {2002},

}

TY - JOUR

AU - Monnot, Jérôme

TI - Differential approximation of NP-hard problems with equal size feasible solutions

JO - RAIRO - Operations Research - Recherche Opérationnelle

PY - 2002

PB - EDP-Sciences

VL - 36

IS - 4

SP - 279

EP - 297

AB - In this paper, we focus on some specific optimization problems from graph theory, those for which all feasible solutions have an equal size that depends on the instance size. Once having provided a formal definition of this class of problems, we try to extract some of its basic properties; most of these are deduced from the equivalence, under differential approximation, between two versions of a problem $\pi $ which only differ on a linear transformation of their objective functions. This is notably the case of maximization and minimization versions of $\pi $, as well as general minimization and minimization with triangular inequality versions of $\pi $. Then, we prove that some well known problems do belong to this class, such as special cases of both spanning tree and vehicles routing problems. In particular, we study the strict rural postman problem (called $SRPP$) and show that both the maximization and the minimization versions can be approximately solved, in polynomial time, within a differential ratio bounded above by $1/2$. From these results, we derive new bounds for standard ratio when restricting edge weights to the interval $[a,ta]$ (the $SRPP[t]$ problem): we respectively provide a $2/(t+1)$- and a $(t+1)/2t$-standard approximation for the minimization and the maximization versions.

LA - eng

KW - approximate algorithms; differential ratio; performance ratio; analysis of algorithms; Approximate algorithms

UR - http://eudml.org/doc/245799

ER -

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