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Lower Bounds for the Average and Smoothed Number of Pareto Optima
Smoothed analysis of multiobjective 0-1 linear optimization has drawn
considerable attention recently. The number of Pareto-optimal solutions (i.e.,
solutions with the property that no other solution is at least as good in all
the coordinates and better in at least one) for multiobjective optimization
problems is the central object of study. In this paper, we prove several lower
bounds for the expected number of Pareto optima. Our basic result is a lower
bound of \Omega_d(n^(d-1)) for optimization problems with d objectives and n
variables under fairly general conditions on the distributions of the linear
objectives. Our proof relates the problem of lower bounding the number of
Pareto optima to results in geometry connected to arrangements of hyperplanes.
We use our basic result to derive (1) To our knowledge, the first lower bound
for natural multiobjective optimization problems. We illustrate this for the
maximum spanning tree problem with randomly chosen edge weights. Our technique
is sufficiently flexible to yield such lower bounds for other standard
objective functions studied in this setting (such as, multiobjective shortest
path, TSP tour, matching). (2) Smoothed lower bound of min {\Omega_d(n^(d-1.5)
\phi^{(d-log d) (1-\Theta(1/\phi))}), 2^{\Theta(n)}}$ for the 0-1 knapsack
problem with d profits for phi-semirandom distributions for a version of the
knapsack problem. This improves the recent lower bound of Brunsch and Roeglin
Enumerating Polytropes
Polytropes are both ordinary and tropical polytopes. We show that tropical
types of polytropes in are in bijection with cones of a
certain Gr\"{o}bner fan in restricted
to a small cone called the polytrope region. These in turn are indexed by
compatible sets of bipartite and triangle binomials. Geometrically, on the
polytrope region, is the refinement of two fans: the fan of
linearity of the polytrope map appeared in \cite{tran.combi}, and the bipartite
binomial fan. This gives two algorithms for enumerating tropical types of
polytropes: one via a general Gr\"obner fan software such as \textsf{gfan}, and
another via checking compatibility of systems of bipartite and triangle
binomials. We use these algorithms to compute types of full-dimensional
polytropes for , and maximal polytropes for .Comment: Improved exposition, fixed error in reporting the number maximal
polytropes for , fixed error in definition of bipartite binomial
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