Approximation Algorithms for the Directed k-Tour and k-Stroll Problems

We consider two natural generalizations of the Asymmetric Traveling Salesman problem: the k-Stroll and the k-Tour problems. The input to the k-Stroll problem is a directed n-vertex graph with nonnegative edge lengths, an integer k, as well as two special vertices s and t. The goal is to find a minimum-length s-t walk, containing at least k distinct vertices (including the endpoints s,t). The k-Tour problem can be viewed as a special case of k-Stroll, where s=t. That is, the walk is required to be a tour, containing some pre-specified vertex s. When k=n, the k-Stroll problem becomes equivalent to Asymmetric Traveling Salesman Path, and k-Tour to Asymmetric Traveling Salesman. Our main result is a polylogarithmic approximation algorithm for the k-Stroll problem. Prior to our work, only bicriteria (O(log² k),3)-approximation algorithms have been known, producing walks whose length is bounded by 3OPT, while the number of vertices visited is Ω(k/log² k). We also show a simple O(log² n/loglogn)-approximation algorithm for the k-Tour problem. The best previously known approximation algorithms achieved min(O(log³ k),O(log² n?logk/loglogn)) approximation in polynomial time, and O(log² k) approximation in quasipolynomial time.     

An Improved Approximation Algorithm for the Most Points Covering Problem

In this paper we present a polynomial time approximation scheme for the most points covering problem. In the most points covering problem, n points in R ², r>0, and an integer m>0 are given and the goal is to cover the maximum number of points with m disks with radius r. The dual of the most points covering problem is the partial covering problem in which n points in R ² are given, and we try to cover at least p≤n points of these n points with the minimum number of disks. Both these problems are NP-hard. To solve the most points covering problem, we use the solution of the partial covering problem to obtain an upper bound for the problem and then we generate a valid solution for the most points covering problem by a careful modification of the partial covering solution. We first present an improved approximation algorithm for the partial covering problem which has a better running time than the previous algorithm for this problem. Using this algorithm, we attain a \((1 - \frac{{2\varepsilon }}{{1 +\varepsilon }})\)-approximation algorithm for the most points covering problem. The running time of our algorithm is \(O((1+\varepsilon )mn+\epsilon^{-1}n^{4\sqrt{2}\epsilon^{-1}+2}) \) which is polynomial with respect to both m and n, whereas the previously known algorithm for this problem runs in \(O(n \log n +n\epsilon^{-6m+6} \log (\frac{1}{\epsilon}))\) which is exponential regarding m.     

Maximum Series-Parallel Subgraph

Consider the NP-hard problem of, given a simple graph?G, to find a series-parallel subgraph of?G with the maximum number of edges. The algorithm that, given a connected graph?G, outputs a spanning tree of?G, is a $\frac{1}{2}$ -approximation. Indeed, if n is the number of vertices in G, any spanning tree in G has?n?1 edges and any series-parallel graph on?n vertices has at most?2n?3 edges. We present a $\frac{7}{12}$ -approximation for this problem and results showing the limits of our approach.     

The Outer-connected Domination Number of Sierpiński-like Graphs

An outer-connected dominating set in a graph G = (V, E) is a set of vertices D ? V satisfying the condition that, for each vertex v ? D, vertex v is adjacent to some vertex in D and the subgraph induced by V?D is connected. The outer-connected dominating set problem is to find an outer-connected dominating set with the minimum number of vertices which is denoted by \(\tilde {\gamma }_{c}(G)\). In this paper, we determine \(\tilde {\gamma }_{c}(S(n,k))\), \(\tilde {\gamma }_{c}(S^{+}(n,k))\), \(\tilde {\gamma }_{c}(S^{++}(n,k))\), and \(\tilde {\gamma }_{c}(S_{n})\), where S(n, k), S ⁺(n, k), S ⁺⁺(n, k), and S _n are Sierpi\(\acute {\mathrm {n}}\)ski-like graphs.     

About the Decidability of Polyhedral Separability in the Lattice $$\mathbb {Z}^d$$

The recognition of primitives in digital geometry is deeply linked with separability problems. This framework leads us to consider the following problem of pattern recognition : given a finite lattice set \(S\subset \mathbb {Z}^d\) and a positive integer n, is it possible to separate S from \(\mathbb {Z}^d \setminus S\) by n half-spaces? In other words, does there exist a polyhedron P defined by at most n half-spaces satisfying \(P\cap \mathbb {Z}^d = S\)? The difficulty comes from the infinite number of constraints generated by all the points of \(\mathbb {Z}^d\setminus S\). It makes the decidability of the problem non-straightforward since the classical algorithms of polyhedral separability can not be applied in this framework. We conjecture that the problem is nevertheless decidable and prove it under some assumptions: in arbitrary dimension, if the interior of the convex hull of S contains at least one lattice point or if the dimension d is 2 or if the dimension \(d=3\) and S is not in a specific configuration of lattice width 0 or 1. The proof strategy is to reduce the set of outliers \(\mathbb {Z}^d\setminus S\) to its minimal elements according to a partial order “is in the shadow of.” These minimal elements are called the lattice jewels of S. We prove that under some assumptions, the set S admits only a finite number of lattice jewels. The result about the decidability of the problem is a corollary of this fundamental property.     

Fully Dynamic Geometric Spanners

In this paper we present an algorithm for maintaining a spanner over a dynamic set of points in constant doubling dimension metric spaces. For a set S of points in ? ^d , a t-spanner is a sparse graph on the points of S such that there is a path in the spanner between any pair of points whose total length is at most t times the distance between the points. We present the first fully dynamic algorithm for maintaining a spanner whose update time depends solely on the number of points in S. In particular, we show how to maintain a (1+ε)-spanner with O(n/ε ^d ) edges, where points can be inserted to S in an amortized update time of O(log?n) and deleted from S in an amortized update time of $\tilde{O}(n^{1/3})$ . As a by-product of our techniques we obtain a simple incremental algorithm for constructing a (1+ε)-spanner with O(n/ε ^d ) edges in constant doubling dimension metric spaces whose running time is O(nlog?n).     

Parallel computation of disease transforms

Distance transforms are an important computational tool for the processing of binary images. For ann ×n image, distance transforms can be computed in time \(\mathcal{O}\) (n) on a mesh-connected computer and in polylogarithmic time on hypercube related structures. We investigate the possibilities of computing distance transforms in polylogarithmic time on the pyramid computer and the mesh of trees. For the pyramid, we obtain a polynomial lower bound using a result by Miller and Stout, so we turn our attention to the mesh of trees. We give a very simple \(\mathcal{O}\) (logn) algorithm for the distance transform with respect to theL ₁-metric, an \(\mathcal{O}\) (log² n) algorithm for the transform with respect to theL _∞-metric, and find that the Euclidean metric is much more difficult. Based on evidence from number theory, we conjecture the impossibility of computing the Euclidean distance transform in polylogarithmic time on a mesh of trees. Instead, we approximate the distance transform up to a given error. This works for anyL _k -metric and takes time \(\mathcal{O}\) (log³ n).     

On Fixed Cost k-Flow Problems

In the Fixed Cost k-Flow problem, we are given a graph G = (V, E) with edge-capacities {u _e ∣e ∈ E} and edge-costs {c _e ∣e ∈ E}, source-sink pair s, t ∈ V, and an integer k. The goal is to find a minimum cost subgraph H of G such that the minimum capacity of an st-cut in H is at least k. By an approximation-preserving reduction from Group Steiner Tree problem to Fixed Cost k-Flow, we obtain the first polylogarithmic lower bound for the problem; this also implies the first non-constant lower bounds for the Capacitated Steiner Network and Capacitated Multicommodity Flow problems. We then consider two special cases of Fixed Cost k-Flow. In the Bipartite Fixed-Cost k-Flow problem, we are given a bipartite graph G = (A ∪ B, E) and an integer k > 0. The goal is to find a node subset S ? A ∪ B of minimum size |S| such G has k pairwise edge-disjoint paths between S ∩ A and S ∩ B. We give an \(O(\sqrt {k\log k})\) approximation for this problem. We also show that we can compute a solution of optimum size with Ω(k/polylog(n)) paths, where n = |A| + |B|. In the Generalized-P2P problem we are given an undirected graph G = (V, E) with edge-costs and integer charges {b _v : v ∈ V}. The goal is to find a minimum-cost spanning subgraph H of G such that every connected component of H has non-negative charge. This problem originated in a practical project for shift design [11]. Besides that, it generalizes many problems such as Steiner Forest, k-Steiner Tree, and Point to Point Connection. We give a logarithmic approximation algorithm for this problem. Finally, we consider a related problem called Connected Rent or Buy Multicommodity Flow and give a log^3+?? n approximation scheme for it using Group Steiner Tree techniques.     

On Cutwidth Parameterized by Vertex Cover

We study the Cutwidth problem, where the input is a graph G, and the objective is find a linear layout of the vertices that minimizes the maximum number of edges intersected by any vertical line inserted between two consecutive vertices. We give an algorithm for Cutwidth with running time O(2 ^k n ^O(1)). Here k is the size of a minimum vertex cover of the input graph G, and n is the number of vertices in G. Our algorithm gives an O(2 ^n/2 n ^O(1)) time algorithm for Cutwidth on bipartite graphs as a corollary. This is the first non-trivial exact exponential time algorithm for Cutwidth on a graph class where the problem remains NP-complete. Additionally, we show that Cutwidth parameterized by the size of the minimum vertex cover of the input graph does not admit a polynomial kernel unless NP?coNP/poly. Our kernelization lower bound contrasts with the recent results of Bodlaender et al. (ICALP, Springer, Berlin, 2011; SWAT, Springer, Berlin, 2012) that both Treewidth and Pathwidth parameterized by vertex cover do admit polynomial kernels.     

Drawing planar graphs using the canonical ordering

We introduce a new method to optimize the required area, minimum angle, and number of bends of planar graph drawings on a grid. The main tool is a new type of ordering on the vertices and faces of triconnected planar graphs. Using this method linear-time-and-space algorithms can be designed for many graph-drawing problems. Our main results are as follows:

Every triconnected planar graphG admits a planar convex grid drawing with straight lines on a (2n?4)×(n?2) grid, wheren is the number of vertices.

Every triconnected planar graph with maximum degree 4 admits a planar orthogonal grid drawing on ann×n grid with at most [3n/2]+4 bends, and ifn>6, then every edge has at most two bends.

Every planar graph with maximum degree 3 admits a planar orthogonal grid drawing with at most [n/2]+1 bends on an [n/2]×[n/2] grid.

Every triconnected planar graphG admits a planar polyline grid drawing on a (2n?6)×(3n?9) grid with minimum angle larger than 2/d radians and at most 5n?15 bends, withd the maximum degree.

These results give in some cases considerable improvements over previous results, and give new bounds in other cases. Several other results, e.g., concerning visibility representations, are included.     

Online Coloring of Bipartite Graphs with and without Advice

In the online version of the well-known graph coloring problem, the vertices appear one after the other together with the edges to the already known vertices and have to be irrevocably colored immediately after their appearance. We consider this problem on bipartite, i.e., two-colorable graphs. We prove that at least ?1.13746?log₂(n)?0.49887? colors are necessary for any deterministic online algorithm to be able to color any given bipartite graph on n vertices, thus improving on the previously known lower bound of ?log₂ n?+1 for sufficiently large n. Recently, the advice complexity was introduced as a method for a fine-grained analysis of the hardness of online problems. We apply this method to the online coloring problem and prove (almost) tight linear upper and lower bounds on the advice complexity of coloring a bipartite graph online optimally or using 3 colors. Moreover, we prove that \(O(\sqrt{n})\) advice bits are sufficient for coloring any bipartite graph on n vertices with at most ?log₂ n? colors.     

Independence Numbers of Random Subgraphs of Some Distance Graph

The distance graph G(n, 2, 1) is a graph where vertices are identified with twoelement subsets of {1, 2,..., n}, and two vertices are connected by an edge whenever the corresponding subsets have exactly one common element. A random subgraph G _p (n, 2, 1) in the Erd?os–Rényi model is obtained by selecting each edge of G(n, 2, 1) with probability p independently of other edges. We find a lower bound on the independence number of the random subgraph G_1/2(n, 2, 1).     

The Complexity of König Subgraph Problems and Above-Guarantee Vertex Cover

A graph is König-Egerváry if the size of a minimum vertex cover equals that of a maximum matching in the graph. These graphs have been studied extensively from a graph-theoretic point of view. In this paper, we introduce and study the algorithmic complexity of finding König-Egerváry subgraphs of a given graph. In particular, given a graph G and a nonnegative integer k, we are interested in the following questions:

1. 1.
   does there exist a set of k vertices (edges) whose deletion makes the graph König-Egerváry?
    
2. 2.
   does there exist a set of k vertices (edges) that induce a König-Egerváry subgraph?
    

We show that these problems are NP-complete and study their complexity from the points of view of approximation and parameterized complexity. Towards this end, we first study the algorithmic complexity of Above Guarantee Vertex Cover, where one is interested in minimizing the additional number of vertices needed beyond the maximum matching size for the vertex cover. Further, while studying the parameterized complexity of the problem of deleting k vertices to obtain a König-Egerváry graph, we show a number of interesting structural results on matchings and vertex covers which could be useful in other contexts.

     

Organization of self-diagnosis of the discrete multicomponent systems a structure like bipartite quasicomplete graphs

Organization of an efficient self-diagnosis of the multicomponent computer and communication systems of diverse structures always attracted attention of the researchers and engineers. A method to solve these problems is presented in the paper by way of the example of a system whose structure is modeled by a uniform ordinary bipartite graph of diameter d = 3, any degree s > 1, and any number n of vertices, where n = s(s ? 1) + 1. The method requires checking of (s ? 1)³ graph loops of length eight each, which is smaller than the number s ²(s ? 1) + s of checks of single graph edges.     

Planar Feedback Vertex Set and Face Cover: Combinatorial Bounds and Subexponential Algorithms

The Planar Feedback Vertex Set problem asks whether an n-vertex planar graph contains at most k vertices meeting all its cycles. The Face Cover problem asks whether all vertices of a plane graph G lie on the boundary of at most k faces of G. Standard techniques from parameterized algorithm design indicate that both problems can be solved by sub-exponential parameterized algorithms (where k is the parameter). In this paper we improve the algorithmic analysis of both problems by proving a series of combinatorial results relating the branchwidth of planar graphs with their face cover. Combining this fact with duality properties of branchwidth, allows us to derive analogous results on feedback vertex set. As a consequence, it follows that Planar Feedback Vertex Set and Face Cover can be solved in \(O(2^{15.11\cdot\sqrt{k}}+n^{2})\) and \(O(2^{10.1\cdot\sqrt {k}}+n^{2})\) steps, respectively.     

On fractional faces of the metric polytope

The integrality recognition problem is considered on a sequence M _{n, k} of nested relaxations of a Boolean quadric polytope, including the rooted semimetric M _n and metric M _{n, 3} polytopes. The constraints of the metric polytope cut off all faces of the rooted semimetric polytope that contain only fractional vertices. This makes it possible to solve the integrality recognition problem on M _n in polynomial time. To solve the integrality recognition problem on the metric polytope, we consider the possibility of cutting off all fractional faces of M _{n, 3} by a certain relaxation M _{n, k} . The coordinates of points of the metric polytope are represented in homogeneous form as a three-dimensional block matrix. We show that in studying the question of cutting off the fractional faces of the metric polytope, it is sufficient to consider only constraints in the form of triangle inequalities.     

Optimal parallel algorithms for rectilinear link-distance problems

We provide optimal parallel solutions to several link-distance problems set in trapezoided rectilinear polygons. All our main parallel algorithms are deterministic and designed to run on the exclusive read exclusive write parallel random access machine (EREW PRAM). LetP be a trapezoided rectilinear simple polygon withn vertices. InO(logn) time usingO(n/logn) processors we can optimally compute:

1. Minimum réctilinear link paths, or shortest paths in theL ₁ metric from any point inP to all vertices ofP.
2. Minimum rectilinear link paths from any segment insideP to all vertices ofP.
3. The rectilinear window (histogram) partition ofP.
4. Both covering radii and vertex intervals for any diagonal ofP.
5. A data structure to support rectilinear link-distance queries between any two points inP (queries can be answered optimally inO(logn) time by uniprocessor).

Our solution to 5 is based on a new linear-time sequential algorithm for this problem which is also provided here. This improves on the previously best-known sequential algorithm for this problem which usedO(n logn) time and space.5 We develop techniques for solving link-distance problems in parallel which are expected to find applications in the design of other parallel computational geometry algorithms. We employ these parallel techniques, for example, to compute (on a CREW PRAM) optimally the link diameter, the link center, and the central diagonal of a rectilinear polygon.     

Approximation and Tidying—A Problem Kernel for?s-Plex Cluster Vertex Deletion

We introduce the NP-hard graph-based data clustering problem s-Plex Cluster Vertex Deletion, where the task is to delete at most?k vertices from a graph so that the connected components of the resulting graph are s-plexes. In an s-plex, every vertex has an edge to all but at most s?1?other vertices; cliques are 1-plexes. We propose a new method based on “approximation and tidying” for kernelizing vertex deletion problems whose goal graphs can be characterized by forbidden induced subgraphs. The method exploits polynomial-time approximation results and thus provides a useful link between approximation and kernelization. Employing “approximation and tidying”, we develop data reduction rules that, in?O(ksn ²) time, transform an s-Plex Cluster Vertex Deletion instance with n vertices into an equivalent instance with O(k ² s ³)?vertices, yielding a problem kernel. To this end, we also show how to exploit structural properties of the specific problem in order to significantly improve the running time of the proposed kernelization method.