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Absolute bounds on optimal cost for a class of set covering problems

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Zusammenfassung

Wir untersuchen Mengenüberdeckungsprobleme mitn Variablen undm Nebenbedingungen, in denen nicht mehr alsk Variable in jeder Restriktion auftreten. Seid die maximale Spaltensumme,r=[m/d]−1 undZ g der Wert einer Greedy-Lösung. Wir zeigen

$$\begin{gathered} Zg \leqslant 1 + r + m - d - \left[ {mk \cdot MAX\left\{ {\frac{{2r}}{{2n - r - 1}},\ln \frac{n}{{n - r}}} \right\}} \right. \hfill \\ \left. { - kd \cdot MIN\left\{ {\frac{{r(r + 1)}}{{2(n - r)}},n \cdot \ln \frac{{n - 1}}{{n - r - 1}} - 1} \right\}} \right]. \hfill \\ \end{gathered} $$

Dies ist die erste nichttriviale oberea-priori-Schranke für Mengenüberdeckungsprobleme. Ein Beispiel zeigt, daß dieser Wert sowohl von der Greedy-Lösung wie auch von der Optimallösung angenommen werden kann. Ferner belegen numerische Beispiele, daß diese Schranke für viele spezielle Probleme wesentlich besser ist als bereits bekannte Schranken. Eine wichtige Teilklasse, für die dies zutrifft, umfaßt Probleme, deren Restriktionsmatrix eine Zirkulante undk=d=[αn] für ein 0<α<1 ist. In diesem Falle zeigen wir

$$\mathop {\lim }\limits_{n \to \infty } Zg/n \leqslant \frac{{\alpha ^2 }}{2}[1/\alpha ][1/\alpha ].$$

Abstract

We study the class of (m constraint,n variable) set covering problems which have no more thank variables represented in each constraint. Letd denote the maximum column sum in the constraint matrix, letr=[m/d]−1, and letZ g denote the cost of a greedy heuristic solution. Then we prove

$$\begin{gathered} Zg \leqslant 1 + r + m - d - \left[ {mk \cdot MAX\left\{ {\frac{{2r}}{{2n - r - 1}},\ln \frac{n}{{n - r}}} \right\}} \right. \hfill \\ \left. { - kd \cdot MIN\left\{ {\frac{{r(r + 1)}}{{2(n - r)}},n \cdot \ln \frac{{n - 1}}{{n - r - 1}} - 1} \right\}} \right]. \hfill \\ \end{gathered} $$

This provides the firsta priori nontrivial upper bound discovered on heuristic solution cost (and thus on optimal solution cost) for the set covering problem. An example demonstrates that this bound is attainable, both for a greedy heuristic solution and for the optimal solution. Numerical examples show that this bound is substantially better than existing bounds for many problem instances. An important subclass of these problems occurs when the constraint matrix is a circulant, in which casem=n andk=d=[αη] for some 0<α<1. For this subclass we prove

$$\mathop {\lim }\limits_{n \to \infty } Zg/n \leqslant \frac{{\alpha ^2 }}{2}[1/\alpha ][1/\alpha ].$$

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References

  1. Balas E (1981) A class of location, distribution and scheduling problems modeling and solution methods. Paper presented at the Chinese-American Symposium on Systems Analysis and Engineering, Xian

  2. Balas E, Ho A (1980) Set covering algorithms using cutting planes, heuristics, and subgradient optimization: a computational study. Mathematical Programming Study 12:37–60

    Google Scholar 

  3. Bartholdi JJ (1981) A guaranteed-accuracy round-off algorithm for cyclic scheduling and set covering. Operations Research 29:501–510

    Google Scholar 

  4. Chvatal V (1979) A greedy heuristic for the set covering problem. Mathematics of Operations Research 4:233–235

    Google Scholar 

  5. Clapham CRJ (1973) Introduction to mathematical analysis. Routledge & Kegan Paul

  6. Garey MR, Johnson DS (1979) Computers and intractability; a guide to the theory of NP-completeness. Freeman, San Francisco

    Google Scholar 

  7. Hochbaum DS (1985) Easy solutions for the k-center problem or the dominating set problem on random graphs. Annals of Discrete Mathematics 25:189–210

    Google Scholar 

  8. Johnson DS (1974) Approximation algorithms for combinatorial problems. J Computer & Systems Sciences 9:256–278

    Google Scholar 

  9. Lovasz L (1975) On the ratio of optimal integral and fractional covers. Discrete Mathematics 13:383–390

    Google Scholar 

  10. Rubin J (1973) A technique for the solution of massive set covering problems with applications to airline crew scheduling. Transportation Science 7:34–48

    Google Scholar 

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Hall, N.G., Vohra, R.V. Absolute bounds on optimal cost for a class of set covering problems. ZOR - Methods and Models of Operations Research 33, 181–192 (1989). https://doi.org/10.1007/BF01423649

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  • DOI: https://doi.org/10.1007/BF01423649

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