155 research outputs found
The Challenge of Unifying Semantic and Syntactic Inference Restrictions
While syntactic inference restrictions don't play an important role for SAT, they are an essential reasoning technique for more expressive logics, such as first-order logic, or fragments thereof. In particular, they can result in short proofs or model representations. On the other hand, semantically guided inference systems enjoy important properties, such as the generation of solely non-redundant clauses. I discuss to what extend the two paradigms may be unifiable
A Note on Non-Degenerate Integer Programs with Small Sub-Determinants
The intention of this note is two-fold. First, we study integer optimization
problems in standard form defined by and present
an algorithm to solve such problems in polynomial-time provided that both the
largest absolute value of an entry in and are constant. Then, this is
applied to solve integer programs in inequality form in polynomial-time, where
the absolute values of all maximal sub-determinants of lie between and
a constant
Mixed-integer Quadratic Programming is in NP
Mixed-integer quadratic programming is the problem of optimizing a quadratic
function over points in a polyhedral set where some of the components are
restricted to be integral. In this paper, we prove that the decision version of
mixed-integer quadratic programming is in NP, thereby showing that it is
NP-complete. This is established by showing that if the decision version of
mixed-integer quadratic programming is feasible, then there exists a solution
of polynomial size. This result generalizes and unifies classical results that
quadratic programming is in NP and integer linear programming is in NP
The Width and Integer Optimization on Simplices With Bounded Minors of the Constraint Matrices
In this paper, we will show that the width of simplices defined by systems of
linear inequalities can be computed in polynomial time if some minors of their
constraint matrices are bounded. Additionally, we present some
quasi-polynomial-time and polynomial-time algorithms to solve the integer
linear optimization problem defined on simplices minus all their integer
vertices assuming that some minors of the constraint matrices of the simplices
are bounded.Comment: 12 page
Vectors in a Box
For an integer d>=1, let tau(d) be the smallest integer with the following
property: If v1,v2,...,vt is a sequence of t>=2 vectors in [-1,1]^d with
v1+v2+...+vt in [-1,1]^d, then there is a subset S of {1,2,...,t} of indices,
2<=|S|<=tau(d), such that \sum_{i\in S} vi is in [-1,1]^d. The quantity tau(d)
was introduced by Dash, Fukasawa, and G\"unl\"uk, who showed that tau(2)=2,
tau(3)=4, and tau(d)=Omega(2^d), and asked whether tau(d) is finite for all d.
Using the Steinitz lemma, in a quantitative version due to Grinberg and
Sevastyanov, we prove an upper bound of tau(d) <= d^{d+o(d)}, and based on a
construction of Alon and Vu, whose main idea goes back to Hastad, we obtain a
lower bound of tau(d)>= d^{d/2-o(d)}.
These results contribute to understanding the master equality polyhedron with
multiple rows defined by Dash et al., which is a "universal" polyhedron
encoding valid cutting planes for integer programs (this line of research was
started by Gomory in the late 1960s). In particular, the upper bound on tau(d)
implies a pseudo-polynomial running time for an algorithm of Dash et al. for
integer programming with a fixed number of constraints. The algorithm consists
in solving a linear program, and it provides an alternative to a 1981 dynamic
programming algorithm of Papadimitriou.Comment: 12 pages, 1 figur
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
FPT-algorithms for some problems related to integer programming
In this paper, we present FPT-algorithms for special cases of the shortest
lattice vector, integer linear programming, and simplex width computation
problems, when matrices included in the problems' formulations are near square.
The parameter is the maximum absolute value of rank minors of the corresponding
matrices. Additionally, we present FPT-algorithms with respect to the same
parameter for the problems, when the matrices have no singular rank
sub-matrices.Comment: arXiv admin note: text overlap with arXiv:1710.00321 From author:
some minor corrections has been don
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