By Alexander K. Hartmann, Martin Weigt
A concise, entire creation to the subject of statistical physics of combinatorial optimization, bringing jointly theoretical ideas and algorithms from machine technology with analytical equipment from physics. the outcome bridges the distance among statistical physics and combinatorial optimization, investigating difficulties taken from theoretical computing, corresponding to the vertex-cover challenge, with the suggestions and strategies of theoretical physics. The authors hide swift advancements and analytical equipment which are either tremendous complicated and unfold via word-of-mouth, delivering the entire valuable fundamentals in required element. all through, the algorithms are proven with examples and calculations, whereas the proofs are given in a manner compatible for graduate scholars, post-docs, and researchers. excellent for rookies to this younger, multidisciplinary box.
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Extra info for Phase transitions in combinatorial optimization problems: basics, algorithms and statistical mechanics
Example text
Yn ) = (c)(d) = c2n/2 + d . Then the multiplication reads as follows: xy = (a2n/2 + b)(c2n/2 + d) = ac2n + (ad + bc)2n/2 + bd = ac2n + ([a + b][c + d] − ac − bd)2n/2 + bd . Please note that in the second line, we use only three different multiplications ac, bd and [a + b][c + d] instead of four as in the first line. We will see that this is crucial for obtaining an asymptotically fast algorithm. Our idea is to use only multiplications of n/2-bit numbers. The only remaining difficulty is that [a + b] and [c + d] may be (n/2 + 1)-bit numbers due to a carry arising from the addition.
Xn/2 )(xn/2+1 . . xn ) = (a)(b) = a2n/2 + b y = (y1 . . yn/2 )(yn/2+1 . . yn ) = (c)(d) = c2n/2 + d . Then the multiplication reads as follows: xy = (a2n/2 + b)(c2n/2 + d) = ac2n + (ad + bc)2n/2 + bd = ac2n + ([a + b][c + d] − ac − bd)2n/2 + bd . Please note that in the second line, we use only three different multiplications ac, bd and [a + b][c + d] instead of four as in the first line. We will see that this is crucial for obtaining an asymptotically fast algorithm. Our idea is to use only multiplications of n/2-bit numbers.
For these graphs, the chromatic number can be found easily, using the following famous theorem: Theorem: Every planar graph is 4-colorable. This had already been conjectured in 1852, but the final solution was only given in 1976 by Appel and Haken [3]. Their proof is, however, quite unusual: Appel and Haken “reduced” the original question to more than 2000 cases according to the graph structure. These cases were finally colored numerically, using about 1200 hours of computational time. Many mathematicians were quite dissatisfied by this proof, but up to now nobody has come up with a “hand-written” one.