Degree-anonymization using edge rotations

Highlights

• A min number of edge rotations transforming a graph of order n into a k-degree-anonymous.

• NP-hard for $k=n/q$, $q\ge 3$.

• Polynomial-time 2-approximable under some constraints.

• Polynomial-time solvable when $k=n$ and for trees when $k=\theta (n)$.

Abstract

The Min Anonymous-Edge-Rotation problem asks for an input graph G and a positive integer k to find a minimum number of edge rotations that transform G into a graph such that for each vertex there are at least $k-1$ other vertices of the same degree (a k-degree-anonymous graph). In this paper, we prove that the Min Anonymous-Edge-Rotation problem is NP-hard even for $k=n/q$, where n is the order of a graph and q any positive integer, $q\ge 3$. We argue that under some constrains on the number of edges in a graph and k, Min Anonymous-Edge-Rotation is polynomial-time 2-approximable. Furthermore, we show that the problem is solvable in polynomial time for any graph when $k=n$ and for trees when $k=\theta (n)$. Additionally, we establish sufficient conditions for an input graph and k such that a solution for Min Anonymous-Edge-Rotation exists.

Introduction

In recent years huge amounts of personal data has been collected on various networks as e.g. Facebook, Instagram, Twitter or LinkedIn. Ensuring the privacy of network users is one of the main research tasks. One possible model to formalise these issues was introduced by Liu and Terzi [18] who transferred the k-degree-anonymity concept from tabular data in databases [11] to graphs which are often used as a representation of networks. Following this study a graph is called k-degree-anonymous if for its each vertex there are at least $k-1$ other vertices with the same degree. The parameter k represents the number of vertices that are mixed together and thus the increasing value of k increases the level of anonymity. In [21], Wu et al. presented a survey of different anonymization models and some of their weaknesses. Casas Roma et al. [4] proposed a survey of several graph-modification techniques for privacy-preserving on networks. In this paper we consider the k-degree-anonymous concept of Liu and Terzi [18].

The main study problem related to k-degree anonymous graphs is to find a minimum number of graph operations to transform an input graph to a k-degree anonymous graph.

Different graph operations of transforming a graph into a k-degree-anonymous one are considered in research papers where the operations maybe the following: delete vertex/edge, add vertex/edge, or add/delete edge (see more details later). One advantage in the approaches based on vertex/edge deletion/adding is that a solution always exists since in the worst case scenario one can consider the empty or the complete graph that is k-degree-anonymous for any k (at most the number of vertices of the graph). However, the basic graph parameters as the number of vertices and edges could be modified with such transformations.

Vertex/edge modification versions associated to k-degree-anonymity have been relatively well studied. Hartung et al. [15], [16] studied the edge adding modification as proposed by Liu and Terzi [18]. For this type of modification Chester et al. [8] established a polynomial time algorithm for bipartite graphs.

The variant of adding vertices instead of edges was studied by Chester et al. in [7] where they presented an approximation algorithm with an additive error. Bredereck et al. [2] investigated the parameterized complexity of several variants of vertex adding which differ in the way the inserted vertices can be adjacent to existing vertices. Concerning the vertex deletion variant, Bazgan et al. [1] showed the NP-hardness even on very restricted graph classes such as trees, split graphs, or trivially perfect graphs. Moreover, in [1] the vertex and edge deletion variants are proved intractable from the approximability and parameterized complexity point of view.

Several papers study the basic properties of edge rotations, including some bounds for the minimum number of edge rotations between two graphs [5], [6], [10], [13], [17].

In this paper we consider the version of transforming a graph into a k-degree-anonymous one using edge rotations which don't modify the number of vertices/edges. It should be noticed that in such case a solution may not always exist, as we discuss in Section 3.

To the best of our knowledge the problem of transforming a graph to a k-degree anonymous graph using the edge rotations has not been fully explored. In some particular cases some research has been done in [19] where the authors study the edge rotation distance and various metric between the degree sequences to find a “closest” regular graph. In paper [3] the authors proposed an heuristic to compute the edge rotation distance to a k-degree anonymous graph.

Our results. In this paper we study the various aspects of the Min Anonymous-Edge-Rotation problem. An input to the problem is an undirected graph $G=(V,E)$ with n vertices and m edges and an integer $k\le n$. The goal is to find a shortest sequence of edge rotations that transforms G into a k-degree-anonymous graph, if such a sequence exists. We first show that when $\frac{n}{2}\le m\le \frac{n(n-3)}{2}$ and $k\le \frac{n}{4}$ a solution always exists. Moreover for trees a solution exists if and only if $\frac{2m}{n}$ is an integer. We prove that Min Anonymous-Edge-Rotation is NP-hard even when $k=\frac{n}{q}$ and $q\ge 3$ is a fixed positive integer. On the positive side we provide a polynomial-time 2-approximable algorithm under some constraints. Finally, we demonstrate that Min Anonymous-Edge-Rotation is solvable in polynomial time for trees when $k=\theta (n)$ and for any graph when $k=n$.

Our paper is organized as follows. Some preliminaries about edge rotations and our formal definitions are given in Section 2. The study of feasibility is initiated in Section 3. Section 4 presents the NP-hardness proof. In Section 5 we study properties of the specific k-degree anonymous degree sequences that are used in Section 6 to present a polynomial-time 2-approximation algorithm and in Section 7 to establish a polynomial time algorithm for trees. Moreover in Section 7 we consider the case $k=n$ in general graphs. Some conclusions are given at the end of the paper.