Single machine scheduling with controllable processing times by submodular optimization

In scheduling with controllable processing times the actual processing time of each job is to be chosen from the interval between the smallest (compressed or fully crashed) value and the largest (decompressed or uncrashed) value. In the problems under consideration, the jobs are processed on a single machine and the quality of a schedule is measured by two functions: the maximum cost (that depends on job completion times) and the total compression cost. Our main model is bicriteria and is related to determining an optimal trade-off between these two objectives. Additionally, we consider a pair of associated single criterion problems, in which one of the objective functions is bounded while the other one is to be minimized. We reduce the bicriteria problem to a series of parametric linear programs defined over the intersection of a submodular polyhedron with a box. We demonstrate that the feasible region is represented by a so-called base polyhedron and the corresponding problem can be solved by the greedy algorithm that runs two orders of magnitude faster than known previously. For each of the associated single criterion problems, we develop algorithms that deliver the optimum faster than it can be deduced from a solution to the bicriteria problem

Maximum st-flow in directed planar graphs via shortest paths

Minimum cuts have been closely related to shortest paths in planar graphs via planar duality - so long as the graphs are undirected. Even maximum flows are closely related to shortest paths for the same reason - so long as the source and the sink are on a common face. In this paper, we give a correspondence between maximum flows and shortest paths via duality in directed planar graphs with no constraints on the source and sink. We believe this a promising avenue for developing algorithms that are more practical than the current asymptotically best algorithms for maximum st-flow.Comment: 20 pages, 4 figures. Short version to be published in proceedings of IWOCA'1

On the Complexity of Local Search for Weighted Standard Set Problems

In this paper, we study the complexity of computing locally optimal solutions for weighted versions of standard set problems such as SetCover, SetPacking, and many more. For our investigation, we use the framework of PLS, as defined in Johnson et al., [JPY88]. We show that for most of these problems, computing a locally optimal solution is already PLS-complete for a simple neighborhood of size one. For the local search versions of weighted SetPacking and SetCover, we derive tight bounds for a simple neighborhood of size two. To the best of our knowledge, these are one of the very few PLS results about local search for weighted standard set problems

Capacitated max-Batching with Interval Graph Compatibilities

We consider the problem of partitioning interval graphs into cliques of bounded size. Each interval has a weight, and the cost of a clique is the maximum weight of any interval in the clique. This natural graph problem can be interpreted as a batch scheduling problem. Solving an open question from [7, 4, 5], we show NP-hardness, even if the bound on the clique sizes is constant. Moreover, we give a PTAS based on a novel dynamic programming technique for this case.

Approximation Algorithms for the Max-Buying Problem with Limited Supply

We consider the Max-Buying Problem with Limited Supply, in which there are $n$ items, with $C_i$ copies of each item $i$, and $m$ bidders such that every bidder $b$ has valuation $v_{ib}$ for item $i$. The goal is to find a pricing $p$ and an allocation of items to bidders that maximizes the profit, where every item is allocated to at most $C_i$ bidders, every bidder receives at most one item and if a bidder $b$ receives item $i$ then $p_i \leq v_{ib}$. Briest and Krysta presented a 2-approximation for this problem and Aggarwal et al. presented a 4-approximation for the Price Ladder variant where the pricing must be non-increasing (that is, $p_1 \geq p_2 \geq \cdots \geq p_n$). We present an $e/(e-1)$-approximation for the Max-Buying Problem with Limited Supply and, for every $\varepsilon > 0$, a $(2+\varepsilon)$-approximation for the Price Ladder variant

Total variation denoising in l1l^1 anisotropy

We aim at constructing solutions to the minimizing problem for the variant of Rudin-Osher-Fatemi denoising model with rectilinear anisotropy and to the gradient flow of its underlying anisotropic total variation functional. We consider a naturally defined class of functions piecewise constant on rectangles (PCR). This class forms a strictly dense subset of the space of functions of bounded variation with an anisotropic norm. The main result shows that if the given noisy image is a PCR function, then solutions to both considered problems also have this property. For PCR data the problem of finding the solution is reduced to a finite algorithm. We discuss some implications of this result, for instance we use it to prove that continuity is preserved by both considered problems.Comment: 34 pages, 9 figure

Packing While Traveling: Mixed Integer Programming for a Class of Nonlinear Knapsack Problems

Packing and vehicle routing problems play an important role in the area of supply chain management. In this paper, we introduce a non-linear knapsack problem that occurs when packing items along a fixed route and taking into account travel time. We investigate constrained and unconstrained versions of the problem and show that both are NP-hard. In order to solve the problems, we provide a pre-processing scheme as well as exact and approximate mixed integer programming (MIP) solutions. Our experimental results show the effectiveness of the MIP solutions and in particular point out that the approximate MIP approach often leads to near optimal results within far less computation time than the exact approach

Bright and Fast Multicoloured Voltage Reporters via Electrochromic FRET

Genetically encoded fluorescent reporters of membrane potential promise to reveal aspects of neural function not detectable by other means. We present a palette of multicoloured brightly fluorescent genetically encoded voltage indicators with sensitivities from 8–13% ΔF/F per 100 mV, and half-maximal response times from 4–7 ms. A fluorescent protein is fused to an archaerhodopsin-derived voltage sensor. Voltage-induced shifts in the absorption spectrum of the rhodopsin lead to voltage-dependent nonradiative quenching of the appended fluorescent protein. Through a library screen, we identify linkers and fluorescent protein combinations that report neuronal action potentials in cultured rat hippocampal neurons with a single-trial signal-to-noise ratio from 7 to 9 in a 1 kHz imaging bandwidth at modest illumination intensity. The freedom to choose a voltage indicator from an array of colours facilitates multicolour voltage imaging, as well as combination with other optical reporters and optogenetic actuators.Chemistry and Chemical BiologyEngineering and Applied SciencesPhysic