Computing shortest paths is one of the most fundamental algorithmic graph problems. It is known since decades that this problem can be solved in near-linear time if all weights are nonnegative. A recent break-through by [Bernstein, Nanongkai, Wulff-Nilsen '22] presented a randomized near-linear time algorithm for this problem. A subsequent improvement in [Bringmann, Cassis, Fischer '23] significantly reduced the number of logarithmic factors and thereby also simplified the algorithm. It is surprising and exciting that both of these algorithms are combinatorial and do not contain any fundamental obstacles for being practical. We launch the, to the best of our knowledge, first extensive investigation towards a practical implementation of [Bringmann, Cassis, Fischer '23]. To this end, we give an accessible overview of the algorithm, discussing what adaptions are necessary to obtain a fast algorithm in practice. We manifest these adaptions in an efficient implementation. We test our implementation on a benchmark data set that is adapted to be more difficult for our implementation in order to allow for a fair comparison. As in [Bringmann, Cassis, Fischer '23] as well as in our implementation there are multiple parameters to tune, we empirically evaluate their effect and thereby determine the best choices. Our implementation is then extensively compared to one of the state-of-the-art algorithms for this problem [Goldberg, Radzik '93]. On the hardest instance type, we are faster by up to almost two orders of magnitude.

The problem of computational design for menu systems has been addressed in some specific cases such as the linear menu (list). The classical approach has been to model this problem as an assignment task, where commands are assigned to menu positions while optimizing for users’ selection performance and grouping of associated items. However, we show that this approach fails with larger, hierarchically organized menus because it does not take into account the ways in which users navigate hierarchical structures. This paper addresses the computational menu design problem by presenting a novel integer programming formulation that yields usable, well-ordered command hierarchies from a single model. First, it introduces a novel objective function based on information foraging theory, which minimizes navigation time in a hierarchical structure. Second, it models the hierarchical menu design problem as a combination of the exact set covering problem and the assignment problem, organizing commands into ordered groups of ordered groups. The approach is efficient for large, representative instances of the problem. In a controlled usability evaluation, the performance of computationally designed menus was ∼25% faster to use than existing commercial designs. We discuss applications of this approach for personalization and adaptation.

We present a method for solving the transshipment problem-also known as uncapacitated minimum cost flow-up to a multiplicative error of 1+ε in undirected graphs with nonnegative edge weights using a tailored gradient descent algorithm. Using O(\cdot ) to hide polylogarithmic factors in n (the number of nodes in the graph), our gradient descent algorithm takes O(ε 2) iterations, and in each iteration it solves an instance of the transshipment problem up to a multiplicative error of polylog n. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. Using a randomized rounding scheme, we can further extend the method to finding approximate solutions for the single-source shortest paths (SSSP) problem. As a consequence, we improve upon prior works by obtaining the following results: (1) Broadcast CONGEST model: (1 + ε)-approximate SSSP using O(( n + D)ε 3) rounds, where D is the (hop) diameter of the network. (2) Broadcast Congested Clique model: (1 + ε)-approximate transshipment and SSSP using O (ε 2) rounds. (3) Multipass Streaming model: (1 + ε)-approximate transshipment and SSSP using O(n) space and O(ε 2) passes. The previously fastest SSSP algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume nonnegative edge weights that are polynomially bounded in n; for general nonnegative weights, there is an additional multiplicative overhead equal to the logarithm of the maximum ratio between nonzero weights. Our algorithms can also handle asymmetric costs for traversing edges in opposite directions. In this case, we obtain an additional multiplicative dependence of the maximum ratio between the two costs on some edge.

Computing systems have been facing severe technology challenges in recent years with regard to power consumption, circuit reliability, and high performance. For many years, the issues of power consumption and performance have been addressed with the use of technology scaling.However, as Dennard’s scaling tends toward an end, it has become difficult to further improve the performance under the same power constraints. In addition to power, reliability also becomes a critical issue when the feature size of the complementary metal-oxide–semiconductor (CMOS) technology is reduced below 7 nm. Thus, ensuring the complete accuracy of the signal has become increasingly challenging in recent years.

The graphical user interface (GUI) has become the prime means for interacting with computing systems. It leverages human perceptual and motor capabilities for elementary tasks such as command exploration and invocation, information search, and multitasking. For designing a GUI, numerous interconnected decisions must be made such that the outcome strikes a balance between human factors and technical objectives. Normally, design choices are specified manually and coded within the software by professional designers and developers. This article surveys combinatorial optimization as a flexible and powerful tool for computational generation and adaptation of GUIs. As recently as 15 years ago, applications were limited to keyboards and widget layouts. The obstacle has been the mathematical definition of design tasks, on the one hand, and the lack of objective functions that capture essential aspects of human behavior, on the other. This article presents definitions of layout design problems as integer programming tasks, a coherent formalism that permits identification of problem types, analysis of their complexity, and exploitation of known algorithmic solutions. It then surveys advances in formulating evaluative functions for common design-goal foci such as user performance and experience. The convergence of these two advances has expanded the range of solvable problems. Approaches to practical deployment are outlined with a wide spectrum of applications. This article concludes by discussing the position of this application area within optimization and human-computer interaction research and outlines challenges for future work.

Weighted correlation clustering is hard to solve and hard to approximate for general graphs. Its applications in network analysis and computer vision call for efficient algorithms. To this end, we make three contributions: We establish partial optimality conditions that can be checked efficiently, and doing so recursively solves the problem for series-parallel graphs to optimality, in linear time. We exploit the packing dual of the problem to compute a heuristic, but non-trivial lower bound faster than that of a canonical linear program relaxation. We introduce a re-weighting with the dual solution by which efficient local search algorithms converge to better feasible solutions. The effectiveness of our methods is demonstrated empirically on a number of benchmark instances.

We present a near-optimal distributed algorithm for $(1+o(1))$-approximation of single-commodity maximum flow in undirected weighted networks that runs in $(D+\sqrt{n})\cdot n^{o(1)}$ communication rounds in the CONGEST model. Here, $n$ and $D$ denote the number of nodes and the network diameter, respectively. This is the first improvement over the trivial bound of $O(n^2)$, and it nearly matches the $\tilde{\Omega}(D+\sqrt{n})$-round complexity lower bound. The development of the algorithm entails two subresults of independent interest: (i) A $(D+\sqrt{n})\cdot n^{o(1)}$-round distributed construction of a spanning tree of average stretch $n^{o(1)}$. (ii) A $(D+\sqrt{n})\cdot n^{o(1)}$-round distributed construction of an $n^{o(1)}$-congestion approximator consisting of the cuts induced by $O(\log n)$ virtual trees. The distributed representation of the cut approximator allows for evaluation in $(D+\sqrt{n})\cdot n^{o(1)}$ rounds. All our algorithms make use of randomization and succeed with high probability.

We present a method for solving the shortest transshipment problem-also known as uncapacitated minimum cost flow-up to a multiplicative error of 1 + ε in undirected graphs with non-negative integer edge weights using a tailored gradient descent algorithm. Our gradient descent algorithm takes ε-3 polylog n iterations, and in each iteration it needs to solve an instance of the transshipment problem up to a multiplicative error of polylog n, where n is the number of nodes. In particular, this allows us to perform a single iteration by computing a solution on a sparse spanner of logarithmic stretch. Using a careful white-box analysis, we can further extend the method to finding approximate solutions for the single-source shortest paths (SSSP) problem. As a consequence, we improve prior work by obtaining the following results: 1. Broadcast CONGEST model: (1+")-approximate SSSP using Õ((√ n+D) · ε-O(1)) rounds, 1 where D is the (hop) diameter of the network. 2. Broadcast congested clique model: (1+ε)-approximate shortest transshipment and SSSP using Õ (ε-O(1)) rounds. 3. Multipass streaming model: (1+ε)-approximate shortest transshipment and SSSP using Õ (n) space and Õ(ε-O(1)) passes. The previously fastest SSSP algorithms for these models leverage sparse hop sets. We bypass the hop set construction; computing a spanner is sufficient with our method. The above bounds assume non-negative integer edge weights that are polynomially bounded in n; for general nonnegative weights, running times scale with the logarithm of the maximum ratio between non-zero weights. In case of asymmetric costs for traversing an edge in opposite directions, running times scale with the maximum ratio between the costs of both directions over all edges.

Designing interactive technology entails several objectives, one of which is identifying and selecting appropriate functionality. Given candidate functionalities such as “print,” “bookmark,” and “share,” a designer has to choose which functionalities to include and which to leave out. Such choices critically affect the acceptability, productivity, usability, and experience of the design. However, designers may overlook reasonable designs because there is an exponential number of functionality sets and multiple factors to consider. This article is the first to formally define this problem and propose an algorithmic method to support designers to explore alternative functionality sets in early stage design. Based on interviews of professional designers, we mathematically define the task of identifying functionality sets that strike the best balance among four objectives: usefulness, satisfaction, ease of use, and profitability. We develop an integer linear programming solution that can efficiently solve very large instances (set size over 1,300) on a regular computer. Further, we build on techniques of robust optimization to search for diverse and surprising functionality designs. Empirical results from a controlled study and field deployment are encouraging. Most designers rated computationally created sets to be of the comparable or superior quality than their own. Designers reported gaining better understanding of available functionalities and the design space.

We present a novel algorithm for the min-cost flow problem that is competitive with recent third-party implementations of well-known algorithms for this problem and even outperforms them on certain realistic instances. We formally prove correctness of our algorithm and show that the worst-case running time is in O(‖b‖1(m + n log n)) where b is the vector of demands. Combined with standard scaling techniques, this pseudo-polynomial bound can be made polynomial in a straightforward way. Furthermore, we evaluate our approach experimentally. Our empirical findings indeed suggest that the running time does not significantly depend on the costs and that a linear dependence on ‖b‖1 is overly pessimistic.

Given a bipartite graph G, we consider the decision problem called BicliqueCover for a fixed positive integer parameter k where we are asked whether the edges of G can be covered with at most k complete bipartite subgraphs (a.k.a. bicliques). In the BicliquePartition problem, we have the additional constraint that each edge should appear in exactly one of the k bicliques. These problems are both known to be NP-complete but fixed parameter tractable. However, the known FPT algorithms have a running time that is doubly exponential in k, and the best known kernel for both problems is exponential in k. We build on this kernel and improve the running time for BicliquePartition to O*(2^{2k^2+k*log(k)+k}) by exploiting a linear algebraic view on this problem. On the other hand, we show that no such improvement is possible for BicliqueCover unless the Exponential Time Hypothesis (ETH) is false by proving a doubly exponential lower bound on the running time. We achieve this by giving a reduction from 3SAT on n variables to an instance of BicliqueCover with k=O(log(n)). As a further consequence of this reduction, we show that there is no subexponential kernel for BicliqueCover unless P=NP. Finally, we point out the significance of the exponential kernel mentioned above for the design of polynomial-time approximation algorithms for the optimization versions of both problems. That is, we show that it is possible to obtain approximation factors of n/log(n) for both problems, whereas the previous best approximation factor was n/sqrt(log(n)).

Imagine a wooden plate with a set of non-overlapping geometric objects painted on it. How many of them can a carpenter cut out using a panel saw making guillotine cuts, i.e., only moving forward through the material along a straight line until it is split into two pieces? Already fifteen years ago, Pach and Tardos investigated whether one can always cut out a constant fraction if all objects are axis-parallel rectangles. However, even for the case of axis-parallel squares this question is still open. In this paper, we answer the latter affirmatively. Our result is constructive and holds even in a more general setting where the squares have weights and the goal is to save as much weight as possible. We further show that when solving the more general question for rectangles affirmatively with only axis-parallel cuts, this would yield a combinatorial O(1)-approximation algorithm for the Maximum Independent Set of Rectangles problem, and would thus solve a long-standing open problem. In practical applications, like the mentioned carpentry and many other settings, we can usually place the items freely that we want to cut out, which gives rise to the two-dimensional guillotine knapsack problem: Given a collection of axis-parallel rectangles without presumed coordinates, our goal is to place as many of them as possible in a square-shaped knapsack respecting the constraint that the placed objects can be separated by a sequence of guillotine cuts. Our main result for this problem is a quasi-PTAS, assuming the input data to be quasi-polynomially bounded integers. This factor matches the best known (quasi-polynomial time) result for (non-guillotine) two-dimensional knapsack.

We present a near-optimal distributed algorithm for (1+o(1))-approximation of single-commodity maximum flow in undirected weighted networks that runs in (D+ √n)⋅ no(1) communication rounds in the Congest model. Here, n and D denote the number of nodes and the network diameter, respectively. This is the first improvement over the trivial O(m) time bound, and it nearly matches the Ω(D+√n) round complexity lower bound.

Keyboard optimization is concerned with the design of keyboards for different terminals, languages, user groups, and tasks. Previous work in HCI has used random search based methods, such as simulated annealing. These "black box" approaches are convenient, because good solutions are found quickly and no assumption must be made about the objective function. This paper contributes by developing integer programming (IP) as a complementary approach. To this end, we present IP formulations for the letter assignment problem and solve them by branch-and-bound. Although computationally expensive, we show that IP offers two strong benefits. First, its structured non-random search approach improves the out- comes. Second, it guarantees bounds, which increases the designer's confidence over the quality of results. We report improvements to three keyboard optimization cases.

Motivated by an application in image processing, we introduce the grid-leveling problem. It turns out to be the dual of a minimum cost flow problem for an apex graph with a grid graph as its basis. We present an O(n3/2) algorithm for this problem. The optimum solution recovers missing DC coefficients from image and video coding by Discrete Cosine Transform used in popular standards like JPEG and MPEG. Generally, we prove that there is an O(n3/2) min-cost flow algorithm for networks that, after removing one node, are planar, have bounded degrees, and have bounded capacities. The costs may be arbitrary.

A general method for recovering missing DCT coefficients in DCT-transformed images is presented in this work. We model the DCT coefficients recovery problem as an optimization problem and recover all missing DCT coefficients via linear programming. The visual quality of the recovered image gradually decreases as the number of missing DCT coefficients increases. For some images, the quality is surprisingly good even when more than 10 most significant DCT coefficients are missing. When only the DC coefficient is missing, the proposed algorithm outperforms existing methods according to experimental results conducted on 200 test images. The proposed recovery method can be used for cryptanalysis of DCT based selective encryption schemes and other applications.

BACKGROUND: Protein conformation and protein/protein interaction can be elucidated by solution-phase Hydrogen/Deuterium exchange (sHDX) coupled to high-resolution mass analysis of the digested protein or protein complex. In sHDX experiments mutant proteins are compared to wild-type proteins or a ligand is added to the protein and compared to the wild-type protein (or mutant). The number of deuteriums incorporated into the polypeptides generated from the protease digest of the protein is related to the solvent accessibility of amide protons within the original protein construct. RESULTS: In this work, sHDX data was collected on a 14.5 T FT-ICR MS. An algorithm was developed based on combinatorial optimization that predicts deuterium exchange with high spatial resolution based on the sHDX data of overlapping proteolytic fragments. Often the algorithm assigns deuterium exchange with single residue resolution. CONCLUSIONS: With our new method it is possible to automatically determine deuterium exchange with higher spatial resolution than the level of digested fragments.

We consider an optimization problem arising in the design of controllers for OLED displays. Our objective is to minimize amplitude of the electrical current through the diodes which has a direct impact on the lifetime of such a display. Modeling the problem in mathematical terms yields a class of network flow problems where we group the arcs and pay in each group only for the arc carrying the maximum flow. We develop (fully) combinatorial approximation heuristics suitable for being implemented in the hardware of a control device that drives an OLED display.

Consider the following illumination problem: given a stage represented by a line segment L and a set of lightsources represented by a set of points S in the plane, assign powers to the lightsources such that every point on the stage receives a sufficient amount -- let's say one unit -- of light while minimizing the overall power consumption. By assuming that the amount of light arriving from a fixed lightsource decreases rapidly with the distance from the lightsource, this becomes an interesting optimization problem.We propose to reconsider the classical illumination problems as known from computational geometry literature (e.g. [12]) under this light attenuation model. This paper examines the simple problem introduced above and presents different solutions, based on convex optimization, discretization and linear programming, as well as a purely combinatorial approximation algorithm. Some experimental results are also provided.

In an industry project with a German car manufacturer we are faced with the challenge of placing a maximum number of uniform rigid rectangular boxes in the interior of a car trunk. The problem is of practical importance due to a European industry norm which requires car manufacturers to state the trunk volume according to this measure.No really satisfactory automated solution for this problem has been known in the past. In spite of its NP hardness, combinatorial optimization techniques, which consider only grid-aligned placements, produce solutions which are very close to the one achievable by a human expert in several hours of tedious work. The remaining gap is mostly due to the constraints imposed by the chosen grid.In this paper we present a new approach which combines the grid-based combinatorial method with Simulated Annealing on a continuous model. This allows us to explore arbitrary orientations and placements of boxes, hence closing the gap even further, and - in some cases - even surpass the manual expert solution.The implemented software system allows our industrial partner to incorporate the trunk volume in a very early stage of the car design process without relying on a repeated and cumbersome manual evaluation of the volume.