David Adjiashvili

Minimizing the number of switch instances on a flexible machine in polynomial time

We revisit the tool switching problem on a flexible manufacturing machine. We present a polynomial algorithm for the problem of finding a switching plan that minimizes the number of tool switch instances on the machine, given a fixed job sequence. We prove tight hardness results for the variable sequence case with the same objective function, as well as a new objective function naturally arising in multi-feeder mailroom inserting systems.

Bulk-Robust combinatorial optimization

We commence an algorithmic study of Bulk-Robustness, a new model of robustness in combinatorial optimization. Unlike most existing models, Bulk-Robust combinatorial optimization features a highly nonuniform failure model. Instead of an interdiction budget, Bulk-Robust counterparts provide an explicit list of interdiction sets, comprising the admissible set of scenarios, thus allowing to model correlations between failures of different components in the system, interdiction sets of variable cardinality and more. The resulting model is suitable for capturing failures of complex structures in the system. We provide complexity results and approximation algorithms for Bulk-Robust counterparts of the Minimum Matroid Basis problems and the Shortest Path problem. Our results rely on various techniques, and outline the rich and heterogeneous combinatorial structure of Bulk-Robust optimization.

Labeling Schemes for Bounded Degree Graphs

We investigate adjacency labeling schemes for graphs of bounded degree Δ = O(1). In particular, we present an optimal (up to an additive constant) logn + O(1) adjacency labeling scheme for bounded degree trees. The latter scheme is derived from a labeling scheme for bounded degree outerplanar graphs. Our results complement a similar bound recently obtained for bounded depth trees [Fraigniaud and Korman, SODA 2010], and may provide new insights for closing the long standing gap for adjacency in trees [Alstrup and Rauhe, FOCS 2002]. We also provide improved labeling schemes for bounded degree planar graphs. Finally, we use combinatorial number systems and present an improved adjacency labeling schemes for graphs of bounded degree Δ with \((e+1)\sqrt{n} < \Delta \leq n/5\).

Firefighting on trees beyond integrality gaps

The Firefighter problem and a variant of it, known as Resource Minimization for Fire Containment (RMFC), are natural models for optimal inhibition of harmful spreading processes. Despite considerable progress on several fronts, the approximability of these problems is still badly understood. This is the case even when the underlying graph is a tree, which is one of the most- studied graph structures in this context and the focus of this paper. In their simplest version, a fire spreads from one fixed vertex step by step from burning to adjacent non-burning vertices, and at each time step B many non-burning vertices can be protected from catching fire. The Firefighter problem asks, for a given B, to maximize the number of vertices that will not catch fire, whereas RMFC (on a tree) asks to find the smallest B that allows for saving all leaves of the tree. Prior to this work, the best known approximation ratios were an O(1)-approximation for the Firefighter problem and an O(log* n)-approximation for RMFC, both being LP-based and essentially matching the integrality gaps of two natural LP relaxations. We improve on both approximations by presenting a PTAS for the Firefighter problem and an O(1)- approximation for RMFC, both qualitatively matching the known hardness results. Our results are obtained through a combination of the known LPs with several new techniques, which allow for efficiently enumerating over super-constant size sets of constraints to strengthen the natural LPs.

Exact and Approximation Algorithms for Optimal Equipment Selection in Deploying In-Building Distributed Antenna Systems

We consider a combinatorial optimization problem in passive In-Building Distributed Antenna Systems (IB-DAS) deployment for indoor mobile broadband service. These systems have a tree topology, in which a central base station is connected to a number of antennas located at tree leaves via cables represented by the tree edges. Each inner node corresponds to a power equipment, of which the available types differ in the number of output ports and/or by power gain at the ports. This paper focuses on the equipment selection problem that amounts to, for a given passive DAS tree topology, selecting a power equipment type for each inner node and assigning the outgoing edges of the node to the equipment ports. The performance metric is the power deviation at the antennas from the target values. We consider as objective function the minimization of either the total or the largest power deviation over all antennas. Our contributions are the development of exact pseudo-polynomial time algorithms and (additive) fully-polynomial time approximation schemes for both objectives. Numerical results are provided to illustrate the algorithms. We also extend some results to account for equipment cost.

Robust Assignments via Ear Decompositions and Randomized Rounding.

Many real-life planning problems require making a priori decisions before all parameters of the problem have been revealed. An important special case of such problem arises in scheduling problems, where a set of tasks needs to be assigned to the available set of machines or personnel (resources), in a way that all tasks have assigned resources, and no two tasks share the same resource. In its nominal form, the resulting computational problem becomes the \emph{assignment problem} on general bipartite graphs. This paper deals with a robust variant of the assignment problem modeling situations where certain edges in the corresponding graph are \emph{vulnerable} and may become unavailable after a solution has been chosen. The goal is to choose a minimum-cost collection of edges such that if any vulnerable edge becomes unavailable, the remaining part of the solution contains an assignment of all tasks. We present approximation results and hardness proofs for this type of problems, and establish several connections to well-known concepts from matching theory, robust optimization and LP-based techniques.

A Polyhedral Frobenius Theorem with Applications to Integer Optimization

We prove a representation theorem of projections of sets of integer points by an integer matrix

Non-Uniform Robust Network Design in Planar Graphs.

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Time-Expanded Packings

. Our result can be seen as a polyhedral analogue of several classical and recent results related to the Frobenius problem. Our result is motivated by a large class of nonlinear integer optimization problems in variable dimension. Concretely, we aim to optimize

Removing redundant quadratic constraints

f(Wx)