146 research outputs found
Transversal numbers over subsets of linear spaces
Let be a subset of . It is an important question in the
theory of linear inequalities to estimate the minimal number such that
every system of linear inequalities which is infeasible over has a
subsystem of at most inequalities which is already infeasible over
This number is said to be the Helly number of In view of Helly's
theorem, and, by the theorem due to Doignon, Bell and
Scarf, We give a common extension of these equalities
showing that We show that
the fractional Helly number of the space (with the
convexity structure induced by ) is at most as long as
is finite. Finally we give estimates for the Radon number of mixed
integer spaces
Note on the Complexity of the Mixed-Integer Hull of a Polyhedron
We study the complexity of computing the mixed-integer hull
of a polyhedron .
Given an inequality description, with one integer variable, the mixed-integer
hull can have exponentially many vertices and facets in . For fixed,
we give an algorithm to find the mixed integer hull in polynomial time. Given
and fixed, we compute a vertex description of
the mixed-integer hull in polynomial time and give bounds on the number of
vertices of the mixed integer hull
Proximity results and faster algorithms for Integer Programming using the Steinitz Lemma
We consider integer programming problems in standard form where , and . We show that such an integer program can be solved in time , where is an upper bound on each
absolute value of an entry in . This improves upon the longstanding best
bound of Papadimitriou (1981) of , where in addition,
the absolute values of the entries of also need to be bounded by .
Our result relies on a lemma of Steinitz that states that a set of vectors in
that is contained in the unit ball of a norm and that sum up to zero can
be ordered such that all partial sums are of norm bounded by . We also use
the Steinitz lemma to show that the -distance of an optimal integer and
fractional solution, also under the presence of upper bounds on the variables,
is bounded by . Here is again an
upper bound on the absolute values of the entries of . The novel strength of
our bound is that it is independent of . We provide evidence for the
significance of our bound by applying it to general knapsack problems where we
obtain structural and algorithmic results that improve upon the recent
literature.Comment: We achieve much milder dependence of the running time on the largest
entry in $b
The distributions of functions related to parametric integer optimization
We consider the asymptotic distribution of the IP sparsity function, which
measures the minimal support of optimal IP solutions, and the IP to LP distance
function, which measures the distance between optimal IP and LP solutions. We
create a framework for studying the asymptotic distribution of general
functions related to integer optimization. There has been a significant amount
of research focused around the extreme values that these functions can attain,
however less is known about their typical values. Each of these functions is
defined for a fixed constraint matrix and objective vector while the right hand
sides are treated as input. We show that the typical values of these functions
are smaller than the known worst case bounds by providing a spectrum of
probability-like results that govern their overall asymptotic distributions.Comment: Accepted for journal publicatio
Intractability of approximate multi-dimensional nonlinear optimization on independence systems
We consider optimization of nonlinear objective functions that balance
linear criteria over -element independence systems presented by
linear-optimization oracles. For , we have previously shown that an
-best approximate solution can be found in polynomial time. Here, using an
extended Erd\H{o}s-Ko-Rado theorem of Frankl, we show that for , finding a
-best solution requires exponential time
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