Inverse of the Gomory corner relaxation of integer programs

What is a minimal set of tuples to delete from a database in order to eliminate all query answers? This problem is called "the resilience of a query" and is one of the key algorithmic problems underlying various forms of reverse data management, such as view maintenance, deletion propagation and causal responsibility. A long-open question is determining the conjunctive queries (CQs) for which resilience can be solved in PTIME. We shed new light on this problem by proposing a unified Integer Linear Programming (ILP) formulation. It is unified in that it can solve both previously studied restrictions (e.g., self-join-free CQs under set semantics that allow a PTIME solution) and new cases (all CQs under set or bag semantics). It is also unified in that all queries and all database instances are treated with the same approach, yet the algorithm is guaranteed to terminate in PTIME for all known PTIME cases. In particular, we prove that for all known easy cases, the optimal solution to our ILP is identical to a simpler Linear Programming (LP) relaxation, which implies that standard ILP solvers return the optimal solution to the original ILP in PTIME. Our approach allows us to explore new variants and obtain new complexity results. 1) It works under bag semantics, for which we give the first dichotomy results in the problem space. 2) We extend our approach to the related problem of causal responsibility and give a more fine-grained analysis of its complexity. 3) We recover easy instances for generally hard queries, including instances with read-once provenance and instances that become easy because of Functional Dependencies in the data. 4) We solve an open conjecture about a unified hardness criterion from PODS 2020 and prove the hardness of several queries of previously unknown complexity. 5) Experiments confirm that our findings accurately predict the asymptotic running times, and that our universal ILP is at times even quicker than a previously proposed dedicated flow algorithm.

This paper considers two-dimensional (2-D) retiming, which is the problem of retiming circuits that operate on 2-D signals. We begin by discussing two types of parallelism available in 2-D data processing, which we call inter-iteration parallelism and inter-operation parallelism. We then present two novel techniques for 2-D retiming that can be used to extract inter-operation parallelism. These two techniques are designed to minimize the amount of memory required to implement a 2-D data-flow graph while maintaining a desired clock rate for the circuit. The first technique is based on an integer linear programming (ILP) formulation of the problem, and is called ILP 2-D retiming. This technique considers the entire 2-D retiming problem as a whole, but long central processing unit times are required if the circuit is large. The second technique, called orthogonal 2-D retiming, is a linear programming formulation which is derived by partitioning ILP 2-D retiming into two parts called s- and a-retiming. This technique finds a solution in polynomial time and is much faster than the ILP 2-D retiming technique, but the two sub problems (s- and a-retiming) can give results which are not compatible with one another. To solve this incompatibility problem, a variation of orthogonal 2-D retiming called integer orthogonal 2-D retiming is developed. This technique runs in polynomial time and the s-retiming and a-retiming steps are guaranteed to give compatible results. We show that the techniques presented in this paper can result in memory hardware savings of 50% compared to previously published 2-D retiming techniques.

The symmetric groups Sn and the cyclic groups Cn essentially are the only examples for symmetry groups of linear or integer programs that have been discussed in the literature, see e.g. [5] and [6]. In [4], Bodi, Herr, and Joswig developed some ideas to tackle linear and integer programs with arbitrary groups of symmetries. However, the question remained whether or not there are linear (integer) programs with groups of symmetries other than Sn and Cn. Indeed, we show in this short note that every finite permutation group is the full symmetry group of a suitable linear or integer program. Some of our constructions are based on graph theory.

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Side-chain positioning is a central component of homology modeling and protein design. In a common formulation of the problem, the backbone is fixed, side-chain conformations come from a rotamer library, and a pairwise energy function is optimized. It is NP-complete to find even a reasonable approximate solution to this problem. We seek to put this hardness result into practical context. We present an integer linear programming (ILP) formulation of side-chain positioning that allows us to tackle large problem sizes. We relax the integrality constraint to give a polynomial-time linear programming (LP) heuristic. We apply LP to position side chains on native and homologous backbones and to choose side chains for protein design. Surprisingly, when positioning side chains on native and homologous backbones, optimal solutions using a simple, biologically relevant energy function can usually be found using LP. On the other hand, the design problem often cannot be solved using LP directly; however, optimal solutions for large instances can still be found using the computationally more expensive ILP procedure. While different energy functions also affect the difficulty of the problem, the LP/ILP approach is able to find optimal solutions. Our analysis is the first large-scale demonstration that LP-based approaches are highly effective in finding optimal (and successive near-optimal) solutions for the side-chain positioning problem.

As we approach exascale systems, power is turning from an optimization goal to a critical operating constraint. With power bounds imposed by both stakeholders and the limitations of existing infrastructure, we need to develop new techniques that work with limited power to extract maximum performance. In this paper, we explore this area and provide an approach to find the theoretical upper bound of computational performance on a per-application basis in hybrid MPI + OpenMP applications.

Job shops are an important production environment for low-volume high-variety manufacturing. Its scheduling has recently been formulated as an integer linear programming (ILP) problem to take advantages of popular mixed-integer linear programming (MILP) methods, e.g., branch-and-cut. When considering a large number of parts, MILP methods may experience difficulties. To address this, a critical but much overlooked issue is formulation tightening. The idea is that if problem constraints can be transformed to directly delineate the problem convex hull in the data preprocessing stage, then a solution can be obtained by using linear programming (LP) methods without combinatorial difficulties. The tightening process, however, is fundamentally challenging because of the existence of integer variables. In this article, an innovative and systematic approach is established for the first time to tighten the formulations of individual parts, each with multiple operations, in the data preprocessing stage. It is a major advancement of our previous work on problems with binary and continuous variables to integer variables. The idea is to first link integer variables to binary variables by innovatively combining constraints so that the integer variables are uniquely determined by the binary variables. With binary and continuous variables only, it is proved that the vertices of the convex hull can be obtained based on vertices of the LP problem after relaxing binary requirements. These vertices are then converted to tightened constraints for general use. This approach significantly improves our previous results on tightening individual operations. Numerical results demonstrate significant benefits on solution quality and computational efficiency. This approach also applies to other complex ILP and MILP problems with similar characteristics and fundamentally changes the way how such problems are formulated and solved. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Scheduling is an important but difficult problem in planning and operation of job shops. The problem has been recently formulated in an integer linear programming (ILP) form to take advantage of popular mixed-integer linear programming methods. Given an ILP problem, there must exist a linear programming (LP) formulation so that all of its vertices are also the vertices to the ILP problem. If such an LP problem can be found in the data preprocessing stage, then the corresponding ILP problem is tight and can be solved by using an LP method without difficulties. In this article, an innovative and systematic approach is established to tighten the formulations of individual parts, each with one or multiple operations. It is a major advancement of our previous work on problems with binary and continuous variables by novel exploitation of the relationship between integer and binary variables in job-shop scheduling. The resulting tightened constraints are characterized by part parameters and the length of the scheduling horizon and can be easily adjusted for other data sets. Results demonstrate significant benefits on solution quality and computational efficiency. This approach also applies to other complex ILP and MILP problems with similar characteristics and fundamentally changes the way how such problems are formulated and solved.

Introduction and Preliminaries. Problems, Algorithms, and Complexity. LINEAR ALGEBRA. Linear Algebra and Complexity. LATTICES AND LINEAR DIOPHANTINE EQUATIONS. Theory of Lattices and Linear Diophantine Equations. Algorithms for Linear Diophantine Equations. Diophantine Approximation and Basis Reduction. POLYHEDRA, LINEAR INEQUALITIES, AND LINEAR PROGRAMMING. Fundamental Concepts and Results on Polyhedra, Linear Inequalities, and Linear Programming. The Structure of Polyhedra. Polarity, and Blocking and Anti--Blocking Polyhedra. Sizes and the Theoretical Complexity of Linear Inequalities and Linear Programming. The Simplex Method. Primal--Dual, Elimination, and Relaxation Methods. Khachiyana s Method for Linear Programming. The Ellipsoid Method for Polyhedra More Generally. Further Polynomiality Results in Linear Programming. INTEGER LINEAR PROGRAMMING. Introduction to Integer Linear Programming. Estimates in Integer Linear Programming. The Complexity of Integer Linear Programming. Totally Unimodular Matrices: Fundamental Properties and Examples. Recognizing Total Unimodularity. Further Theory Related to Total Unimodularity. Integral Polyhedra and Total Dual Integrality. Cutting Planes. Further Methods in Integer Linear Programming. References. Indexes.

In this work, we focus on the problem of latency-constrained scheduling with consideration of multiple voltage technologies in High-level synthesis. Without the resource concern, we propose an Integer Linear Programming (ILP) formulation and further relax it to a piecewise Linear Programming (LP) problem, which is optimally solved using the efficient piecewise-linear extended network simplex method(PLNSM). The experimental results showed 80X+ speedup compared to the general LP formulation. Considering the resource usage, we propose a two-stage heuristic Network Simplex Method based Power-efficient Multiple Voltage Scheduling(NPMVS) method. Firstly, the above relaxed LP formulation is modified to perform mobility allocation and delay assignment for the operations so as to minimize the power and the differences between the allocated operation mobilities and the predefined target mobilities. The modified formulation is solved using the PLNSM and iteratively performed to minimize power and resource density variation in control steps by gradually updating the predefined target mobilities. Secondly, with the allocated operation mobilities, we apply dependency-free operation scheduling with the objective of minimizing the resource usage. Experimental results show that the proposed method can produce optimum solutions for all 6 benchmarks with 14 groups of data in a maximum time of 0.25 second.

Partial inverse combinatorial optimization problems are bilevel optimization problems in which the leader aims to incentivize the follower to include respectively not include given sets of elements in the solution of their combinatorial problem. If the sets of required and forbidden elements define a complete follower solution and the follower problem is solvable in polynomial time, then the inverse combinatorial problem is also solvable in polynomial time. In contrast, partial inverse problems can be NP-complete when the follower problem is solvable in polynomial time. This applies e.g. to the partial inverse min cut problem. In this paper, we consider partial inverse combinatorial optimization problems in which weights can only be increased. Furthermore, we assume that the lower-level combinatorial problem can be solved as a linear program. In this setting, we show that the partial inverse shortest path problem on a directed acyclic graph is NP-complete. Moreover, the partial inverse assignment problem is NP-complete. Both results even hold if there is only one required arc or edge, respectively. For solving partial inverse combinatorial optimization problems with only weight increases, we present a novel branch-and-bound scheme that exploits the difference in complexity between complete inverse and partial inverse versions of a problem. For both primal heuristics and node relaxations, we use auxiliary problems that are basically complete inverse problems on similar instances. Branching is done on follower variables. We test our approach on partial inverse shortest path, assignment and min cut problems, and computationally compare it to an MPCC reformulation as well as a decomposition scheme.

In recent decades, the research of energy-aware scheduling on heterogeneous multiprocessor systems is becoming more and more popular. A classic method for real-time task allocation is Linear Programming (LP). However, existing LP formulations are usually regarded as ineffective in solving large-scale allocation problems due to the unacceptable time consumption. In this work, we propose two integer linear programming (ILP) formulations to deal with the allocation problems for large task sets. One exact ILP(1) is formulated to derive an intermediate solution, and the other relaxed ILP(2) is considered to calculate the desired minimum energy. Then the desired minimum energy can be taken as a reference to evaluate the optimality of the intermediate solution. Experimental results on randomly generated task sets demonstrate that our method achieves average 19.2% less energy within limited time than the classic greedy and the state-of-the-art heuristic algorithm.

We consider integer and linear programming problems for which the linear constraints exhibit a (recursive) block-structure: The problem decomposes into independent and efficiently solvable sub-problems if a small number of constraints is deleted. A prominent example are $n$-fold integer programming problems and their generalizations which have received considerable attention in the recent literature. The previously known algorithms for these problems are based on the augmentation framework, a tailored integer programming variant of local search. In this paper we propose a different approach. Our algorithm relies on parametric search and a new proximity bound. We show that block-structured linear programming can be solved efficiently via an adaptation of a parametric search framework by Norton, Plotkin, and Tardos in combination with Megiddo's multidimensional search technique. This also forms a subroutine of our algorithm for the integer programming case by solving a strong relaxation of it. Then we show that, for any given optimal vertex solution of this relaxation, there is an optimal integer solution within $\ell_1$-distance independent of the dimension of the problem. This in turn allows us to find an optimal integer solution efficiently. We apply our techniques to integer and linear programming with $n$-fold structure or bounded dual treedepth, two benchmark problems in this field. We obtain the first algorithms for these cases that are both near-linear in the dimension of the problem and strongly polynomial. Moreover, unlike the augmentation algorithms, our approach is highly parallelizable.

A critical step in the process of creating a relational data base is normalization, whereby a series of tables is created in order to avoid redundancy problems and insertion and deletion anomalies. Several important normalization decision problems are NP-Complete. In this paper we develop a unified approach to data base normalization using linear integer programming. These linear integer programming formulations result from structuring normalization problems as gainfree Leontief substitution flow problems on hypergraphs. We provide a sufficient condition based on the structural properties of the hypergraph which yields a strongly polynomial algorithm for testing whether or not a given decomposition is in third normal form. We also show that under these conditions the key of a relation is unique and a relation in third normal form is also in Boyce-Codd normal form. We also provide linear integer programming formulations for solving normalization decision problems when no special structure is present. We test these linear integer programming models for normalization against existing normalization algorithms and find the mathematical programming approach is several orders of magnitude faster. Our results suggest that the mathematical programming approach is practical for normalizing actual large scale data bases.

Integer Linear Programming (ILP) formulations of multi-label Markov random fields (MRFs) models with global connectivity priors were investigated previously in computer vision. In these works, only Linear Programming (LP) relaxations [1] or simplified versions [2] of the problem were solved. This paper investigates the ILP of MRF with exact connectivity priors via a branch-and-cut method, which provably finds globally optimal solutions. It enforces connectivity priors iteratively by a cutting plane method, and provides feasible solutions with a guarantee on sub-optimality even if we terminate it earlier. The proposed ILP can be applied as a post-processing method on top of any existing multi-label segmentation approach. As it provides globally optimal solution, it can be used off-line to serve as quality check for any fast on-line algorithm. Furthermore, the scribble based model presented in this paper could be potentially used to generate ground-truth proposals for any deep learning based segmentation. We demonstrate the power and usefulness of our model by extensive experiments on the BSDS500 and PASCAL VOC dataset. The experiments show that our proposed model achieves great performance, yielding provably global optimum in most instances and that provably good optimization solutions also provide good segmentation accuracy, even with the limited computing time of few seconds.

Structured prediction is used in areas such as computer vision and natural language processing to predict structured outputs such as segmentations or parse trees. In these settings, prediction is performed by MAP inference or, equivalently, by solving an integer linear program. Because of the complex scoring functions required to obtain accurate predictions, both learning and inference typically require the use of approximate solvers. We propose a theoretical explanation to the striking observation that approximations based on linear programming (LP) relaxations are often tight on real-world instances. In particular, we show that learning with LP relaxed inference encourages integrality of training instances, and that tightness generalizes from train to test data.