Parameterizing cut sets in a graph by the number of their components

For a connected graph G=(V,E), a subset U⊆V is a disconnected cut if U disconnects G and the subgraph G[U] induced by U is disconnected as well. A cut U is a k-cut if G[U] contains exactly k(≥1) components. More specifically, a k-cut U is a (k,?)-cut if V?U induces a subgraph with exactly ?(≥2) components. The Disconnected Cut problem is to test whether a graph has a disconnected cut and is known to be NP-complete. The problems k-Cut and (k,?)-Cut are to test whether a graph has a k-cut or (k,?)-cut, respectively. By pinpointing a close relationship to graph contractibility problems we show that (k,?)-Cut is in P for k=1 and any fixed constant ?≥2, while it is NP-complete for any fixed pair k,?≥2. We then prove that k-Cut is in P for k=1 and NP-complete for any fixed k≥2. On the other hand, for every fixed integer g≥0, we present an FPT algorithm that solves (k,?)-Cut on graphs of Euler genus at most g when parameterized by k+?. By modifying this algorithm we can also show that k-Cut is in FPT for this graph class when parameterized by k. Finally, we show that Disconnected Cut is solvable in polynomial time for minor-closed classes of graphs excluding some apex graph.     

Kernels for below-upper-bound parameterizations of the hitting set and directed dominating set problems

In the Hitting Set problem, we are given a collection F of subsets of a ground set V and an integer p, and asked whether V has a p-element subset that intersects each set in F. We consider two parameterizations of Hitting Set below tight upper bounds, p=m−k and p=n−k. In both cases k is the parameter. We prove that the first parameterization is fixed-parameter tractable, but has no polynomial kernel unless coNP ⊆ NP/poly. The second parameterization is W[1]-complete, but the introduction of an additional parameter, the degeneracy of the hypergraph H=(V,F), makes the problem not only fixed-parameter tractable, but also one with a linear kernel. Here the degeneracy of H=(V,F) is the minimum integer d such that for each X⊂V the hypergraph with vertex set V?X and edge set containing all edges of F without vertices in X, has a vertex of degree at most d.In Nonblocker (Directed Nonblocker), we are given an undirected graph (a directed graph) G on n vertices and an integer k, and asked whether G has a set X of n−k vertices such that for each vertex y∉X there is an edge (arc) from a vertex in X to y. Nonblocker can be viewed as a special case of Directed Nonblocker (replace an undirected graph by a symmetric digraph). Dehne et al. (Proc. SOFSEM 2006) proved that Nonblocker has a linear-order kernel. We obtain a linear-order kernel for Directed Nonblocker.     

Fast Minor Testing in Planar Graphs

Minor Containment is a fundamental problem in Algorithmic Graph Theory used as a subroutine in numerous graph algorithms. A model of a graph H in a graph G is a set of disjoint connected subgraphs of G indexed by the vertices of H, such that if {u,v} is an edge of H, then there is an edge of G between components C _u and C _v . A graph H is a minor of G if G contains a model of H as a subgraph. We give an algorithm that, given a planar n-vertex graph G and an h-vertex graph H, either finds in time $\mathcal{O}(2^{\mathcal{O}(h)} \cdot n +n^{2}\cdot\log n)$ a model of H in G, or correctly concludes that G does not contain H as a minor. Our algorithm is the first single-exponential algorithm for this problem and improves all previous minor testing algorithms in planar graphs. Our technique is based on a novel approach called partially embedded dynamic programming.     

Partial deduction of updateable definite logic programs

In this paper, we study the partial deduction of updateable logic programs. Partial deduction is a transformation technique that, given a normal program P and a normal goal G, produces from P a residual program P′ wrt G. An updateable program is a normal program to which an update can be applied. An update is a sequence of additions of clauses to a program and deletions of clauses from a program. We present algorithms that, given a normal program P, a normal goal G, a residual program P′ wrt G, and an update t for P, compute the corresponding update t′ such that t′(P′) is a residual program of t(P) wrt G. Weprove the correctness of these algorithms. We also describe an application of one of the algorithms to updateable knowledge bases.     

A note on finding all homogeneous set sandwiches

A module is a set of vertices H of a graph G=(V,E) such that each vertex of V?H is either adjacent to all vertices of H or to none of them. A homogeneous set is a nontrivial module. A graph G_s=(V,E_s) is a sandwich for a pair of graphs G_t=(V,E_t) and G=(V,E) if E_t⊆E_s⊆E. In a recent paper, Tang et al. [Inform. Process. Lett. 77 (2001) 17-22] described an O(Δn²) algorithm for testing the existence of a homogeneous set in sandwich graphs of G_t=(V,E_t) and G=(V,E) and then extended it to an enumerative algorithm computing all these possible homogeneous sets. In this paper, we invalidate this latter algorithm by proving there are possibly exponentially many such sets, even if we restrict our attention to strong modules. We then give a correct characterization of a homogeneous set of a sandwich graph.     

A faster algorithm for matching a set of patterns with variable length don't cares

We present a simple and faster solution to the problem of matching a set of patterns with variable length don't cares. Given an alphabet Σ, a pattern p is a word p₁@p₂?@pm, where pi is a string over Σ called a keyword and @∉Σ is a symbol called a variable length don't care (VLDC) symbol. Pattern p matches a text t if t=u₀p₁u₁…um−1pmum for some u₀,…,um∈Σ^∗. The problem addressed in this paper is: given a set of patterns P and a text t, determine whether one of the patterns of P matches t.Kucherov and Rusinowitch (1997) [9] presented an algorithm that solves the problem in time O((|t|+|P|)log|P|), where |P| is the total length of keywords in every pattern of P. We give a new algorithm based on Aho-Corasick automaton. It uses the solutions of Dynamic Marked Ancestor Problem of Chan et al. (2007) [5]. The algorithm takes O((|t|+‖P‖)logκ/loglogκ) time, where ‖P‖ is the total number of keywords in every pattern of P, and κ is the number of distinct keywords in P. The algorithm is faster and simpler than the previous approach.     

On vertex ranking of a starlike graph

A vertex coloring c:V→{1,2,…,t} of a graph G=(V,E) is a vertex t-ranking if for any two vertices of the same color every path between them contains a vertex of larger color. The vertex ranking number χ_r(G) is the smallest value of t such that G has a vertex t-ranking. A χ_r(G)-ranking of G is said to be an optimal vertex ranking. In this paper, we present an O(|V|+|E|) time algorithm for finding an optimal vertex ranking of a starlike graph G=(V,E). Our result implies that an optimal vertex ranking of a split graph can be computed in linear time.     

Spanners of bounded degree graphs

A k-spanner of a graph G is a spanning subgraph of G in which the distance between any pair of vertices is at most k times the distance in G. We prove that for fixed k,w, the problem of deciding if a given graph has a k-spanner of treewidth w is fixed-parameter tractable on graphs of bounded degree. In particular, this implies that finding a k-spanner that is a tree (a tree k-spanner) is fixed-parameter tractable on graphs of bounded degree. In contrast, we observe that if the graph has only one vertex of unbounded degree, then Treek-Spanner is NP-complete for k?4.     

CHAP and rewrite components

We present an additional feature to the Challenge Handshake Authentication Protocol. It makes the protocol resilient to offline brute-force/dictionary attacks. We base our contribution to the protocol on the concept of a rewrite complement for ground term rewrite systems (GTRSs). We also introduce and study the notion of a type-based complement which is a special case of a rewrite complement. We show the following decision results. Given GTRSs A, C, and a reduced GTRS B over some ranked alphabet ??, one can decide whether C is a type-based complement of A for B. Given a GTRS A and a reduced GTRS B over some ranked alphabet ??, one can decide whether there is a GTRS C such that C is a type-based complement of A for B. If the answer is yes, then we can construct such a GTRS C.     

Order-k α-hulls and α-shapes

We introduce order-k α-hulls and α-shapes – generalizations of α-hulls and α-shapes. Being also a generalization of k-hull (known in statistics as “k-depth contour”), order-k α-hull provides a link between shape reconstruction and statistical depth. As a generalization of α-hull, order-k α-hull gives a robust shape estimation by ignoring locally up to k outliers in a point set. Order-k α-shape produces an “inner” shape of the set, with the amount of “digging” into the points controlled by k. As a generalization of k-hull, order-k α-hull is capable of determining “deep” points amidst samples from a multimodal distribution: it correctly identifies points which lie outside clusters of samples.The order-k α-hulls and α-shapes are related to order-k Voronoi diagrams in the same way in which α-hulls and α-shapes are related to Voronoi diagrams. This implies that order-k α-hull and α-shape can be readily built from order-k Voronoi diagram, and that the number of different order-k α-shapes for all possible values of α is proportional to the complexity of order-k Voronoi diagram.     

Low-level dichotomy for quantified constraint satisfaction problems

Building on a result of Larose and Tesson for constraint satisfaction problems (CSPs), we uncover a dichotomy for the quantified constraint satisfaction problem QCSP(B), where B is a finite structure that is a core. Specifically, such problems are either in ALogtime or are L-hard. This involves demonstrating that if CSP(B) is first-order expressible, and B is a core, then QCSP(B) is in ALogtime.We show that the class of B such that CSP(B) is first-order expressible (indeed trivial) is a microcosm for all QCSPs. Specifically, for any B there exists a C — generally not a core — such that CSP(C) is trivial, yet QCSP(B) and QCSP(C) are equivalent under logspace reductions.     

Fault-diameter of Cartesian graph bundles

Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let G be a kG-connected graph and Dc(G) denote the diameter of G after deleting any of its c<kG vertices. We prove that Da+b+1(G)?Da(F)+Db(B)+1 if G is a graph bundle with fibre F over base B, a<kF, and b<kB.     

Max- and Min-Neighborhood Monopolies

Given a graph G=(V,E) and a set of vertices M ? V, a vertex v ∈ V is said to be controlled by M if the majority of v’s neighbors (including itself) belong to M. M is called a monopoly in G if every vertex v∈ V is controlled by M. For a specified M and a given range for edge set E (E ₁ ? E ? E ₂), we try to determine an E such that M is a monopoly in G=(V,E). We first present a polynomial algorithm for testing if such an E exists, by formulating it as a network flow problem. Assuming that a solution for E does exist, we then show that solutions with the maximum and minimum |E| , respectively, can be found in polynomial time, by solving weighted matching problems. In case there is no solution for E, we want to maximize the number of vertices controlled by the given M. Unfortunately, this problem turns out to be NP-hard. We, therefore, design a simple approximation algorithm which guarantees an approximation ratio of 2.     

Improved Algorithms for Constructing Fault-Tolerant Spanners

Let S be a set of n points in a metric space, and let k be a positive integer. Algorithms are given that construct k -fault-tolerant spanners for S . If in such a spanner at most k vertices and/ or edges are removed, then each pair of points in the remaining graph is still connected by a ``short'' path. First, an algorithm is given that transforms an arbitrary spanner into a k -fault-tolerant spanner. For the Euclidean metric in R ^d , this leads to an O(n log n + c ^k n) -time algorithm that constructs a k -fault-tolerant spanner of degree O(c ^k ) , whose total edge length is O(c ^k ) times the weight of a minimum spanning tree of S , for some constant c . For constant values of k , this result is optimal. In the second part of the paper, algorithms are presented for the Euclidean metric in R ^d . These algorithms construct (i) in O(n log n + k ² n) time, a k -fault-tolerant spanner with O(k ² n) edges, and (ii) in O(k n log n) time, such a spanner with O(k n log n) edges.     

(1 + i) - ary GCD Computation inZ[ i ]as an Analogue to the Binary GCD Algorithm

We present a novel algorithm for GCD computation over the ring of Gaussian integersZ[i ], that is similar to the binary GCD algorithm forZ, in which powers of 1  + i are extracted. Our algorithm has a running time of O(n²) bit operations with a small constant hidden in the O -notation if the two input numbers have a length ofO (n) bits. This is noticeably faster than a least remainder version of the Euclidean algorithm inZ[ i ] or the Caviness–Collins GCD algorithm that both have a running time ofO (n·μ(n)) bit operations, whereμ (n) denotes a good upper bound for the multiplication time of n -bit integers. Our new GCD algorithm is also faster by a constant factor than a Lehmer-type GCD algorithm (i.e. in every Euclidean step a small remainder is calculated, but this remainder need not to be a least remainder) inZ[ i ] which achieves a running time of O(n²) bit operations.     

On the Grundy and b-Chromatic Numbers of a Graph

The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-colouring, that is a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertexes of G. The b-chromatic number of a graph G, denoted by χ _b (G), is the largest k such that G has a b-colouring with k colours, that is a colouring in which each colour class contains a b-vertex, a vertex with neighbours in all other colour classes. Trivially χ _b (G),Γ(G)≤Δ(G)+1. In this paper, we show that deciding if Γ(G)≤Δ(G) is NP-complete even for a bipartite graph G. We then show that deciding if Γ(G)≥|V(G)|?k or if χ _b (G)≥|V(G)|?k are fixed parameter tractable problems with respect to the parameter k.     

Restricted arc-connectivity of digraphs

Since interconnection networks are often modeled by graphs or digraphs, the edge-connectivity of a graph or arc-connectivity of a digraph are important measurements for fault tolerance of networks.The restricted edge-connectivity λ^′(G) of a graph G is the minimum cardinality over all edge-cuts S in a graph G such that there are no isolated vertices in G−S. A connected graph G is called λ^′-connected, if λ^′(G) exists.In 1988, Esfahanian and Hakimi [A.H. Esfahanian, S.L. Hakimi, On computing a conditional edge-connectivity of a graph, Inform. Process. Lett. 27 (1988), 195-199] have shown that each connected graph G of order n?4, except a star, is λ^′-connected and satisfies λ^′(G)?ξ(G), where ξ(G) is the minimum edge-degree of G.If D is a strongly connected digraph, then we call in this paper an arc set S a restricted arc-cut of D if D−S has a non-trivial strong component D₁ such that D−V(D₁) contains an arc. The restricted arc-connectivity λ^′(D) is the minimum cardinality over all restricted arc-cuts S.We observe that the recognition problem, whether λ^′(D) exists for a strongly connected digraph D is solvable in polynomial time. Furthermore, we present some analogous results to the above mentioned theorem of Esfahanian and Hakimi for digraphs, and we show that this theorem follows easily from one of our results.     

On the k-tuple domination of de Bruijn and Kautz digraphs

In a digraph G, a vertex u is said to dominate itself and vertices v such that (u,v) is an arc of G. For a positive integer k, a k-tuple dominating set D of a digraph is a subset of vertices such that every vertex is dominated by at least k vertices in D. The k-tuple domination number of a given digraph is the minimum cardinality of a k-tuple dominating set of the digraph. In this letter, we give the exact values of the k-tuple domination number of de Bruijn and Kautz digraphs.     

An interval extension based on occurrence grouping

In interval arithmetics, special care has been brought to the definition of interval extension functions that compute narrow interval images. In particular, when a function f is monotonic w.r.t. a variable in a given domain, it is well-known that the monotonicity-based interval extension of f computes a sharper image than the natural interval extension does. This paper presents a so-called “occurrence grouping” interval extension [f] _og of a function f. When f is not monotonic w.r.t. a variable x in a given domain, we try to transform f into a new function f ^og that is monotonic w.r.t. two subsets x _a and x _b of the occurrences of x: f ^og is increasing w.r.t. x _a and decreasing w.r.t. x _b . [f] _og is the interval extension by monotonicity of f ^og and produces a sharper interval image than the natural extension does. For finding a good occurrence grouping, we propose a linear program and an algorithm that minimize a Taylor-based over-estimate of the image diameter of [f] _og . Experiments show the benefits of this new interval extension for solving systems of nonlinear equations.