LP-Based Algorithms for Capacitated Facility Location

Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees were based on a single technique, local search, and no linear programming relaxation was known to efficiently approximate the problem. In this paper, we present a linear programming relaxation with constant integrality gap for capacitated facility location. We demonstrate that the fundamental theories of multi-commodity flows and matchings provide key insights that lead to the strong relaxation. Our algorithmic proof of integrality gap is obtained by finally accessing the rich toolbox of LP-based methodologies: we present a constant factor approximation algorithm based on LP-rounding.Comment: 25 pages, 6 figures; minor revision

An (MI)LP-based Primal Heuristic for 3-Architecture Connected Facility Location in Urban Access Network Design

We investigate the 3-architecture Connected Facility Location Problem arising in the design of urban telecommunication access networks. We propose an original optimization model for the problem that includes additional variables and constraints to take into account wireless signal coverage. Since the problem can prove challenging even for modern state-of-the art optimization solvers, we propose to solve it by an original primal heuristic which combines a probabilistic fixing procedure, guided by peculiar Linear Programming relaxations, with an exact MIP heuristic, based on a very large neighborhood search. Computational experiments on a set of realistic instances show that our heuristic can find solutions associated with much lower optimality gaps than a state-of-the-art solver.Comment: This is the authors' final version of the paper published in: Squillero G., Burelli P. (eds), EvoApplications 2016: Applications of Evolutionary Computation, LNCS 9597, pp. 283-298, 2016. DOI: 10.1007/978-3-319-31204-0_19. The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-31204-0_1

Exploration of the scalability of LocFaults approach for error localization with While-loops programs

A model checker can produce a trace of counterexample, for an erroneous program, which is often long and difficult to understand. In general, the part about the loops is the largest among the instructions in this trace. This makes the location of errors in loops critical, to analyze errors in the overall program. In this paper, we explore the scala-bility capabilities of LocFaults, our error localization approach exploiting paths of CFG(Control Flow Graph) from a counterexample to calculate the MCDs (Minimal Correction Deviations), and MCSs (Minimal Correction Subsets) from each found MCD. We present the times of our approach on programs with While-loops unfolded b times, and a number of deviated conditions ranging from 0 to n. Our preliminary results show that the times of our approach, constraint-based and flow-driven, are better compared to BugAssist which is based on SAT and transforms the entire program to a Boolean formula, and further the information provided by LocFaults is more expressive for the user

Mapping constrained optimization problems to quantum annealing with application to fault diagnosis

Current quantum annealing (QA) hardware suffers from practical limitations such as finite temperature, sparse connectivity, small qubit numbers, and control error. We propose new algorithms for mapping boolean constraint satisfaction problems (CSPs) onto QA hardware mitigating these limitations. In particular we develop a new embedding algorithm for mapping a CSP onto a hardware Ising model with a fixed sparse set of interactions, and propose two new decomposition algorithms for solving problems too large to map directly into hardware. The mapping technique is locally-structured, as hardware compatible Ising models are generated for each problem constraint, and variables appearing in different constraints are chained together using ferromagnetic couplings. In contrast, global embedding techniques generate a hardware independent Ising model for all the constraints, and then use a minor-embedding algorithm to generate a hardware compatible Ising model. We give an example of a class of CSPs for which the scaling performance of D-Wave's QA hardware using the local mapping technique is significantly better than global embedding. We validate the approach by applying D-Wave's hardware to circuit-based fault-diagnosis. For circuits that embed directly, we find that the hardware is typically able to find all solutions from a min-fault diagnosis set of size N using 1000N samples, using an annealing rate that is 25 times faster than a leading SAT-based sampling method. Further, we apply decomposition algorithms to find min-cardinality faults for circuits that are up to 5 times larger than can be solved directly on current hardware.Comment: 22 pages, 4 figure

Sparse sum-of-squares (SOS) optimization: A bridge between DSOS/SDSOS and SOS optimization for sparse polynomials

Optimization over non-negative polynomials is fundamental for nonlinear systems analysis and control. We investigate the relation between three tractable relaxations for optimizing over sparse non-negative polynomials: sparse sum-of-squares (SSOS) optimization, diagonally dominant sum-of-squares (DSOS) optimization, and scaled diagonally dominant sum-of-squares (SDSOS) optimization. We prove that the set of SSOS polynomials, an inner approximation of the cone of SOS polynomials, strictly contains the spaces of sparse DSOS/SDSOS polynomials. When applicable, therefore, SSOS optimization is less conservative than its DSOS/SDSOS counterparts. Numerical results for large-scale sparse polynomial optimization problems demonstrate this fact, and also that SSOS optimization can be faster than DSOS/SDSOS methods despite requiring the solution of semidefinite programs instead of less expensive linear/second-order cone programs.Comment: 9 pages, 3 figure