Vassilis Zissimopoulos

A simulated annealing approach for the circular cutting problem

We propose a heuristic for the constrained and the unconstrained circular cutting problem based upon simulated annealing. We define an energy function, the small values of which provide a good concentration of the circular pieces on the left bottom corner of the initial rectangle. Such values of the energy correspond to configurations where pieces are placed in the rectangle without overlapping. Appropriate software has been devised and computational results and comparisons with some other algorithms are also provided and discussed.

A constant approximation algorithm for the densest k-subgraph problem on chordal graphs

The densest k-subgraph (DkS) problem asks for a k-vertex subgraph of a given graph with the maximum number of edges. The DkS problem is NP-hard even for special graph classes including bipartite, planar, comparability and chordal graphs, while no constant approximation algorithm is known for any of these classes. In this paper we present a 3-approximation algorithm for the class of chordal graphs. The analysis of our algorithm is based on a graph theoretic lemma of independent interest.

On the social cost of distributed selfish content replication

We study distributed content replication networks formed voluntarily by selfish autonomous users, seeking access to information objects that originate from distant servers. Each user caters to minimization of its individual access cost by replicating locally (up to constrained storage capacity) a subset of objects, and accessing the rest from the nearest possible location. We show existence of stable networks by proving existence of pure strategy Nash equilibria for a game-theoretic formulation of this situation. Social (overall) cost of stable networks is measured by the average or by the maximum access cost experienced by any user. We study socially most and least expensive stable networks by means of tight bounds on the ratios of the Price of Anarchy and Stability respectively. Although in the worst case the ratios may coincide, we identify cases where they differ significantly. We comment on simulations exhibiting occurence of cost-efficient stable networks on average.

The densest k-subgraph problem on clique graphs

Abstract The Densest k-Subgraph (DkS) problem asks for a k-vertex subgraph of a given graph with the maximum number of edges. The problem is strongly NP-hard, as a generalization of the well known Clique problem and we also know that it does not admit a Polynomial Time Approximation Scheme (PTAS). In this paper we focus on special cases of the problem, with respect to the class of the input graph. Especially, towards the elucidation of the open questions concerning the complexity of the problem for interval graphs as well as its approximability for chordal graphs, we consider graphs having special clique graphs. We present a PTAS for stars of cliques and a dynamic programming algorithm for trees of cliques.

Steiner forests on stochastic metric graphs

We consider the problem of connecting given vertex pairs over a stochastic metric graph, each vertex of which has a probability of presence independently of all other vertices. Vertex pairs requiring connection are always present with probability 1. Our objective is to satisfy the connectivity requirements for every possibly materializable subgraph of the given metric graph, so as to optimize the expected total cost of edges used. This is a natural problem model for cost-efficient Steiner Forests on stochastic metric graphs, where uncertain availability of intermediate nodes requires fast adjustments of traffic forwarding. For this problem we allow a priori design decisions to be taken, that can be modified efficiently when an actual subgraph of the input graph materializes. We design a fast (almost linear time in the number of vertices) modification algorithm whose outcome we analyze probabilistically, and show that depending on the a priori decisions this algorithm yields 2 or 4 approximation factors of the optimum expected cost. We also show that our analysis of the algorithm is tight.

Absolute o(log m ) error in approximating random set covering: an average case analysis

This work concerns average case analysis of simple solutions for random set covering (SC) instances. Simple solutions are constructed via an O(nm) algorithm. At first an analytical upper bound on the expected solution size is provided. The bound in combination with previous results yields an absolute asymptotic approximation result of o(log m) order. An upper bound on the variance of simple solution values is calculated. Sensitivity analysis performed on simple solutions for random SC instances shows that they are highly robust, in the sense of maintaining their feasibility against augmentation of the input data with additional random constraints.

Measures of Intrinsic Hardness for Constraint Satisfaction Problem Instances

Our aim is to investigate the factors which determine the intrinsic hardness of constructing a solution to any particular constraint satisfaction problem instance, regardless of the algorithm employed. The line of reasoning is roughly the following: There exists a set of distinct, possibly overlapping, trajectories through the states of the search space, which start at the unique initial state and terminate at complete feasible assignments. These trajectories are named solution paths. The entropy of the distribution of solution paths among the states of each level of the search space provides a measure of the amount of choice available for selecting a solution path at that level. This measure of choice is named solution path diversity. Intrinsic instance hardness is identified with the deficit in solution path diversity and is shown to be linked to the distribution of instance solutions as well as constrainedness, an established hardness measure.

Optimal data placement on networks with a constant number of clients

We introduce optimal algorithms for the problems of data placement (DP) and page placement (PP) in networks with a constant number of clients each of which has limited storage availability and issues requests for data objects. The objective for both problems is to efficiently utilize each client’s storage (deciding where to place replicas of objects) so that the total incurred access and installation cost over all clients are minimized. In the PP problem an extra constraint on the maximum number of clients served by a single client must be satisfied. Our algorithms solve both problems optimally when all objects have uniform lengths. When object lengths are non-uniform we also find the optimal solution, albeit a small, asymptotically tight violation of each client’s storage size by elmax where lmax is the maximum length of the objects and e some arbitrarily small positive constant. We make no assumption on the underlying topology of the network (metric, ultrametric, etc.), thus obtaining the first non-trivial results for non-metric data placement problems.

An approximation scheme for scheduling independent jobs into subcubes of a hypercube of fixed dimension

Abstract We study the problem of scheduling independent jobs in a hypercube where jobs are executed in subcubes of various dimensions. The problem being NP-complete, several approximation algorithms based on list scheduling have been proposed, having approximation ratio of order of 2. In this paper, a linear time e-approximation algorithm for the problem is provided when the size of the hypercube is fixed. We use a reduction to a special strip-packing (or two-dimensional packing) problem with bounded number of distinct pieces. Then, we transform the strip-packing solution into a feasible one for the initial scheduling problem with a small loss in performance. Finally, we provide an improvement which leads to significant reduction of the size of the strip-packing problem.

On the Complexity of Minimizing the Total Calibration Cost

Bender et al. (SPAA 2013) proposed a theoretical framework for testing in contexts where safety mistakes must be avoided. Testing in such a context is made by machines that need to be often calibrated. Since calibrations have non negligible cost, it is important to study policies minimizing the calibration cost while performing all the necessary tests. We focus on the single-machine setting and we study the complexity status of different variants of the problem. First, we extend the model by considering that the jobs have arbitrary processing times and that the preemption of jobs is allowed. For this case, we propose an optimal polynomial time algorithm. Then, we study the case where there is many types of calibrations with different lengths and costs. We prove that the problem becomes NP-hard for arbitrary processing times even when the preemption of the jobs is allowed. Finally, we focus on the case of unit-time jobs and we show that a more general problem, where the recalibration of the machine is not instantaneous, can be solved in polynomial time.