Anonymizing binary and small tables is hard to approximate

The problem of publishing personal data without giving up privacy is becoming increasingly important. An interesting formalization recently proposed is the k-anonymity. This approach requires that the rows in a table are clustered in sets of size at least k and that all the rows in a cluster become the same tuple, after the suppression of some records. The natural optimization problem, where the goal is to minimize the number of suppressed entries, is known to be NP-hard when the values are over a ternary alphabet, k=3 and the rows length is unbounded. In this paper we give a lower bound on the approximation factor that any polynomial-time algorithm can achieve on two restrictions of the problem, namely (i) when the records values are over a binary alphabet and k=3, and (ii) when the records have length at most 8 and k=4, showing that these restrictions of the problem are APX-hard.     

Fixed-parameter tractability of anonymizing data by suppressing entries

A popular model for protecting privacy when person-specific data is released is k -anonymity. A dataset is k-anonymous if each record is identical to at least (k−1) other records in the dataset. The basic k-anonymization problem, which minimizes the number of dataset entries that must be suppressed to achieve k-anonymity, is NP-hard and hence not solvable both quickly and optimally in general. We apply parameterized complexity analysis to explore algorithmic options for restricted versions of this problem that occur in practice. We present the first fixed-parameter algorithms for this problem and identify key techniques that can be applied to this and other k-anonymization problems.     

Possible winner problems on partial tournaments: a parameterized study

We study possible winner problems related to the uncovered set and the Banks set on partial tournaments from the viewpoint of parameterized complexity. We first study a problem where given a partial tournament D and a subset X of vertices, we are asked to add some arcs to D such that all vertices in X are included in the uncovered set. We focus on two parameterizations: parameterized by |X| and parameterized by the number of arcs to be added. In addition, we study a parameterized variant of the problem which is to determine whether all vertices of X can be included in the uncovered set by reversing at most k arcs. Finally, we study some parameterizations of a possible winner problem on partial tournaments, where we are given a partial tournament D and a distinguished vertex p, and asked whether D has a maximal transitive subtournament with p being the 0-indegree vertex. These parameterized problems are related to the Banks set. We achieve \(\mathcal {XP}\) results, \(\mathcal {W}\)-hardness results as well as \(\mathcal {FPT}\) results along with a kernelization lower bound for the problems studied in this paper.     

Minimum common string partition revisited

Minimum Common String Partition (MCSP) has drawn much attention due to its application in genome rearrangement. In this paper, we investigate three variants of MCSP: MCSP ^c , which requires that there are at most c elements in the alphabet; d-MCSP, which requires the occurrence of each element to be bounded by d; and x-balanced MCSP, which requires the length of blocks being in range (n/k?x,n/k+x), where n is the length of the input strings, k is the number of blocks in the optimal common partition and x is a constant integer. We show that MCSP ^c is NP-hard when c≥2. As for d-MCSP, we present an FPT algorithm which runs in O ^?((d!)^2k ) time. As it is still unknown whether an FPT algorithm only parameterized on k exists for the general case of MCSP, we also devise an FPT algorithm for the special case x-balanced MCSP parameterized on both k and x.     

Optimal key tree structure for two-user replacement and deletion problems

We study the optimal tree structure for the key management problem. In the key tree, when two or more leaves are deleted or replaced, the updating cost is defined to be the number of encryptions needed to securely update the remaining keys. The objective is to find the optimal tree structure where the worst case updating cost is minimum. We extend the result of 1-replacement problem in Snoeyink et al. (Proceedings of the twentieth annual IEEE conference on computer communications, pp. 422–431, 2001) to the k-user case. We first show that the k-deletion problem can be solved by handling k-replacement problem. Focusing on the k-replacement problem, we show that the optimal tree can be computed in $O(n^{(k+1)^{2}})$ time given a constant k. Then we derive a tight degree bound for optimal tree when replacing two users. By investigating the maximum number of leaves that can be placed on the tree given a fixed worst case replacement cost, we prove that a balanced tree with certain root degree 5≤d≤7 where the number of leaves in the subtrees differs by at most one and each subtree is a 2-3 tree can always achieve the minimum worst case 2-replacement cost. Thus the optimal tree for two-user replacement problem can be efficiently constructed in O(n) time.     

On nonlinear multi-covering problems

In this paper we define the exact k-coverage problem, and study it for the special cases of intervals and circular-arcs. Given a set system consisting of a ground set of n points with integer demands \(\{d_0,\dots ,d_{n-1}\}\) and integer rewards, subsets of points, and an integer k, select up to k subsets such that the sum of rewards of the covered points is maximized, where point i is covered if exactly \(d_i\) subsets containing it are selected. Here we study this problem and some related optimization problems. We prove that the exact k-coverage problem with unbounded demands is NP-hard even for intervals on the real line and unit rewards. Our NP-hardness proof uses instances where some of the natural parameters of the problem are unbounded (each of these parameters is linear in the number of points). We show that this property is essential, as if we restrict (at least) one of these parameters to be a constant, then the problem is polynomial time solvable. Our polynomial time algorithms are given for various generalizations of the problem (in the setting where one of the parameters is a constant).     

Parameterized lower bound and inapproximability of polylogarithmic string barcoding

String barcoding is a method that can identify microorganisms by analyzing their genome sequences. In this paper, we study the polylogarithmic string barcoding problem, where the lengths of the substrings in the testing set are polylogarithmically bounded. In particular, we show that the polylogarithmic string barcoding problem remains NP-hard and moreover, for a problem instance with n sequences, it is NP-hard to achieve an approximate ratio within dln n in polynomial time, where d is some constant. We then consider the parameterized polylogarithmic string barcoding problem, where the number of substrings in the test set is considered to be a fixed parameter k. We show that, unless W[2]=FPT, there does not exist a 2 ^O(k) n ^c algorithm that can decide whether a test set of size k exists or not, where c is a constant independent of n and k.     

Exact combinatorial algorithms and experiments for?finding maximum k-plexes

We propose new practical algorithms to find maximum-cardinality k-plexes in graphs. A k-plex denotes a vertex subset in a graph inducing a subgraph where every vertex has edges to all but at most k vertices in the k-plex. Cliques are 1-plexes. In analogy to the special case of finding maximum-cardinality cliques, finding maximum-cardinality k-plexes is NP-hard. Complementing previous work, we develop exact combinatorial algorithms, which are strongly based on methods from parameterized algorithmics. The experiments with our freely available implementation indicate the competitiveness of our approach, for many real-world graphs outperforming the previously used methods.     

Generalized perfect domination in graphs

Let k be a positive integer and G=(V,E) be a graph. A vertex subset D of a graph G is called a perfect k-dominating set of G, if every vertex v of G, not in D, is adjacent to exactly k vertices of D. The minimum cardinality of a perfect k-dominating set of G is the perfect k-domination number γ _kp (G). In this paper, we give characterizations of graphs for which γ _kp (G)=γ(G)+k?2 and prove that the perfect k-domination problem is NP-complete even when restricted to bipartite graphs and chordal graphs. Also, by using dynamic programming techniques, we obtain an algorithm to determine the perfect k-domination number of trees.     

Online scheduling on uniform machines with two hierarchies

In this paper we study online scheduling problem on m parallel uniform machines with two hierarchies. The objective is to minimize the maximum completion time (makespan). Machines are provided with different capability. The machines with speed s can schedule all jobs, while the other machines with speed 1 can only process partial jobs. Online algorithms for any 0<s<∞ are provided in the paper. For the case of k=1 and m=2, and the case of some values of s, k=1 and m=3, the algorithms are the best possible, where k is the number of machines with hierarchy 1, and m is the number of machines. Lower bounds for some special cases are also presented.     

Finding paths with minimum shared edges

Motivated by a security problem in geographic information systems, we study the following graph theoretical problem: given a graph G, two special nodes s and t in G, and a number k, find k paths from s to t in G so as to minimize the number of edges shared among the paths. This is a generalization of the well-known disjoint paths problem. While disjoint paths can be computed efficiently, we show that finding paths with minimum shared edges is NP-hard. Moreover, we show that it is even hard to approximate the minimum number of shared edges within a factor of $2^{\log^{1-\varepsilon}n}$ , for any constant ε>0. On the positive side, we show that there exists a (k?1)-approximation algorithm for the problem, using an adaption of a network flow algorithm. We design some heuristics to improve the quality of the output, and provide empirical results.     

On the vertex cover $$P_3$$ problem parameterized by treewidth

Consider a graph G. A subset of vertices, F, is called a vertex cover \(P_t\) (\(VCP_t\)) set if every path of order t contains at least one vertex in F. Finding a minimum \(VCP_t\) set in a graph is is NP-hard for any integer \(t\ge 2\) and is called the \(MVCP_3\) problem. In this paper, we study the parameterized algorithms for the \(MVCP_3\) problem when the underlying graph G is parameterized by the treewidth. Given an n-vertex graph together with its tree decomposition of width at most p, we present an algorithm running in time \(4^{p}\cdot n^{O(1)}\) for the \(MVCP_3\) problem. Moreover, we show that for the \(MVCP_3\) problem on planar graphs, there is a subexponential parameterized algorithm running in time \(2^{O(\sqrt{k})}\cdot n^{O(1)}\) where k is the size of the optimal solution.     

Relay node placement in two-tiered wireless sensor networks with base stations

This paper addresses the relay node placement problem in two-tiered wireless sensor networks with base stations, which aims to deploy a minimum number of relay nodes to achieve certain coverage and connectivity requirement. Under the assumption that the communication range of the sensor nodes is no more than that of the relay node, we present a polynomial time (5+?)-approximation algorithm for the 1-coverage 1-connected problem. Furthermore, we consider the fault tolerant problem in the network, we present a polynomial time (20+?)-approximation algorithm for the 2-coverage 2-connected problem, where ? is any given positive constant. For the k-coverage 2-connected situation, we present a polynomial time (15k?10+?)-approximation algorithm.     

Critical edges/nodes for the minimum spanning tree problem: complexity and approximation

In this paper, we study the complexity and the approximation of the k most vital edges (nodes) and min edge (node) blocker versions for the minimum spanning tree problem (MST). We show that the k most vital edges MST problem is NP-hard even for complete graphs with weights 0 or 1 and 3-approximable for graphs with weights 0 or 1. We also prove that the k most vital nodes MST problem is not approximable within a factor n ^1?? , for any ?>0, unless NP=ZPP, even for complete graphs of order n with weights 0 or 1. Furthermore, we show that the min edge blocker MST problem is NP-hard even for complete graphs with weights 0 or 1 and that the min node blocker MST problem is NP-hard to approximate within a factor 1.36 even for graphs with weights 0 or 1.     

Parameterized dominating set problem in chordal graphs: complexity and lower bound

In this paper, we study the parameterized dominating set problem in chordal graphs. The goal of the problem is to determine whether a given chordal graph G=(V,E) contains a dominating set of size k or not, where k is an integer parameter. We show that the problem is W[1]-hard and it cannot be solved in time unless 3SAT can be solved in subexponential time. In addition, we show that the upper bound of this problem can be improved to when the underlying graph G is an interval graph.     

Note on the hardness of generalized connectivity

Let G be a nontrivial connected graph of order n and let k be an integer with 2??k??n. For a set S of k vertices of G, let ??(S) denote the maximum number ? of edge-disjoint trees T ₁,T ₂,??,T _? in G such that V(T _i )??V(T _j )=S for every pair i,j of distinct integers with 1??i,j???. Chartrand et al. generalized the concept of connectivity as follows: The k-connectivity, denoted by ?? _k (G), of G is defined by ?? _k (G)=min{??(S)}, where the minimum is taken over all k-subsets S of V(G). Thus ?? ₂(G)=??(G), where ??(G) is the connectivity of G, for which there are polynomial-time algorithms to solve it. This paper mainly focus on the complexity of determining the generalized connectivity of a graph. At first, we obtain that for two fixed positive integers k ₁ and k ₂, given a graph G and a k ₁-subset S of V(G), the problem of deciding whether G contains k ₂ internally disjoint trees connecting S can be solved by a polynomial-time algorithm. Then, we show that when k ₁ is a fixed integer of at least 4, but k ₂ is not a fixed integer, the problem turns out to be NP-complete. On the other hand, when k ₂ is a fixed integer of at least 2, but k ₁ is not a fixed integer, we show that the problem also becomes NP-complete.     

FPT algorithms for Connected Feedback Vertex Set

We study the recently introduced Connected Feedback Vertex Set (CFVS) problem from the view-point of parameterized algorithms. CFVS is the connected variant of the classical Feedback Vertex Set problem and is defined as follows: given a graph G=(V,E) and an integer k, decide whether there exists F?V, |F|??k, such that G[V?F] is a forest and G[F] is connected. We show that Connected Feedback Vertex Set can be solved in time O(2 ^O(k) n ^O(1)) on general graphs and in time $O(2^{O(\sqrt{k}\log k)}n^{O(1)})$ on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses, as a subroutine, a parameterized algorithm for Group Steiner Tree, a well studied variant of Steiner Tree. We find the algorithm for Group Steiner Tree of independent interest and believe that it could be useful for obtaining parameterized algorithms for other connectivity problems.     

The bandpass problem: combinatorial optimization and library of problems

A combinatorial optimization problem, called the Bandpass Problem, is introduced. Given a rectangular matrix A of binary elements {0,1} and a positive integer B called the Bandpass Number, a set of B consecutive non-zero elements in any column is called a Bandpass. No two bandpasses in the same column can have common rows. The Bandpass problem consists of finding an optimal permutation of rows of the matrix, which produces the maximum total number of bandpasses having the same given bandpass number in all columns. This combinatorial problem arises in considering the optimal packing of information flows on different wavelengths into groups to obtain the highest available cost reduction in design and operating the optical communication networks using wavelength division multiplexing technology. Integer programming models of two versions of the bandpass problems are developed. For a matrix A with three or more columns the Bandpass problem is proved to be NP-hard. For matrices with two or one column a polynomial algorithm solving the problem to optimality is presented. For the general case fast performing heuristic polynomial algorithms are presented, which provide near optimal solutions, acceptable for applications. High quality of the generated heuristic solutions has been confirmed in the extensive computational experiments. As an NP-hard combinatorial optimization problem with important applications the Bandpass problem offers a challenge for researchers to develop efficient computational solution methods. To encourage the further research a Library of Bandpass Problems has been developed. The Library is open to public and consists of 90 problems of different sizes (numbers of rows, columns and density of non-zero elements of matrix A and bandpass number B), half of them with known optimal solutions and the second half, without.     

Degree-constrained orientations of embedded graphs

We investigate the problem of orienting the edges of an embedded graph in such a way that the resulting digraph fulfills given in-degree specifications both for the vertices and for the faces of the embedding. This primal-dual orientation problem was first proposed by Frank for the case of planar graphs, in conjunction with the question for a good characterization of the existence of such orientations. We answer this question by showing that a feasible orientation of a planar embedding, if it exists, can be constructed by combining certain parts of a primally feasible orientation and a dually feasible orientation. For the general case of arbitrary embeddings, we show that the number of feasible orientations is bounded by \(2^{2g}\), where \(g\) is the genus of the embedding. Our proof also yields a fixed-parameter algorithm for determining all feasible orientations parameterized by the genus. In contrast to these positive results, however, we also show that the problem becomes \(N\!P\)-complete even for a fixed genus if only upper and lower bounds on the in-degrees are specified instead of exact values.     

Parameterized algorithms for min–max 2-cluster editing

For a given graph and an integer t, the Min–Max 2-Clustering problem asks if there exists a modification of a given graph into two maximal disjoint cliques by inserting or deleting edges such that the number of the editing edges incident to each vertex is at most t. It has been shown that the problem can be solved in polynomial time for \(t<n/4\), where n is the number of vertices. In this paper, we design parameterized algorithms for different ranges of t. Let \(k=t-n/4\). We show that the problem is polynomial-time solvable when roughly \(k<\sqrt{n/32}\). When \(k\in o(n)\), we design a randomized and a deterministic algorithm with sub-exponential time parameterized complexity, i.e., the problem is in SUBEPT. We also show that the problem can be solved in \(O({2}^{n/r}\cdot n^2)\) time for \(k<n/12\) and in \(O(n^2\cdot 2^{3n/4+k})\) time for \(n/12\le k< n/4\), where \(r=2+\lfloor (n/4-3k-2)/(2k+1) \rfloor \ge 2\).