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그래프 지름은 2차원 이상의 polyhedron에서 중요하며, simplex 알고리즘의 효율성을 위해 다방면의 연구를 받았다.
이 논문에서는 고리모노미얼 upper bound를 사용하여 지름의 upper bound를 nlog d + 2로 제시했다.
한편, 지름의 lower bound에 대한 Hirsch conjecture은 아직 해결되지 않았다.
다음은 영어 요약이다:
We prove a quasi-polynomial upper bound for the diameter of graphs of polyhedra, improving on earlier exponential bounds in terms of dimension.
The diameter of a graph is a measure of the maximum distance between any two vertices and has important applications in computational geometry and linear programming.
The paper presents an improved upper bound on this diameter, specifically nlog d + 2, using quasi-polynomial techniques.
This is an improvement over earlier exponential bounds in terms of dimension.
However, the Hirsch conjecture regarding a lower bound for the diameter remains open.
Note: Quasi-polynomial refers to a class of functions that are polynomial in the logarithm of their input size. In this case, the upper bound is nlog d + 2, which grows polynomially with respect to log d.
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arXiv:math/9204233v1 [math.MG] 1 Apr 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 26, Number 2, April 1992, Pages 315-316A QUASI-POLYNOMIAL BOUND FOR THE DIAMETEROF GRAPHS OF POLYHEDRAGil Kalai and Daniel J. KleitmanAbstract. The diameter of the graph of a d-dimensional polyhedron with n facetsis at most nlog d+2Let P be a convex polyhedron.
The graph of P denoted by G(P) is an abstractgraph whose vertices are the extreme points of P and two vertices u and v areadjacent if the interval [v, u] is an extreme edge (= 1-dimensional face) of P. Thediameter of the graph of P is denoted by δ(P).Let∆(d, n)bethemaximaldiameterofthegraphsofd-dimen-sional polyhedra P with n facets. (A facet is a (d −1)-dimensional face.) Thus, Pis the set of solutions of n linear inequalities in d variables.
It is an old standingproblem to determine the behavior of the function ∆(d, n). The value of ∆(d, n) isa lower bound for the number of iterations needed for Dantzig’s simplex algorithmfor linear programming with any pivot rule.In 1957 Hirsch conjectured [2] that ∆(d, n) ≤n−d.
Klee and Walkup [6] showedthat the Hirsch conjecture is false for unbounded polyhedra. They proved that forn ≥2d, ∆(d, n) ≥n −d + [d/5].
This is the best known lower bound for ∆(d, n).The statement of the Hirsch conjecture for bounded polyhedra is still open. For arecent survey on the Hirsch conjecture and its relatives, see [5].In 1967 Barnette proved [1, 3] that ∆(d, n) ≤n3d−3.
An improved upper bound,∆(d, n) ≤n2d−3, was proved in 1970 by Larman [7]. Barnette’s and Larman’sbounds are linear in n but exponential in the dimension d. In 1990 the first author[4] proved a subexponential bound ∆(d, n) ≤2√(n−d) log(n−d).The purpose of this paper is to announce and to give a complete proof of a quasi-polynomial upper bound for ∆(d, n).
Such a bound was proved by the first authorin March 1991. The proof presented here is a substantial simplification that wassubsequently found by the second author.
See [4] for the original proof and relatedresults. The existence of a polynomial (or even linear) upper bound for ∆(d, n)is still open.
Recently, the first author found a randomized pivot rule for linearprogramming which requires an expected n4√d (or less) arithmetic operations forevery linear programming problem with d variables and n constraints.1991 Mathematics Subject Classification. Primary 52A25, 90C05.Received by the editors July 1, 1991The first author was supported in part by a BSF grant by a GIF grant.
The second authorwas supported by an AFOSR grantc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1
2GIL KALAI AND D. J. KLEITMANTheorem 1. (1)∆(d, n) ≤nlog d+2.Proof.
Let P be a d-dimensional polyhedron with n facets, and let v and u be twovertices of P. Let kv [ku] be the maximal positive number such that the union ofall vertices in all paths in G(P) starting from v [u] of length at most kv [ku] areincident to at most n/2 facets. Clearly, there is a facet F of P so that we can reachF by a path of length kv + 1 from v and a path of length ku + 1 from u.
We claimnow that kv ≤∆(d, [n/2]). Indeed, let Q be the polyhedron obtained from P byignoring all the inequalities that correspond to facets that cannot be reached from vby a path of length at most kv.
Let ω be a vertex in G(P) whose distance from v iskv. We claim that the distance of ω from v in G(Q) is also kv.
To see this considerthe shortest path between v and ω in G(Q). If the length of this path is smallerthan kv there must be an edge in the path that is not an edge of P. Consider thefirst such edge E. Since E is not an edge of P, it intersects a hyperplane H thatcorresponds to one of the inequalities that was ignored.
This gives a path in Pof length smaller than kv from v to the facet of P determined by H, which is acontradiction.We obtained that the distance from v to u is at most ∆(d−1, n−1)+2∆(d, [n/2])+2. This gives the inequality ∆(d, n) ≤∆(d −1, n −1) +2∆(d, [n/2]) + 2, whichimplies the statement of the theorem.References1.
D. W. Barnette, Wv paths on 3-polytopes, J. Combin. Theory 7 (1969), 62–70.2.
G. B. Dantzig, Linear programming and extensions, Princeton Univ. Press, Princeton, NJ1963.3.
B. Gr¨ubaum, Convex polytopes, Wiley Interscience, London, 1967.4. G. Kalai, Upper bounds for the diameter and height of polytopes, Discrete Comput.
Geom.7 (1992) (to appear).5. V. Klee and P. Kleinschmidt, The d-steps conjecture and its relatives, Math.
OperationResearch 12 (1987), 718–755.6. V. Klee and D. Walkup, The d-step conjecture for polyhedra of dimension d < 6, ActaMath.
133 (1967), 53–78.7. D. G. Larman, Paths on polytopes, Proc.
London Math. Soc.
(3) 20 (1970), 161–178.Institute of Mathematics, Hebrew University of Jerusalem, Jerusalem, Israeland IBM Almaden Research Center, San Jose, California 95120Department of Mathematics, Massachusetts Institute of Technology, Cambridge,Massachusetts 02139
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