Linear programming relaxation

Summary

In mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable. For example, in a 0–1 integer program, all constraints are of the form :x_i\in{0,1}. The relaxation of the original integer program instead uses a collection of linear constraints :0 \le x_i \le 1. The resulting relaxation is a linear program, hence the name. This relaxation technique transforms an NP-hard optimization problem (integer programming) into a related problem that is solvable in polynomial time (linear programming); the solution to the relaxed linear program can be used to gain information about the solution to the original integer program. Example Consider the set cover problem, the linear programming relaxation of which was first considered by . In this problem, one is given as input a family of sets F = {S0, S1, ...}; the task is to find a subfamily, with as few sets as possible,

Official source

https://en.wikipedia.org/wiki/Linear_programming_relaxation

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High-dimensional Data Cubes

Sachin Basil John, Christoph Koch

This paper introduces an approach to supporting high-dimensional data cubes at interactive query speeds and moderate storage cost. The approach is based on binary(-domain) data cubes that are judiciously partially materialized; the missing information can be quickly reconstructed using statistical or linear programming techniques. This enables new applications such as exploratory data analysis for feature engineering and other fields of data science. Moreover, it allows us to use data cubes in data warehouses and data lakes without design regret. It removes the need to compromise when building a data cube — all columns that we might ever wish to use can be included as dimensions. As a collateral benefit, our approach also speeds up certain dice, roll-up, and drill-down operations on data cubes with hierarchical dimensions compared to traditional data cubes.

2022

Strengths and Limitations of Linear Programming Relaxations

Abbas Bazzi

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than

2

and

\omega(1)

, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than

1.5

for the Vertex Cover Problem, and better than

2

for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of

2-\varepsilon

for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

EPFL2017

Stochastic optimization is a popular modeling paradigm for decision-making under uncertainty and has a wide spectrum of applications in management science, economics and engineering. However, the stochastic optimization models one faces in practice are intractable, and numerical solutions necessitate approximations. The mainstream approach for making a stochastic optimization model amenable to numerical solution is to discretize the probability distribution of the uncertain problem parameters. However, both the accuracy of the approximation as well as the computational burden of solving the approximate problem scale with the number of scenarios of the approximate distribution. An effective means to ease the computational burden is to use scenario reduction, which replaces an accurate initial distribution accommodating many scenarios with a simpler distribution supported on only few scenarios that is close to the initial distribution with respect to a probability metric. Using the Wasserstein distance as measure of proximity between distributions, we provide new insights into the fundamental limitations of scenario reduction, and we propose the first polynomial-time constant-factor approximations for a popular scenario reduction problem from the literature. As scenario reduction is equivalent to clustering, it suffers from two well-known shortcomings. Namely, it suffers from outlier sensitivity and may produce highly unbalanced clusters. To mitigate both shortcomings, we formulate a joint outlier detection and clustering problem, where the clusters must satisfy certain cardinality constraints. We cast this problem as a mixed-integer linear program (MILP) that admits tractable semidefinite and linear programming relaxations. We propose deterministic rounding schemes that transform the relaxed solutions to feasible solutions for the MILP. We also prove that these solutions are optimal in the MILP if a cluster separation condition holds. Finally, we develop a highly efficient scenario reduction method for a large-scale hydro scheduling problem. Specifically, we study the optimal operation of a fleet of interconnected hydropower plants that sell energy on both the spot and the reserve markets, and we propose a two-layer stochastic programming framework for its solution. The outer layer problem (the planner's problem) optimizes the end-of-day reservoir filling levels over one year, whereas the inner layer problem (the trader's problem) selects optimal hourly market bids within each day. We prove that the trader's problem admits a scenario reduction that dramatically reduces its complexity without loss of optimality, which in turn facilitates an efficient approximation of the planner's problem.

EPFL2020

Strengths and Limitations of Linear Programming Relaxations

Abbas Bazzi

2

and

\omega(1)

1.5

for the Vertex Cover Problem, and better than

2

2-\varepsilon

EPFL2017

EPFL2020