ABSTRACT The minimum ‐‐cut problem is one of the most‐studied problems in discrete optimization and has a unique complexity status in multi‐objective optimization. Even though the single‐objective version of the problem can be solved in polynomial time, it has been shown in the seminal work of Papadimitriou and Yannakakis (2000) that there does not exist a multi‐objective fully polynomial‐time approximation scheme (MFPTAS) for the minimum ‐‐cut problem unless . This holds both for the case of objective functions with arc capacities in and for objective functions with general capacities, and even for tractable instances where the number of non‐dominated points is only quadratic in the input size. In this article, we strengthen these results by showing that, assuming , there does not exist an MFPTAS for the minimum ‐‐cut problem with two objectives and arc capacities in , nor for the minimum ‐‐cut problem with two objectives and arc capacities in . This advancement is particularly interesting since the considered problem variants are the only known problems in multi‐objective optimization that do not admit an MFPTAS even though their single‐objective versions are solvable in polynomial time and the problems are tractable , that is, the numbers of non‐dominated points are polynomial (even linear) in the input size. Furthermore, we complement this result by showing that, on graphs of bounded tree‐width, the minimum ‐‐cut problem with polynomially bounded arc capacities can be solved exactly in polynomial time for any constant number of objectives.

Convex approximation sets for multiobjective optimization problems are a well-studied relaxation of the common notion of approximation sets. Instead of approximating each image of a feasible solution by the image of some solution in the approximation set up to a multiplicative factor in each component, a convex approximation set only requires this multiplicative approximation to be achieved by some convex combination of finitely many images of solutions in the set. This makes convex approximation sets efficiently computable for a wide range of multiobjective problems: even for many problems for which (classic) approximations sets are hard to compute. In this article, we propose a polynomial-time algorithm to compute convex approximation sets that builds on an exact or approximate algorithm for the weighted sum scalarization and is therefore applicable to a large variety of multiobjective optimization problems. The provided convex approximation quality is arbitrarily close to the approximation quality of the underlying algorithm for the weighted sum scalarization. In essence, our algorithm can be interpreted as an approximate version of the dual variant of Benson’s outer approximation algorithm. Thus, in contrast to existing convex approximation algorithms from the literature, information on solutions obtained during the approximation process is utilized to significantly reduce both the practical running time and the cardinality of the returned solution sets while still guaranteeing the same worst-case approximation quality. We underpin these advantages by the first comparison of all existing convex approximation algorithms on several instances of the triobjective knapsack problem and the triobjective symmetric metric traveling salesman problem. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research was supported by the German Research Foundation [Project 398572517]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0220 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0220 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Abstract Linear-parametric optimization, where multiple objectives are combined into a single objective using linear combinations with parameters as coefficients, has numerous links to other fields in optimization and a wide range of application areas. In this survey, we provide a comprehensive overview of structural results and algorithmic strategies for solving linear-parametric optimization problems exactly and approximately. Transferring concepts from related areas such as multi-objective optimization provides further relevant results. The survey consists of two parts: First, we list strategies that work in a general fashion and do not rely on specific problem structures. Second, we look at well-studied parametric optimization problems and cover both important theoretical results and specialized algorithmic approaches for these problems. Among these problems are parametric variants of shortest path problems, minimum cost flow and maximum flow problems, spanning tree problems, the knapsack problem, and matching problems. Overall, we cover the results from 128 publications (and refer to 35 supplemental works) published between 1963 and 2024.

Eye tracking technology in adaptive learning systems enhances diagnostic capabilities by providing valuable insights into cognitive processes. This information can be leveraged to identify and address difficulties. So far, there have been only few attempts of realizing this. Studies are usually only about recognizing correctness of answers and the evaluation is complex and difficult to transfer due to features depending on Areas of Interests (AOIs). We close this gap and present a time-dynamic approach to identify specific difficulties based on raw gaze data. The eye tracking data of 139 students while solving a math problem serve as a sample. Difficulties that arose during the solution process are known. A temporal convolutional network (TCN) is trained to perform a multiclass classification on sequential data. On this basis we present an algorithm which simulates a dynamic classification in an adaptive real-time system. We evaluate this procedure achieving an accuracy of almost 80%.

Automated detection of individual problem-solving strategies is mandatory for the realization of adaptive learning systems (ALS). In this context, we present a new algorithm which is able to detect dynamic geometric structures (slope triangles) in gaze data of 62 learners using automated fixation-clustering and temporal networks. Preliminary results show a promising performance for fixed bandwidth parameters of the clustering algorithm (average recall: 0.60, maximum precision: 0.66).

Abstract Scalarization is a common technique to transform a multiobjective optimization problem into a scalar-valued optimization problem. This article deals with the weighted Tchebycheff scalarization applied to multiobjective discrete optimization problems. This scalarization consists of minimizing the weighted maximum distance of the image of a feasible solution to some desirable reference point. By choosing a suitable weight, any Pareto optimal image can be obtained. In this article, we provide a comprehensive theory of this set of eligible weights. In particular, we analyze the polyhedral and combinatorial structure of the set of all weights yielding the same Pareto optimal solution as well as the decomposition of the weight set as a whole. The structural insights are linked to properties of the set of Pareto optimal solutions, thus providing a profound understanding of the weighted Tchebycheff scalarization method and, as a consequence, also of all methods for multiobjective optimization problems using this scalarization as a building block.

Abstract This article investigates the approximation quality achievable for biobjective minimization problems with respect to the Pareto cone by solutions that are (approximately) optimal with respect to larger ordering cones. When simultaneously considering $$\alpha $$ α -approximations for all closed convex ordering cones of a fixed inner angle $$\gamma \in \left[ \frac{\pi }{2}, \pi \right] $$ γ ∈ π 2 , π , an approximation guarantee between $$\alpha $$ α and $$2 \alpha $$ 2 α is achieved, which depends continuously on $$\gamma $$ γ . The analysis is best-possible for any inner angle and it generalizes and unifies the known results that the set of supported solutions is a 2-approximation and that the efficient set itself is a 1-approximation. Moreover, it is shown that, for maximization problems, no approximation guarantee is achievable in general by considering larger ordering cones in the described fashion, which again generalizes a known result about the set of supported solutions.

Abstract We determine the power of the weighted sum scalarization with respect to the computation of approximations for general multiobjective minimization and maximization problems. Additionally, we introduce a new multi-factor notion of approximation that is specifically tailored to the multiobjective case and its inherent trade-offs between different objectives. For minimization problems, we provide an efficient algorithm that computes an approximation of a multiobjective problem by using an exact or approximate algorithm for its weighted sum scalarization. In case that an exact algorithm for the weighted sum scalarization is used, this algorithm comes arbitrarily close to the best approximation quality that is obtainable by supported solutions – both with respect to the common notion of approximation and with respect to the new multi-factor notion. Moreover, the algorithm yields the currently best approximation results for several well-known multiobjective minimization problems. For maximization problems, however, we show that a polynomial approximation guarantee can, in general, not be obtained in more than one of the objective functions simultaneously by supported solutions.

Algorithms for approximating the nondominated set of multiobjective optimization problems are reviewed. The approaches are categorized into general methods that are applicable under mild assumptions and, thus, to a wide range of problems, and into algorithms that are specifically tailored to structured problems. All in all, this survey covers 52 articles published within the last 41 years, that is, between 1979 and 2020. Summary of Contribution: In many problems in operations research, several conflicting objective functions have to be optimized simultaneously, and one is interested in finding Pareto optimal solutions. Because of the high complexity of finding Pareto optimal solutions and their usually very large number, however, the exact solution of such multiobjective problems is often very difficult, which motivates the study of approximation algorithms for multiobjective optimization problems. This research area uses techniques and methods from algorithmics and computing in order to efficiently determine approximate solutions to many well-known multiobjective problems from operations research. Even though approximation algorithms for multiobjective optimization problems have been investigated for more than 40 years and more than 50 research articles have been published on this topic, this paper provides the first survey of this important area at the intersection of computing and operations research.

Abstract Papadimitriou and Yannakakis (Proceedings of the 41st annual IEEE symposium on the Foundations of Computer Science (FOCS), pp 86–92, 2000) show that the polynomial-time solvability of a certain auxiliary problem determines the class of multiobjective optimization problems that admit a polynomial-time computable $$(1+\varepsilon , \dots , 1+\varepsilon )$$ ( 1 + ε , ⋯ , 1 + ε ) -approximate Pareto set (also called an $$\varepsilon $$ ε -Pareto set). Similarly, in this article, we characterize the class of multiobjective optimization problems having a polynomial-time computable approximate $$\varepsilon $$ ε -Pareto set that is exact in one objective by the efficient solvability of an appropriate auxiliary problem. This class includes important problems such as multiobjective shortest path and spanning tree, and the approximation guarantee we provide is, in general, best possible. Furthermore, for biobjective optimization problems from this class, we provide an algorithm that computes a one-exact $$\varepsilon $$ ε -Pareto set of cardinality at most twice the cardinality of a smallest such set and show that this factor of 2 is best possible. For three or more objective functions, however, we prove that no constant-factor approximation on the cardinality of the set can be obtained efficiently.

Abstract This article is dedicated to the weight set decomposition of a multiobjective (mixed-)integer linear problem with three objectives. We propose an algorithm that returns a decomposition of the parameter set of the weighted sum scalarization by solving biobjective subproblems via Dichotomic Search which corresponds to a line exploration in the weight set. Additionally, we present theoretical results regarding the boundary of the weight set components that direct the line exploration. The resulting algorithm runs in output polynomial time, i.e. its running time is polynomial in the encoding length of both the input and output. Also, the proposed approach can be used for each weight set component individually and is able to give intermediate results, which can be seen as an “approximation” of the weight set component. We compare the running time of our method with the one of an existing algorithm and conduct a computational study that shows the competitiveness of our algorithm. Further, we give a state-of-the-art survey of algorithms in the literature.

The alternating direction method of multipliers has recently been adapted for linear programming decoding of low-density parity-check codes. The computation of the projection onto the parity polytope is the core of this algorithm and usually involves a sorting operation, which is the main effort of the projection. In this paper, we present an algorithm with low complexity to compute this projection. The algorithm relies on new findings in the recursive structure of the parity polytope and iteratively fixes selected components. As shown in our realistic simulation setup, it requires up to 37% less arithmetical operations compared with state-of-the-art projections. Additionally, it does not involve a sorting operation, which is needed in all exact state-of-the-art projection algorithms. These two benefits make it appealing for efficient hardware and software implementations.

We consider the (linear) parametric 0–1 knapsack problem in which the profits of the items are affine-linear functions of a real-valued parameter and the task is to compute a solution for all values of the parameter. For this problem, it is known that the piecewise linear convex function mapping the parameter to the optimal objective value of the corresponding instance (called the optimal value function ) can have exponentially many breakpoints (points of slope change), which implies that every optimal algorithm for the problem must output a number of solutions that is exponential in the number of items. We provide the first (parametric) polynomial time approximation scheme (PTAS) for the parametric 0–1 knapsack problem. Moreover, we exploit the connection between the parametric problem and the bicriteria problem in order to show that the parametric 0–1 knapsack problem admits a parametric FPTAS when the parameter is restricted to the positive real line and the slopes and intercepts of the affine-linear profit functions of the items are nonnegative. The method used to obtain this result applies to many linear parametric optimization problems and provides a general connection between bicriteria and linear parametric optimization problems.

The hypervolume subset selection problem consists of finding a subset, with a given cardinality k, of a set of nondominated points that maximizes the hypervolume indicator. This problem arises in selection procedures of evolutionary algorithms for multiobjective optimization, for which practically efficient algorithms are required. In this article, two new formulations are provided for the two-dimensional variant of this problem. The first is a (linear) integer programming formulation that can be solved by solving its linear programming relaxation. The second formulation is a k-link shortest path formulation on a special digraph with the Monge property that can be solved by dynamic programming in [Formula: see text] time. This improves upon the result of [Formula: see text] in Bader ( 2009 ), and slightly improves upon the result of [Formula: see text] in Bringmann et al. ( 2014b ), which was developed independently from this work using different techniques. Numerical results are shown for several values of n and k.

In this work, we consider efficient maximum-likelihood decoding of linear block codes for small-to-moderate block lengths. The presented approach is a branch-and-bound algorithm using the cutting-plane approach of Zhang and Siegel (IEEE Trans. Inf. Theory, 2012) for obtaining lower bounds. We have compared our proposed algorithm to the state-of-the-art commercial integer program solver CPLEX, and for all considered codes our approach is faster for both low and high signal-to-noise ratios. For instance, for the benchmark (155, 64) Tanner code our algorithm is more than 11 times as fast as CPLEX for an SNR of 1.0 dB on the additive white Gaussian noise channel. By a small modification, our algorithm can be used to calculate the minimum distance, which we have again verified to be much faster than using the CPLEX solver.

An efficient LP decoder is the key building block for a maximum likelihood decoder based on integer programming. In this paper we propose to employ a variant of the simplex algorithm for LP decoding, called the dual simplex algorithm. This algorithm has two advantages: It inherently uses the received LLRs to generate a close to optimum starting solution and it allows to reuse former LP solutions if an adaptive LP decoding scheme is used. It is shown, that the dual simplex algorithm outperforms the standard (primal) simplex by a factor of 15-20 in runtime. This allows for efficient future hardware implementations. Furthermore the use of fixed-point instead of floating-point numbers is investigated to further reduce hardware complexity.

We present a novel algorithm that solves the turbo code LP decoding problem in a finite number of steps by Euclidean distance minimizations, which in turn rely on repeated shortest path computations in the trellis graph representing the turbo code. Previous attempts to exploit the combinatorial graph structure only led to algorithms which are either of heuristic nature or do not guarantee finite convergence. A numerical study shows that our algorithm clearly beats the running time, up to a factor of 100, of generic commercial LP solvers for medium-sized codes, especially for high SNR values.

Abstract In a dynamic network, the quickest path problem asks for a path such that a given amount of flow can be sent from source to sink via this path in minimal time. In practical settings, for example, in evacuation or transportation planning, the problem parameters might not be known exactlya priori. It is therefore of interest to consider robust versions of these problems in which travel times and/or capacities of arcs depend on a certain scenario. In this article, min–max versions of robust quickest path problems are investigated and, depending on their complexity status, exact algorithms or fully polynomial‐time approximation schemes are proposed. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012

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 paper reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.

We present an exact algorithm for computing an earliest arrival flow in a discrete time setting on series-parallel graphs. In contrast to previous results for the earliest arrival flow problem this algorithm runs in polynomial time. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 57(2), 169–173 2011

For binary linear codes with short and medium block length ML decoding can be achieved by solving the associated integer programming (IP) problem with a general purpose solver. IP also offers algorithms for computing the minimum distance. In this article, we present several IP formulations and computationally compare them on various LDPC and BCH codes. Most of these formulations are obtained by forcing integrality on linear programming (LP) decoding formulations proposed in the literature.

Maximum likelihood (ML) decoding is the optimal decoding algorithm for arbitrary linear block codes and can be written as an integer programming (IP) problem. Feldman relaxed this IP problem and presented linear programming (LP) based decoding. In this paper, we propose a new separation algorithm to improve the error-correcting performance of LP decoding for binary linear block codes. We use an IP formulation with indicator variables that help in detecting the violated parity checks. We derive Gomory cuts from the IP and use them in our separation algorithm. An efficient method of finding cuts induced by redundant parity checks (RPC) is also proposed. Under certain circumstances we can guarantee that these RPC cuts are valid and cut off the fractional optimal solutions of LP decoding. It is demonstrated on three LDPC codes and two BCH codes that our separation algorithm performs significantly better than LP decoding and belief propagation (BP) decoding.

We study an integer programming (IP) based separation approach to find the maximum likelihood (ML) codeword for binary linear codes. An algorithm introduced in Tanatmis et al. is extended and improved with respect to decoding performance without increasing the worst case complexity. This is demonstrated on the LDPC and the BCH code classes. Moreover, we propose an integer programming formulation to calculate the minimum distance of a binary linear code. We exemplarily compute the minimum distance of the (204, 102) LDPC code and the (576, 288) WIMAX code. Using the minimum distance of a code, a new class of valid inequalities is introduced.