776 research outputs found
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
The Polyhedron-Hitting Problem
We consider polyhedral versions of Kannan and Lipton's Orbit Problem (STOC
'80 and JACM '86)---determining whether a target polyhedron V may be reached
from a starting point x under repeated applications of a linear transformation
A in an ambient vector space Q^m. In the context of program verification, very
similar reachability questions were also considered and left open by Lee and
Yannakakis in (STOC '92). We present what amounts to a complete
characterisation of the decidability landscape for the Polyhedron-Hitting
Problem, expressed as a function of the dimension m of the ambient space,
together with the dimension of the polyhedral target V: more precisely, for
each pair of dimensions, we either establish decidability, or show hardness for
longstanding number-theoretic open problems
On the Exact Solution to a Smart Grid Cyber-Security Analysis Problem
This paper considers a smart grid cyber-security problem analyzing the
vulnerabilities of electric power networks to false data attacks. The analysis
problem is related to a constrained cardinality minimization problem. The main
result shows that an relaxation technique provides an exact optimal
solution to this cardinality minimization problem. The proposed result is based
on a polyhedral combinatorics argument. It is different from well-known results
based on mutual coherence and restricted isometry property. The results are
illustrated on benchmarks including the IEEE 118-bus and 300-bus systems
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows
In this paper, we develop a simultaneous column-and-row generation algorithm that could be applied to a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints, which are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on-the-fly within an efficient solution approach. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm. These assumptions are general enough and cover all problems with column-dependent-rows studied in the literature up until now to
the best of our knowledge. We then introduce in detail a set of pricing subproblems, which are used within the proposed column-and-row generation algorithm. This is followed by a formal discussion on the optimality of the algorithm. To illustrate the proposed approach, the paper is concluded by applying the proposed framework to the multi-stage cutting stock and the quadratic set covering problems
Stabilized Benders methods for large-scale combinatorial optimization, with appllication to data privacy
The Cell Suppression Problem (CSP) is a challenging Mixed-Integer Linear Problem arising in statistical tabular data protection. Medium sized instances of CSP involve thousands of binary variables and million of continuous variables and constraints. However, CSP has the typical
structure that allows application of the renowned Benders’ decomposition method: once the “complicating” binary variables are fixed, the problem decomposes into a large set of linear subproblems on the “easy” continuous ones. This allows to project away the easy variables, reducing to a master problem in the complicating ones where the value functions of the subproblems are approximated with the standard cutting-plane approach. Hence, Benders’ decomposition suffers from the same drawbacks of the cutting-plane method, i.e., oscillation and slow convergence, compounded with the fact that the master problem is combinatorial. To overcome this drawback we present a stabilized Benders decomposition whose master is restricted to a neighborhood of successful candidates by local branching constraints, which are dynamically adjusted, and even dropped, during the iterations. Our experiments with randomly generated and real-world CSP instances with up to 3600 binary variables, 90M continuous variables and 15M inequality constraints show that our approach is competitive with both the current state-of-the-art (cutting-plane-based) code for cell suppression, and the Benders implementation in CPLEX 12.7. In some instances, stabilized Benders is able to quickly provide a very good solution in less than one minute, while the other approaches were not able to find any feasible solution in one hour.Peer ReviewedPreprin
Reformulation and decomposition of integer programs
In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm
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