arXiv subjects

Jannik Matuschke

Publications and source records attributed to Jannik Matuschke.

34 records · Page 2Linked to original sources

arXiv2020

Malleable scheduling beyond identical machines

In malleable job scheduling, jobs can be executed simultaneously on multiple machines with the processing time depending on the number of allocated machines. In this setting, jobs are required to be executed non-preemptively and in unison, in the sense that they occupy, during their execution, the same time interval over all the machines of the allocated set. In this work, we study generalizations of malleable job scheduling inspired by standard scheduling on unrelated machines. Specifically, we introduce a general model of malleable job scheduling, where each machine has a (possibly different) speed for each job, and the processing time of a job $j$ on a set of allocated machines $S$ depends on the total speed of $S$ with respect to $j$. For machines with unrelated speeds, we show that the optimal makespan cannot be approximated within a factor less than $\frac{e}{e-1}$, unless $P = NP$. On the positive side, we present polynomial-time algorithms with approximation ratios $\frac{2e}{e-1}$ for machines with unrelated speeds, $3$ for machines with uniform speeds, and $7/3$ for restricted assignments on identical machines. Our algorithms are based on deterministic LP rounding. They result in sparse schedules, in the sense that each machine shares at most one job with other machines. We also prove lower bounds on the integrality gap of $1+φ$ for unrelated speeds ($φ$ is the golden ratio) and $2$ for uniform speeds and restricted assignments. To indicate the generality of our approach, we show that it also yields constant factor approximation algorithms for a variant where we determine the effective speed of a set of allocated machines based on the $L_p$ norm of their speeds.

cs.DS↗

arXiv2019

Popular Branchings and Their Dual Certificates

Let $G$ be a digraph where every node has preferences over its incoming edges. The preferences of a node extend naturally to preferences over branchings, i.e., directed forests; a branching $B$ is popular if $B$ does not lose a head-to-head election (where nodes cast votes) against any branching. Such popular branchings have a natural application in liquid democracy. The popular branching problem is to decide if $G$ admits a popular branching or not. We give a characterization of popular branchings in terms of dual certificates and use this characterization to design an efficient combinatorial algorithm for the popular branching problem. When preferences are weak rankings, we use our characterization to formulate the popular branching polytope in the original space and also show that our algorithm can be modified to compute a branching with least unpopularity margin. When preferences are strict rankings, we show that "approximately popular" branchings always exist.

cs.DS↗

arXiv2019

Exact and approximation algorithms for the expanding search problem

Suppose a target is hidden in one of the vertices of an edge-weighted graph according to a known probability distribution. The expanding search problem asks for a search sequence of the vertices so as to minimize the expected time for finding the target, where the time for reaching the next vertex is determined by its distance to the region that was already searched. This problem has numerous applications, such as searching for hidden explosives, mining coal, and disaster relief. In this paper, we develop exact algorithms and heuristics, including a branch-and-cut procedure, a greedy algorithm with a constant-factor approximation guarantee, and a novel local search procedure based on a spanning tree neighborhood. Computational experiments show that our branch-and-cut procedure outperforms all existing methods for general instances and both heuristics compute near-optimal solutions with little computational effort.

cs.DM↗

arXiv2019

Maintaining Perfect Matchings at Low Cost

The min-cost matching problem suffers from being very sensitive to small changes of the input. Even in a simple setting, e.g., when the costs come from the metric on the line, adding two nodes to the input might change the optimal solution completely. On the other hand, one expects that small changes in the input should incur only small changes on the constructed solutions, measured as the number of modified edges. We introduce a two-stage model where we study the trade-off between quality and robustness of solutions. In the first stage we are given a set of nodes in a metric space and we must compute a perfect matching. In the second stage $2k$ new nodes appear and we must adapt the solution to a perfect matching for the new instance. We say that an algorithm is $(α,β)$-robust if the solutions constructed in both stages are $α$-approximate with respect to min-cost perfect matchings, and if the number of edges deleted from the first stage matching is at most $βk$. Hence, $α$ measures the quality of the algorithm and $β$ its robustness. In this setting we aim to balance both measures by deriving algorithms for constant $α$ and $β$. We show that there exists an algorithm that is $(3,1)$-robust for any metric if one knows the number $2k$ of arriving nodes in advance. For the case that $k$ is unknown the situation is significantly more involved. We study this setting under the metric on the line and devise a $(10,2)$-robust algorithm that constructs a solution with a recursive structure that carefully balances cost and redundancy.

cs.DM↗

arXiv2018

New and simple algorithms for stable flow problems

Stable flows generalize the well-known concept of stable matchings to markets in which transactions may involve several agents, forwarding flow from one to another. An instance of the problem consists of a capacitated directed network, in which vertices express their preferences over their incident edges. A network flow is stable if there is no group of vertices that all could benefit from rerouting the flow along a walk. Fleiner established that a stable flow always exists by reducing it to the stable allocation problem. We present an augmenting-path algorithm for computing a stable flow, the first algorithm that achieves polynomial running time for this problem without using stable allocation as a black-box subroutine. We further consider the problem of finding a stable flow such that the flow value on every edge is within a given interval. For this problem, we present an elegant graph transformation and based on this, we devise a simple and fast algorithm, which also can be used to find a solution to the stable marriage problem with forced and forbidden edges. Finally, we study the stable multicommodity flow model introduced by Király and Pap. The original model is highly involved and allows for commodity-dependent preference lists at the vertices and commodity-specific edge capacities. We present several graph-based reductions that show equivalence to a significantly simpler model. We further show that it is NP-complete to decide whether an integral solution exists.

cs.DM↗

arXiv2018

Electric Vehicle Valet

We propose a novel way to use Electric Vehicles (EVs) as dynamic mobile energy storage with the goal to support grid balancing during peak load times. EVs seeking parking in a busy/expensive inner city area, can get free parking with a valet company in exchange for being utilized for grid support. The valet company would have an agreement with the local utility company to receive varying rewards for discharging EVs at designated times and locations of need (say, where power lines are congested). Given vehicle availabilities, the valet company would compute an optimal schedule of which vehicle to utilize where and when so as to maximize rewards collected. Our contributions are a detailed description of this new concept along with supporting theory to bring it to fruition. On the theory side, we provide new hardness results, as well as efficient algorithms with provable performance guarantees that we also test empirically.

math.OC↗

arXiv2017

The Complexity of Computing a Robust Flow

Robust network flows are a concept for dealing with uncertainty and unforeseen failures in the network infrastructure. They and their dual counterpart, network flow interdiction, have received steady attention within the operations research community over the past years. One of the most basic models is the Maximum Robust Flow Problem: Given a network and an integer k, the task is to find a path flow of maximum robust value, i.e., the guaranteed value of surviving flow after removal of any k arcs in the network. The complexity of this problem appeared to have been settled almost a decade ago: Aneja et al. (2001) showed that the problem can be solved efficiently when k = 1, while an article by Du and Chandrasekaran (2007) established that the problem is NP-hard for any constant value of k larger than 1. We point to a flaw in the proof of the latter result, leaving the complexity for constant k open once again. For the case that k is not bounded by a constant, we present a new hardness proof, establishing that the problem is strongly NP-hard, even when only two different capacity values occur and the number of paths is polynomial in the size of the input. We further show that computing optimal integral solutions is already NP-hard for k = 2 (whereas for k=1, an efficient algorithm is known) and give a positive result for the case that capacities are in {1, 2}.

cs.DM↗

arXiv2017

A Local-Search Algorithm for Steiner Forest

In the Steiner Forest problem, we are given a graph and a collection of source-sink pairs, and the goal is to find a subgraph of minimum total length such that all pairs are connected. The problem is APX-Hard and can be 2-approximated by, e.g., the elegant primal-dual algorithm of Agrawal, Klein, and Ravi from 1995. We give a local-search-based constant-factor approximation for the problem. Local search brings in new techniques to an area that has for long not seen any improvements and might be a step towards a combinatorial algorithm for the more general survivable network design problem. Moreover, local search was an essential tool to tackle the dynamic MST/Steiner Tree problem, whereas dynamic Steiner Forest is still wide open. It is easy to see that any constant factor local search algorithm requires steps that add/drop many edges together. We propose natural local moves which, at each step, either (a) add a shortest path in the current graph and then drop a bunch of inessential edges, or (b) add a set of edges to the current solution. This second type of moves is motivated by the potential function we use to measure progress, combining the cost of the solution with a penalty for each connected component. Our carefully-chosen local moves and potential function work in tandem to eliminate bad local minima that arise when using more traditional local moves.

cs.DS↗

arXiv2017

Robust randomized matchings

The following game is played on a weighted graph: Alice selects a matching $M$ and Bob selects a number $k$. Alice's payoff is the ratio of the weight of the $k$ heaviest edges of $M$ to the maximum weight of a matching of size at most $k$. If $M$ guarantees a payoff of at least $α$ then it is called $α$-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns a $1/\sqrt{2}$-robust matching, which is best possible. We show that Alice can improve her payoff to $1/\ln(4)$ by playing a randomized strategy. This result extends to a very general class of independence systems that includes matroid intersection, b-matchings, and strong 2-exchange systems. It also implies an improved approximation factor for a stochastic optimization variant known as the maximum priority matching problem and translates to an asymptotic robustness guarantee for deterministic matchings, in which Bob can only select numbers larger than a given constant. Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound.

cs.DM↗

arXiv2017

Rerouting flows when links fail

We introduce and investigate reroutable flows, a robust version of network flows in which link failures can be mitigated by rerouting the affected flow. Given a capacitated network, a path flow is reroutable if after failure of an arbitrary arc, we can reroute the interrupted flow from the tail of that arc to the sink, without modifying the flow that is not affected by the failure. Similar types of restoration, which are often termed "local", were previously investigated in the context of network design, such as min-cost capacity planning. In this paper, our interest is in computing maximum flows under this robustness assumption. An important new feature of our model, distinguishing it from existing max robust flow models, is that no flow can get lost in the network. We also study a tightening of reroutable flows, called strictly reroutable flows, making more restrictive assumptions on the capacities available for rerouting. For both variants, we devise a reroutable-flow equivalent of an s-t-cut and show that the corresponding max flow/min cut gap is bounded by 2. It turns out that a strictly reroutable flow of maximum value can be found using a compact LP formulation, whereas the problem of finding a maximum reroutable flow is NP-hard, even when all capacities are in {1, 2}. However, the tightening can be used to get a 2-approximation for reroutable flows. This ratio is tight in general networks, but we show that in the case of unit capacities, every reroutable flow can be transformed into a strictly reroutable flow of same value. While it is NP-hard to compute a maximal integral flow even for unit capacities, we devise a surprisingly simple combinatorial algorithm that finds a half-integral strictly reroutable flow of value 1, or certifies that no such solutions exits. Finally, we also give a hardness result for the case of multiple arc failures.

cs.DM↗

arXiv2016

Matchings with lower quotas: Algorithms and complexity

We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph $G= (A \cup P, E)$ with weights on the edges in $E$, and with lower and upper quotas on the vertices in $P$. We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in $A$ are incident to at most one matching edge, while vertices in $P$ are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classic polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between NP-hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota $u_{max}$ as basis, and we prove that this dependence is necessary unless FPT = W[1]. The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee $u_{\max}+1$, which is asymptotically best possible unless P = NP. Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas.

cs.DM↗

arXiv2016

Protection of flows under targeted attacks

Due to the importance of robustness in many real-world optimization problems, the field of robust optimization has gained a lot of attention over the past decade. We concentrate on maximum flow problems and introduce a novel robust optimization model which, compared to known models from the literature, features several advantageous properties: (i) We consider a general class of path-based flow problems which can be used to model a large variety of network routing problems (and other packing problems). (ii) We aim at solutions that are robust against targeted attacks by a potent adversary who may attack any flow path of his choice on any edge of the network. (iii) In contrast to previous robust maximum flow models, for which no efficient algorithms are known, optimal robust flows for the most important basic variants of our model can be found in polynomial time. We also consider generalizations where the flow player can spend a budget to protect the network against the interdictor. Here, we show that the problem can be solved efficiently when the interdiction costs are determined by the flow player from scratch. However, the problem becomes hard to approximate when the flow player has to improve an initial protection infrastructure.

cs.DM↗

arXiv2014

Fare Evasion in Transit Networks

Public transit systems in urban areas usually require large state subsidies, primarily due to high fare evasion rates. In this paper, we study new models for optimizing fare inspection strategies in transit networks based on bilevel programming. In the first level, the leader (the network operator) determines probabilities for inspecting passengers at different locations, while in the second level, the followers (the fare-evading passengers) respond by optimizing their routes given the inspection probabilities and travel times. To model the followers' behavior we study both a non-adaptive variant, in which passengers select a path a priori and continue along it throughout their journey, and an adaptive variant, in which they gain information along the way and use it to update their route. For these problems, which are interesting in their own right, we design exact and approximation algorithms and we prove a tight bound of 3/4 on the ratio of the optimal cost between adaptive and non-adaptive strategies. For the leader's optimization problem, we study a fixed-fare and a flexible-fare variant, where ticket prices may or may not be set at the operator's will. For the latter variant, we design an LP based approximation algorithm. Finally, using a local search procedure that shifts inspection probabilities within an initially determined support set, we perform an extensive computational study for all variants of the problem on instances of the Dutch railway and the Amsterdam subway network. This study reveals that our solutions are within 95% of theoretical upper bounds drawn from the LP relaxation.

cs.GT↗

arXiv2013

Reference Point Methods and Approximation in Multicriteria Optimization

Operations research applications often pose multicriteria problems. Mathematical research on multicriteria problems predominantly revolves around the set of Pareto optimal solutions, while in practice, methods that output a single solution are more widespread. In real-world multicriteria optimization, reference point methods are widely used and successful examples of such methods. A reference point solution is the solution closest to a given reference point in the objective space. We study the approximation of reference point solutions. In particular, we establish that approximating reference point solutions is polynomially equivalent to approximating the Pareto set. Complementing these results, we show for a number of general algorithmic techniques in single criteria optimization how they can be lifted to reference point optimization. In particular, we lift the link between dynamic programming and FPTAS, as well as oblivious LP-rounding techniques. The latter applies, e.g., to Set Cover and several machine scheduling problems.

cs.DS↗

arXiv2012

Abstract flows over time: A first step towards solving dynamic packing problems

Flows over time generalize classical network flows by introducing a notion of time. Each arc is equipped with a transit time that specifies how long flow takes to traverse it, while flow rates may vary over time within the given edge capacities. In this paper, we extend this concept of a dynamic optimization problem to the more general setting of abstract flows. In this model, the underlying network is replaced by an abstract system of linearly ordered sets, called "paths" satisfying a simple switching property: Whenever two paths P and Q intersect, there must be another path that is contained in the beginning of P and the end of Q. We show that a maximum abstract flow over time can be obtained by solving a weighted abstract flow problem and constructing a temporally repeated flow from its solution. In the course of the proof, we also show that the relatively modest switching property of abstract networks already captures many essential properties of classical networks.

cs.DM↗

arXiv2012

Lattices and maximum flow algorithms in planar graphs

We show that the left/right relation on the set of s-t-paths of a plane graph induces a so-called submodular lattice. If the embedding of the graph is s-t-planar, this lattice is even consecutive. This implies that Ford and Fulkerson's uppermost path algorithm for maximum flow in such graphs is indeed a special case of a two-phase greedy algorithm on lattice polyhedra. We also show that the properties submodularity and consecutivity cannot be achieved simultaneously by any partial order on the paths if the graph is planar but not s-t-planar, thus providing a characterization of this class of graphs.

cs.DM↗