4,400 research outputs found
An update on the Hirsch conjecture
The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to
George Dantzig. It states that the graph of a d-dimensional polytope with n
facets cannot have diameter greater than n - d.
Despite being one of the most fundamental, basic and old problems in polytope
theory, what we know is quite scarce. Most notably, no polynomial upper bound
is known for the diameters that are conjectured to be linear. In contrast, very
few polytopes are known where the bound is attained. This paper collects
known results and remarks both on the positive and on the negative side of the
conjecture. Some proofs are included, but only those that we hope are
accessible to a general mathematical audience without introducing too many
technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2
and put into the appendix arXiv:0912.423
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Algorithms for network piecewise-linear programs
In this paper a subarea of Piecewise-Linear Programming named network Piecewise-Linear Programming (NPLP) is discussed. Initially the problem formulation, main efinitins and related Concepts are presented. In the sequence of the paper, four specialized algorithms for NPLP, as well as the results of a preliminary computational study, are presented
The Stochastic Shortest Path Problem : A polyhedral combinatorics perspective
In this paper, we give a new framework for the stochastic shortest path
problem in finite state and action spaces. Our framework generalizes both the
frameworks proposed by Bertsekas and Tsitsikli and by Bertsekas and Yu. We
prove that the problem is well-defined and (weakly) polynomial when (i) there
is a way to reach the target state from any initial state and (ii) there is no
transition cycle of negative costs (a generalization of negative cost cycles).
These assumptions generalize the standard assumptions for the deterministic
shortest path problem and our framework encapsulates the latter problem (in
contrast with prior works). In this new setting, we can show that (a) one can
restrict to deterministic and stationary policies, (b) the problem is still
(weakly) polynomial through linear programming, (c) Value Iteration and Policy
Iteration converge, and (d) we can extend Dijkstra's algorithm
Recent progress on the combinatorial diameter of polytopes and simplicial complexes
The Hirsch conjecture, posed in 1957, stated that the graph of a
-dimensional polytope or polyhedron with facets cannot have diameter
greater than . The conjecture itself has been disproved, but what we
know about the underlying question is quite scarce. Most notably, no polynomial
upper bound is known for the diameters that were conjectured to be linear. In
contrast, no polyhedron violating the conjecture by more than 25% is known.
This paper reviews several recent attempts and progress on the question. Some
work in the world of polyhedra or (more often) bounded polytopes, but some try
to shed light on the question by generalizing it to simplicial complexes. In
particular, we include here our recent and previously unpublished proof that
the maximum diameter of arbitrary simplicial complexes is in and
we summarize the main ideas in the polymath 3 project, a web-based collective
effort trying to prove an upper bound of type nd for the diameters of polyhedra
and of more general objects (including, e. g., simplicial manifolds).Comment: 34 pages. This paper supersedes one cited as "On the maximum diameter
of simplicial complexes and abstractions of them, in preparation
Strongly polynomial primal monotonic build-up simplex algorithm for maximal flow problems
The maximum flow problem (MFP) is a fundamental model in operations research. The network simplex algorithm is one of the most efficient solution methods for MFP in practice. The theoretical properties of established pivot algorithms for MFP is less understood. Variants of the primal simplex and dual simplex methods for MFP have been proven strongly polynomial, but no similar result exists for other pivot algorithms like the monotonic build-up or the criss-cross simplex algorithm. The monotonic build-up simplex algorithm (MBUSA) starts with a feasible solution, and fixes the dual feasibility one variable a time, temporarily losing primal feasibility. In the case of maximum flow problems, pivots in one such iteration are all dual degenerate, bar the last one. Using a labelling technique to break these ties we show a variant that solves the maximum flow problem in 2|V||A|2 pivots
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