Counting the types of squares rather than their occurrences, we consider the problem of bounding the number of distinct squares in a string. Fraenkel and Simpson showed in 1998 that a string of length n contains at most 2n distinct squares. Ilie presented in 2007 an asymptotic upper bound of 2n−Θ(log n). We show that a string of length n contains at most 11n/6 distinct squares. This new upper bound is obtained by investigating the combinatorial structure of double squares and showing that a string of length n contains at most 5n/6 particular double squares. In addition, the established structural properties provide a novel proof of Fraenkel and Simpson's result.
Let ∆(d, n) be the maximum possible edge diameter over all d-dimensional polytopes defined by n inequalities. The Hirsch conjecture, formulated in 1957, suggests that ∆(d, n) is no greater than n − d. No polynomial bound is currently known for ∆(d, n), the best one being quasipolynomial due to Kalai and Kleitman in 1992. Goodey showed in 1972 that ∆(4, 10) = 5 and ∆(5, 11) = 6, and more recently, Bremner and Schewe showed ∆(4, 11) = ∆(6, 12) = 6. In this follow-up, we show that ∆(4, 12) = 7 and present strong evidence that ∆(5, 12) = ∆(6, 13) = 7.
Inspired by Bárány's Colourful Carathéodory Theorem [4], we introduce a colourful generalization of Liu's simplicial depth [13]. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d 2 + 1 and that the maximum is d d+1 + 1. We exhibit configurations attaining each of these depths, and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.
By analogy with the conjecture of Hirsch, we conjecture that the order of the largest total curvature of the central path associated to a polytope is the number of inequalities defining the polytope. By analogy with a result of Dedieu, Malajovich and Shub, we conjecture that the average diameter of a bounded cell of an arrangement is less than the dimension. We prove continuous analogues of two results of Holt-Klee and Klee-Walkup: we construct a family of polytopes which attain the conjectured order of the largest total curvature, and we prove that the special case where the number of inequalities is twice the dimension is equivalent to the general case. We substantiate these conjectures in low dimensions and highlight additional links.
Continuous Analogue of the Conjecture of HirschLet P be a full dimensional convex polyhedron defined by m inequalities in dimension n. The diameter δ(P ) is the smallest number such that any two vertices of the polyhedron P can be connected by a path with at most δ(P ) edges. The conjecture of Hirsch, formulated in 1957 and reported in [2], states that the diameter of a polyhedron defined by m inequalities in dimension n is not greater than m − n. The conjecture does not hold for unbounded polyhedra. A polytope is a bounded polyhedron. No polynomial bound is known for the diameter of a polytope.
Conjecture 1.1. (Conjecture of Hirsch for polytopes)The diameter of a polytope defined by m inequalities in dimension n is not greater than m − n.Intuitively, the total curvature [15] is a measure of how far off a certain curve is from being a straight line. Let ψ : [α, β] → R n be a C 2 ((α − ε, β + ε)) map for some ε > 0 with a non-zero derivative in [α, β]. Denote its arc length by l(t) = t α ψ (τ ) dτ , its parametrization by the arc lengthand its curvature at the point t by κ(t) =ψ arc (t). The total curvature is defined asThe requirementψ = 0 insures that any given segment of the curve is traversed only once and allows to define a curvature at any point on the curve.We present one useful proposition. Roughly speaking, it states that two similar curves might not differ greatly in their total curvatures either. This fact is used in Section 3 in proving the analogue of the d-step conjecture for the total curvature of the central path.Proposition 1
By refining a variant of the Klee-Minty example that forces the central path to visit all the vertices of the Klee-Minty n-cube, we exhibit a nearly worst-case example for pathfollowing interior point methods. Namely, while the theoretical iteration-complexity upper bound is O(2 n n 5 2 ), we prove that solving this n-dimensional linear optimization problem requires at least 2 n − 1 iterations.
Summary. We consider a family of linear optimization problems over the ndimensional Klee-Minty cube and show that the central path may visit all of its vertices in the same order as simplex methods do. This is achieved by carefully adding an exponential number of redundant constraints that forces the central path to take at least 2 n −2 sharp turns. This fact suggests that any feasible path-following interior-point method will take at least O(2 n ) iterations to solve this problem, while in practice typically only a few iterations, e.g., 50, suffices to obtain a high quality solution. Thus, the construction potentially exhibits the worst-case iterationcomplexity known to date which almost matches the theoretical iteration-complexity bound for this type of methods. In addition, this construction gives a counterexample to a conjecture that the total central path curvature is O(n).
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