Marek Karpinski

We survey some of the recent results on the complexity of recognizing

n-dimensional linear arrangements and convex polyhedra by randomized

algebraic decision trees. We give also a number of concrete applications

of these results. In particular, we derive first nontrivial, in fact

quadratic, ...
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Valentin E. Brimkov, Stefan S. Dantchev

In the framework of the Blum-Shub-Smale real number model \cite{BSS}, we study the {\em algebraic complexity} of the integer linear programming problem

(ILP$_{\bf R}$) : Given a matrix $A \in {\bf R}^{m \times n}$ and vectors

$b \in {\bf R}^m$, $d \in {\bf R}^n$, decide if there is $x ...
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Albert Atserias, Maria Luisa Bonet, Jordi Levy

We study the Chv\'atal rank of polytopes as a complexity measure of

unsatisfiable sets of clauses. Our first result establishes a

connection between the Chv\'atal rank and the minimum refutation

length in the cutting planes proof system. The result implies that

length lower bounds for cutting planes, or even for ...
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Arist Kojevnikov

We continue a study initiated by Krajicek of

a Resolution-like proof system working with clauses of linear

inequalities, R(CP). For all proof systems of this kind

Krajicek proved an exponential lower bound that depends

on the maximal absolute value of coefficients in the given proof and

the maximal clause width.

Nathan Segerlind

The matrix cuts of Lov{\'{a}}sz and Schrijver are methods for tightening linear relaxations of zero-one programs by the addition of new linear inequalities. We address the question of how many new inequalities are necessary to approximate certain combinatorial problems with strong guarantees, and to solve certain instances of Boolean satisfiability.

... more >>>Russell Impagliazzo, Shachar Lovett, Ramamohan Paturi, Stefan Schneider

We give an exact algorithm for the 0-1 Integer Linear Programming problem with a linear number of constraints that improves over exhaustive search by an exponential factor. Specifically, our algorithm runs in time $2^{(1-\text{poly}(1/c))n}$ where $n$ is the

number of variables and $cn$ is the number of constraints. The key ...
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Noah Fleming, Mika Göös, Russell Impagliazzo, Toniann Pitassi, Robert Robere, Li-Yang Tan, Avi Wigderson

The Stabbing Planes proof system was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas -- certain unsatisfiable systems of linear equations mod 2 -- which are canonical ... more >>>