An overview on polynomial approximation of NP-hard problems

The fact that polynomial time algorithm is very unlikely to be devised for an optimal solving of the NP-hard problems strongly motivates both the researchers and the practitioners to try to solve such problems heuristically, by making a trade-off between computational time and solution's quality. In other words, heuristic computation consists of trying to find not the best solution but one solution which is 'close to' the optimal one in reasonable time. Among the classes of heuristic methods for NP-hard problems, the polynomial approximation algorithms aim at solving a given NP-hard problem in poly-nomial time by computing feasible solutions that are, under some predefined criterion, as near to the optimal ones as possible. The polynomial approximation theory deals with the study of such algorithms. This survey first presents and analyzes time approximation algorithms for some classical examples of NP-hard problems. Secondly, it shows how classical notions and tools of complexity theory, such as polynomial reductions, can be matched with polynomial approximation in order to devise structural results for NP-hard optimization problems. Finally, it presents a quick description of what is commonly called inapproximability results. Such results provide limits on the approximability of the problems tackled

Online Admission Control and Embedding of Service Chains

The virtualization and softwarization of modern computer networks enables the definition and fast deployment of novel network services called service chains: sequences of virtualized network functions (e.g., firewalls, caches, traffic optimizers) through which traffic is routed between source and destination. This paper attends to the problem of admitting and embedding a maximum number of service chains, i.e., a maximum number of source-destination pairs which are routed via a sequence of to-be-allocated, capacitated network functions. We consider an Online variant of this maximum Service Chain Embedding Problem, short OSCEP, where requests arrive over time, in a worst-case manner. Our main contribution is a deterministic O(log L)-competitive online algorithm, under the assumption that capacities are at least logarithmic in L. We show that this is asymptotically optimal within the class of deterministic and randomized online algorithms. We also explore lower bounds for offline approximation algorithms, and prove that the offline problem is APX-hard for unit capacities and small L > 2, and even Poly-APX-hard in general, when there is no bound on L. These approximation lower bounds may be of independent interest, as they also extend to other problems such as Virtual Circuit Routing. Finally, we present an exact algorithm based on 0-1 programming, implying that the general offline SCEP is in NP and by the above hardness results it is NP-complete for constant L.Comment: early version of SIROCCO 2015 pape

Parameterized (in)approximability of subset problems

We discuss approximability and inapproximability in FPT-time for a large class of subset problems where a feasible solution $S$ is a subset of the input data and the value of $S$ is $|S|$. The class handled encompasses many well-known graph, set, or satisfiability problems such as Dominating Set, Vertex Cover, Set Cover, Independent Set, Feedback Vertex Set, etc. In a first time, we introduce the notion of intersective approximability that generalizes the one of safe approximability and show strong parameterized inapproximability results for many of the subset problems handled. Then, we study approximability of these problems with respect to the dual parameter $n-k$ where $n$ is the size of the instance and $k$ the standard parameter. More precisely, we show that under such a parameterization, many of these problems, while W[$\cdot$]-hard, admit parameterized approximation schemata.Comment: 7 page

Provenance Views for Module Privacy

Scientific workflow systems increasingly store provenance information about the module executions used to produce a data item, as well as the parameter settings and intermediate data items passed between module executions. However, authors/owners of workflows may wish to keep some of this information confidential. In particular, a module may be proprietary, and users should not be able to infer its behavior by seeing mappings between all data inputs and outputs. The problem we address in this paper is the following: Given a workflow, abstractly modeled by a relation R, a privacy requirement \Gamma and costs associated with data. The owner of the workflow decides which data (attributes) to hide, and provides the user with a view R' which is the projection of R over attributes which have not been hidden. The goal is to minimize the cost of hidden data while guaranteeing that individual modules are \Gamma -private. We call this the "secureview" problem. We formally define the problem, study its complexity, and offer algorithmic solutions

The traveling salesman problem for lines, balls and planes

We revisit the traveling salesman problem with neighborhoods (TSPN) and propose several new approximation algorithms. These constitute either first approximations (for hyperplanes, lines, and balls in $\mathbb{R}^d$, for $d\geq 3$) or improvements over previous approximations achievable in comparable times (for unit disks in the plane). \smallskip (I) Given a set of $n$ hyperplanes in $\mathbb{R}^d$, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in $O(n)$ time, when $d$ is constant. \smallskip (II) Given a set of $n$ lines in $\mathbb{R}^d$, a TSP tour whose length is at most $O(\log^3 n)$ times the optimal can be computed in polynomial time for all $d$. \smallskip (III) Given a set of $n$ unit balls in $\mathbb{R}^d$, a TSP tour whose length is at most $O(1)$ times the optimal can be computed in polynomial time, when $d$ is constant.Comment: 30 pages, 9 figures; final version to appear in ACM Transactions on Algorithm

A note on the independence number, domination number and related parameters of random binary search trees and random recursive trees

We identify the mean growth of the independence number of random binary search trees and random recursive trees and show normal fluctuations around their means. Similarly we also show normal limit laws for the domination number and variations of it for these two cases of random tree models. Our results are an application of a recent general theorem of Holmgren and Janson on fringe trees in these two random tree models

Networked Slepian-Wolf: theory, algorithms, and scaling laws

Consider a set of correlated sources located at the nodes of a network, and a set of sinks that are the destinations for some of the sources. The minimization of cost functions which are the product of a function of the rate and a function of the path weight is considered, for both the data-gathering scenario, which is relevant in sensor networks, and general traffic matrices, relevant for general networks. The minimization is achieved by jointly optimizing a) the transmission structure, which is shown to consist in general of a superposition of trees, and b) the rate allocation across the source nodes, which is done by Slepian-Wolf coding. The overall minimization can be achieved in two concatenated steps. First, the optimal transmission structure is found, which in general amounts to finding a Steiner tree, and second, the optimal rate allocation is obtained by solving an optimization problem with cost weights determined by the given optimal transmission structure, and with linear constraints given by the Slepian-Wolf rate region. For the case of data gathering, the optimal transmission structure is fully characterized and a closed-form solution for the optimal rate allocation is provided. For the general case of an arbitrary traffic matrix, the problem of finding the optimal transmission structure is NP-complete. For large networks, in some simplified scenarios, the total costs associated with Slepian-Wolf coding and explicit communication (conditional encoding based on explicitly communicated side information) are compared. Finally, the design of decentralized algorithms for the optimal rate allocation is analyzed