Secluded Connectivity Problems

Consider a setting where possibly sensitive information sent over a path in a network is visible to every {neighbor} of the path, i.e., every neighbor of some node on the path, thus including the nodes on the path itself. The exposure of a path $P$ can be measured as the number of nodes adjacent to it, denoted by $N[P]$. A path is said to be secluded if its exposure is small. A similar measure can be applied to other connected subgraphs, such as Steiner trees connecting a given set of terminals. Such subgraphs may be relevant due to considerations of privacy, security or revenue maximization. This paper considers problems related to minimum exposure connectivity structures such as paths and Steiner trees. It is shown that on unweighted undirected $n$-node graphs, the problem of finding the minimum exposure path connecting a given pair of vertices is strongly inapproximable, i.e., hard to approximate within a factor of $O(2^{\log^{1-\epsilon}n})$ for any $\epsilon>0$ (under an appropriate complexity assumption), but is approximable with ratio $\sqrt{\Delta}+3$, where $\Delta$ is the maximum degree in the graph. One of our main results concerns the class of bounded-degree graphs, which is shown to exhibit the following interesting dichotomy. On the one hand, the minimum exposure path problem is NP-hard on node-weighted or directed bounded-degree graphs (even when the maximum degree is 4). On the other hand, we present a polynomial algorithm (based on a nontrivial dynamic program) for the problem on unweighted undirected bounded-degree graphs. Likewise, the problem is shown to be polynomial also for the class of (weighted or unweighted) bounded-treewidth graphs

Approximating Directed Steiner Problems via Tree Embedding

In the k-edge connected directed Steiner tree (k-DST) problem, we are given a directed graph G on n vertices with edge-costs, a root vertex r, a set of h terminals T and an integer k. The goal is to find a min-cost subgraph H of G that connects r to each terminal t by k edge-disjoint r,t-paths. This problem includes as special cases the well-known directed Steiner tree (DST) problem (the case k = 1) and the group Steiner tree (GST) problem. Despite having been studied and mentioned many times in literature, e.g., by Feldman et al. [SODA'09, JCSS'12], by Cheriyan et al. [SODA'12, TALG'14] and by Laekhanukit [SODA'14], there was no known non-trivial approximation algorithm for k-DST for k >= 2 even in the special case that an input graph is directed acyclic and has a constant number of layers. If an input graph is not acyclic, the complexity status of k-DST is not known even for a very strict special case that k= 2 and |T| = 2. In this paper, we make a progress toward developing a non-trivial approximation algorithm for k-DST. We present an O(D k^{D-1} log n)-approximation algorithm for k-DST on directed acyclic graphs (DAGs) with D layers, which can be extended to a special case of k-DST on "general graphs" when an instance has a D-shallow optimal solution, i.e., there exist k edge-disjoint r,t-paths, each of length at most D, for every terminal t. For the case k= 1 (DST), our algorithm yields an approximation ratio of O(D log h), thus implying an O(log^3 h)-approximation algorithm for DST that runs in quasi-polynomial-time (due to the height-reduction of Zelikovsky [Algorithmica'97]). Consequently, as our algorithm works for general graphs, we obtain an O(D k^{D-1} log n)-approximation algorithm for a D-shallow instance of the k-edge-connected directed Steiner subgraph problem, where we wish to connect every pair of terminals by k-edge-disjoint paths

On the Size and the Approximability of Minimum Temporally Connected Subgraphs

We consider temporal graphs with discrete time labels and investigate the size and the approximability of minimum temporally connected spanning subgraphs. We present a family of minimally connected temporal graphs with $n$ vertices and $\Omega(n^2)$ edges, thus resolving an open question of (Kempe, Kleinberg, Kumar, JCSS 64, 2002) about the existence of sparse temporal connectivity certificates. Next, we consider the problem of computing a minimum weight subset of temporal edges that preserve connectivity of a given temporal graph either from a given vertex r (r-MTC problem) or among all vertex pairs (MTC problem). We show that the approximability of r-MTC is closely related to the approximability of Directed Steiner Tree and that r-MTC can be solved in polynomial time if the underlying graph has bounded treewidth. We also show that the best approximation ratio for MTC is at least $O(2^{\log^{1-\epsilon} n})$ and at most $O(\min\{n^{1+\epsilon}, (\Delta M)^{2/3+\epsilon}\})$, for any constant $\epsilon > 0$, where $M$ is the number of temporal edges and $\Delta$ is the maximum degree of the underlying graph. Furthermore, we prove that the unweighted version of MTC is APX-hard and that MTC is efficiently solvable in trees and $2$-approximable in cycles

Approximating Generalized Network Design under (Dis)economies of Scale with Applications to Energy Efficiency

In a generalized network design (GND) problem, a set of resources are assigned to multiple communication requests. Each request contributes its weight to the resources it uses and the total load on a resource is then translated to the cost it incurs via a resource specific cost function. For example, a request may be to establish a virtual circuit, thus contributing to the load on each edge in the circuit. Motivated by energy efficiency applications, recently, there is a growing interest in GND using cost functions that exhibit (dis)economies of scale ((D)oS), namely, cost functions that appear subadditive for small loads and superadditive for larger loads. The current paper advances the existing literature on approximation algorithms for GND problems with (D)oS cost functions in various aspects: (1) we present a generic approximation framework that yields approximation results for a much wider family of requests in both directed and undirected graphs; (2) our framework allows for unrelated weights, thus providing the first non-trivial approximation for the problem of scheduling unrelated parallel machines with (D)oS cost functions; (3) our framework is fully combinatorial and runs in strongly polynomial time; (4) the family of (D)oS cost functions considered in the current paper is more general than the one considered in the existing literature, providing a more accurate abstraction for practical energy conservation scenarios; and (5) we obtain the first approximation ratio for GND with (D)oS cost functions that depends only on the parameters of the resources' technology and does not grow with the number of resources, the number of requests, or their weights. The design of our framework relies heavily on Roughgarden's smoothness toolbox (JACM 2015), thus demonstrating the possible usefulness of this toolbox in the area of approximation algorithms.Comment: 39 pages, 1 figure. An extended abstract of this paper is to appear in the 50th Annual ACM Symposium on the Theory of Computing (STOC 2018

Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

The Directed Steiner Network (DSN) problem takes as input a directed edge-weighted graph $G=(V,E)$ and a set $\mathcal{D}\subseteq V\times V$ of $k$ demand pairs. The aim is to compute the cheapest network $N\subseteq G$ for which there is an $s\to t$ path for each $(s,t)\in\mathcal{D}$. It is known that this problem is notoriously hard as there is no $k^{1/4-o(1)}$-approximation algorithm under Gap-ETH, even when parametrizing the runtime by $k$ [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter $k$. For the bi-DSN$_\text{Planar}$ problem, the aim is to compute a planar optimum solution $N\subseteq G$ in a bidirected graph $G$, i.e., for every edge $uv$ of $G$ the reverse edge $vu$ exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for $k$. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) it is unlikely that a PAS exists for any generalization of bi-DSN$_\text{Planar}$, unless FPT=W[1]. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network $N\subseteq G$ needs to strongly connect a given set of $k$ terminals. It has been observed before that for SCSS a parameterized $2$-approximation exists when parameterized by $k$ [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for $k$ no parameterized $(2-\varepsilon)$-approximation algorithm exists under Gap-ETH. Additionally we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for $k$

Networking - A Statistical Physics Perspective

Efficient networking has a substantial economic and societal impact in a broad range of areas including transportation systems, wired and wireless communications and a range of Internet applications. As transportation and communication networks become increasingly more complex, the ever increasing demand for congestion control, higher traffic capacity, quality of service, robustness and reduced energy consumption require new tools and methods to meet these conflicting requirements. The new methodology should serve for gaining better understanding of the properties of networking systems at the macroscopic level, as well as for the development of new principled optimization and management algorithms at the microscopic level. Methods of statistical physics seem best placed to provide new approaches as they have been developed specifically to deal with non-linear large scale systems. This paper aims at presenting an overview of tools and methods that have been developed within the statistical physics community and that can be readily applied to address the emerging problems in networking. These include diffusion processes, methods from disordered systems and polymer physics, probabilistic inference, which have direct relevance to network routing, file and frequency distribution, the exploration of network structures and vulnerability, and various other practical networking applications.Comment: (Review article) 71 pages, 14 figure