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 $SRPP\left[t\right]$ problem): we respectively provide a $2/(t+1)$- and a $(t+1)/2t$-standard approximation for the minimization and the maximization versions.

@article{RO_2002__36_4_279_0, author = {Monnot, J\'er\^ome}, title = {Differential approximation of {NP-hard} problems with equal size feasible solutions}, journal = {RAIRO - Operations Research - Recherche Op\'erationnelle}, pages = {279--297}, publisher = {EDP-Sciences}, volume = {36}, number = {4}, year = {2002}, doi = {10.1051/ro:2003008}, zbl = {1037.90059}, mrnumber = {1997926}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ro:2003008/} }

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 DA - 2002/// SP - 279 EP - 297 VL - 36 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ro:2003008/ UR - https://zbmath.org/?q=an%3A1037.90059 UR - https://www.ams.org/mathscinet-getitem?mr=1997926 UR - https://doi.org/10.1051/ro:2003008 DO - 10.1051/ro:2003008 LA - en ID - RO_2002__36_4_279_0 ER -

%0 Journal Article %A Monnot, Jérôme %T Differential approximation of NP-hard problems with equal size feasible solutions %J RAIRO - Operations Research - Recherche Opérationnelle %D 2002 %P 279-297 %V 36 %N 4 %I EDP-Sciences %U https://doi.org/10.1051/ro:2003008 %R 10.1051/ro:2003008 %G en %F RO_2002__36_4_279_0

Monnot, Jérôme. Differential approximation of NP-hard problems with equal size feasible solutions. RAIRO - Operations Research - Recherche Opérationnelle, Volume 36 (2002) no. 4, pp. 279-297. doi : 10.1051/ro:2003008. http://archive.numdam.org/articles/10.1051/ro:2003008/

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