Vertex Cover Research Articles

Δ-wavy probability distributions and Potts model

We define the wavy probability distributions on a subset and Δ-wavy probability distributions - two generalizations of the wavy probability distributions. A classification on the Δ-waviness is given. For the Δ-wavy probability distributions having normalization constant, we give a formula for this constant, to compute this constant. We show that the Potts model is a Δ-wavy probability distribution, where Δ is a partition which will be specified. For the normalization constant of Potts model, we give formulas and bounds. As to the formulas for this constant, we give two general formulas, one of them is simple while the other is more complicated, and based on independent sets, a formula for the Potts model on connected separable graphs - closed-form expressions are then obtained in several cases -, and a formula for the Potts model on graphs with a vertex of degree 2 - a recurrence relation is then obtained for the normalization constant of Potts model on Cn; the cycle graph with n vertices; the normalization constant of Ising model on Cn is computed using this relation. As to the bounds for the normalization constant, we present two ways to obtain such bounds; we illustrate these ways giving a general lower bound, and a lower bound and an upper one when the model is the Potts model on Gn,n, the square grid graph, n = 6k, k ≥ 1 - two upper bounds for the free energy per site of this model are then obtained, one of them being in the limit. A sampling method for the Δ-wavy probability distributions is given and, as a result, a sampling method for the Potts is given. This method - that for the Potts model too - has two steps, Step 1 and Step 2, when |Δ| > 1 and one step, Step 2 only, when |Δ| = 1. For the Potts model, Step 1 is, in general, difficult. As to Step 2, for the Potts model too, using the Gibbs sampler in a generalized sense, we obtain an exact (not approximate) sampling method having p + 1 steps (p + 1 substeps of Step 2), where p = |I|; I is an independent set, best, a maximum independent set, best, a maximum independent set - for the Potts model on Gn1,n2,…,nd , the d-dimensional grid graph, d ≥ 1, n1, n2,…, nd ≥ 1, n1n2·…·nd ≥ 2; we obtain an exact sampling method for half or half+1 vertices.

Sun-Safe Zones: Investigating Integrated Shading Strategies for Children's Play Areas in Urban Parks.

Although Open Space Ratio is a critical control in the Development Approval process, there are no ultraviolet radiation (UVR) protection guidelines for urban parks. This study explores key strategies for shade provision in children's play areas in urban parks, aiming to promote sun-safe play environments against alarming skin cancer trends. The literature review identified primary issues affecting UVR exposure in public venues, and the research comprises a shade audit of Beaton Park in Dalkeith. The methods involved using virtual park modeling and Shadow Analysis simulations to generate the daily average number of hours in shade for each month. Our recommendations based on this analysis are (a) a minimum canopy cover representing 50% of the entire ground cover; (b) a minimum diameter for a shade (umbrella) of about 2.5 times the diameter of the table; and (c) an ideal umbrella height of 90 cm from the table surface. This research proposes a potential nexus between landscape design and a UVR protection framework for child-friendly Sun-safe Zones (SsZ).

Above guarantee parameterization for vertex cover on graphs with maximum degree 4

In the vertex cover problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of vertices S of size at most k such that every edge of G is incident on at least one vertex in S. We study the vertex cover problem on graphs with maximum degree 4 and minimum degree at least 2, parameterized by $$r = k-n/3$$ . We give an algorithm for this problem whose running time is $$O^*(1.6253^r)$$ . As a corollary, we obtain an $$O^*(1.2403^k)$$ -time algorithm for vertex cover on graphs with maximum degree 4.

On the upward book thickness problem: Combinatorial and complexity results

Among the vast literature concerning graph drawing and graph theory, linear layouts of graphs have been the subject of intense research over the years, both from a combinatorial and from an algorithmic perspective. In particular, upward book embeddings of directed acyclic graphs (DAGs) form a popular class of linear layouts with notable applications, and the upward book thickness of a DAG is the minimum number of pages required by any of its upward book embeddings.A long-standing conjecture by Heath, Pemmaraju, and Trenk (1999) states that the upward book thickness of outerplanar DAGs is bounded above by a constant. In this paper, we show that the conjecture holds for subfamilies of upward outerplanar graphs, namely those whose underlying graph is an internally-triangulated outerpath or a cactus, and those whose biconnected components are st-outerplanar graphs.On the complexity side, it is known that deciding whether a graph has upward book thickness k is NP-hard for any fixed k≥3. We show that the problem, for any k≥5, remains NP-hard for graphs whose domination number is O(k), but it is fixed-parameter tractable (FPT) in the vertex cover number.

Subexponential Parameterized Algorithms for Planar and Apex-Minor-Free Graphs via Low Treewidth Pattern Covering

We prove the following theorem. Given a planar graph $G$ and an integer $k$, it is possible in polynomial time to randomly sample a subset $A$ of vertices of $G$ with the following properties: (i) $A$ induces a subgraph of $G$ of treewidth $\mathcal{O}(\sqrt{k}\log k)$, and (ii) for every connected subgraph $H$ of $G$ on at most $k$ vertices, the probability that $A$ covers the whole vertex set of $H$ is at least $(2^{\mathcal{O}(\sqrt{k}\log^2 k)}\cdot n^{\mathcal{O}(1)})^{-1}$, where $n$ is the number of vertices of $G$. Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound $2^{\mathcal{O}(\sqrt{k} \log^2 k)} n^{\mathcal{O}(1)}$. The technique can be applied to problems expressible as searching for a small, connected pattern with a prescribed property in a large host graph, examples of such problems include Directed $k$-Path, Weighted $k$-Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up to this point, it was open whether these problems can be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality. Furthermore, all our results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface.

Dominance complexes and vertex cover numbers of graphs

The dominance complex D(G) of a simple graph \(G = (V,E)\) is the simplicial complex consisting of the subsets of V whose complements are dominating. We show that the connectivity of D(G) plus 2 is a lower bound for the vertex cover number \(\tau (G)\) of G.

On the Hilbert function of vertex cover algebras of Cohen-Macaulay bipartite graphs

We study the h−vector and the Hilbert function of the vertex cover algebra (A)G, introduced and first studied by J. Herzog, T. Hibi and N. V. Trung ([6]), for a special class of bipartite graphs, namely for Cohen-Macaulay bipartite graphs.

Nilpotent covers of symmetric and alternating groups

We prove that the symmetric group $S_n$ has a unique minimal cover $\mathcal{M}$ by maximal nilpotent subgroups, and we obtain an explicit and easily computed formula for the size of $\mathcal{M}$. In addition, we prove that the size of $\mathcal{M}$ is equal to the size of a maximal non-nilpotent subset of $S_n$. This cover $\mathcal{M}$ has attractive properties; for instance, it is a normal cover, and the number of conjugacy classes of subgroups in the cover is equal to the number of partitions of $n$ into distinct positive integers. We show that these results contrast with those for the alternating group $A_n$. In particular, we prove that, for all but finitely many values of $n$, no minimal cover of $A_n$ by maximal nilpotent subgroups is a normal cover and the size of a minimal cover of $A_n$ by maximal nilpotent subgroups is strictly greater than the size of a maximal non-nilpotent subset of $A_n$.

P-Edge/vertex-connected vertex cover: Parameterized and approximation algorithms

We introduce and study two natural generalizations of the Connected Vertex Cover (VC) problem: the p-Edge-Connected and p-Vertex-Connected VC problem (where p≥2 is a fixed integer). We obtain an 2O(pk)nO(1)-time algorithm for p-Edge-Connected VC and an 2O(k2)nO(1)-time algorithm for p-Vertex-Connected VC. Thus, like Connected VC, both constrained VC problems are FPT. Furthermore, like Connected VC, neither problem admits a polynomial kernel unless NP ⊆ coNP/poly, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. Finally, we describe a 2(p+1)-approximation algorithm for the p-Edge-Connected VC. The proofs for the new VC problems require more sophisticated arguments than for Connected VC. In particular, for the approximation algorithm we use Gomory-Hu trees and for the approximate kernels a result on small-size spanning p-vertex/edge-connected subgraphs of a p-vertex/edge-connected graph by Nishizeki and Poljak (1994) [30] and Nagamochi and Ibaraki (1992) [27].

Distributed maximal independent set computation driven by finite-state dynamics

A Maximal Independent Set (MIS) is an inclusion maximal set of pairwise non-adjacent vertices. The computation of an MIS is one of the core problems in distributed computing. In this article, we introduce and analyze a finite-state distributed randomized algorithm for computing a Maximal Independent Set (MIS) on arbitrary undirected graphs. Our algorithm is self-stabilizing (reaches a correct output on any initial configuration) and can be implemented on systems with very scarce conditions. We analyze the convergence time of the proposed algorithm, showing that in many cases the algorithm converges in logarithmic time with high probability.

Fractional Decomposition Tree Algorithm: A tool for studying the integrality gap of Integer Programs

We present a new algorithm, Fractional Decomposition Tree (FDT), for finding a feasible solution for an integer program (IP) where all variables are binary. FDT runs in polynomial time and is guaranteed to find a feasible integer solution provided the integrality gap of an instance’s polyhedron, independent of objective function, is bounded. The algorithm gives a construction for Carr and Vempala’s theorem that any feasible solution to the IP’s linear-programming relaxation, when scaled by the instance integrality gap, dominates a convex combination of feasible solutions. FDT is also a tool for studying the integrality gap of IP formulations. The upper bound on the integrality gap of an FDT solution can be exponentially large. However our experiments demonstrate that FDT can be effective in practice. We study the integrality gap of two problems: optimally augmenting a tree to a 2-edge-connected graph and finding a minimum-cost 2-edge-connected multi-subgraph (2EC). We also give a simplified algorithm, DomToIP, that finds a feasible solution to an IP instance, or concludes that it has unbounded integrality gap. We show that FDT’s speed and approximation quality compare well to that of the original feasibility pump heuristic on moderate-sized instances of the vertex cover problem. For a particular set of hard-to-decompose fractional 2EC solutions, FDT always gave a better integer solution than the Best-of-Many Christofides Algorithm (BOMC).

Componentwise linear powers and the $x$-condition

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gröbner basis of the defining ideal $J$ of $A$ we give a condition, called the $x$-condition, which implies that all graded components $A_k$ of $A$ have linear quotients and with additional assumptions are componentwise linear. A typical example of such an algebra is the Rees ring $\mathcal{R}(I)$ of a graded ideal or the symmetric algebra $\textrm{Sym}(M)$ of a module $M$. We apply our criterion to study certain symmetric algebras and the powers of vertex cover ideals of certain classes of graphs.

Minimal vertex covers in infinite hypergraphs

A vertex cover of a hypergraph is a set of vertices which intersects each hyperedge.A hypergraph possesses propertyC(k,ρ) iff |⋂E′|<ρ for each k element set E′ of hyperedges.Komjáth proved that every uniform hypergraph possessing property C(2,r) for some r∈ω has a minimal vertex cover. In this paper we will relax the assumption of uniformity to an assumption that the set of cardinalities of the hyperedges is a “small” set of infinite cardinals, e.g. it is countable, or it does not contain uncountably many limit cardinals.Komjáth also proved that GCH does not decide the following statement: If a hypergraph G possessing propertyC(2,ω)is μ-uniform for someμ≥ω1, then G has a minimal vertex cover.Using Shelah's Revised GCH theorem, we show that if we strengthen the assumption μ≥ω1 to μ≥ℶω, then we can prove the statement in ZFC!We also show that if all the hyperedges of a hypergraph are countably infinite, then instead of C(2,r) the assumption C(k,r) (for some k∈ω) is enough to guarantee the existence of a minimal vertex cover. If every hyperedge has cardinality ω1, then we can only prove that C(3,r) is enough.

Possibilistic Data Cleaning

Classical data cleaning performs a minimal set of operations on the data to satisfy the given integrity constraints. Often, this minimization is equivalent to vertex cover, for example when tuples can be removed due to the violation of functional dependencies. Classically, the uncertainty of tuples and constraints is ignored. We propose not to view data as dirty but the uncertainty information about data. Since probabilities are often unavailable and their treatment is limited due to correlations in the data, we investigate a qualitative approach to uncertainty. Tuples are assigned degrees of possibility with which they occur, and constraints are assigned degrees of certainty that say to which tuples they apply. Our approach is non-invasive to the data as we lower the possibility degree of tuples as little as possible. The new resulting qualitative version of vertex cover remains NP-hard. We establish an algorithm that is fixed-parameter tractable in the size of the qualitative vertex cover. Experiments with synthetic and real-world data show that our algorithm outperforms the classical algorithm proportionally to the available number of uncertainty degrees. By mining the certainty degrees with which constraints hold, our framework becomes applicable even when uncertainty information is unavailable.

Total tessellation cover: Bounds, hardness, and applications

The concept of graph tessellation cover was defined in the context of quantum walk models, and is a current research area in graph theory. In this work, we propose a generalization called total tessellation cover. A tessellation of a graph G is a partition of its vertex set V ( G ) into vertex disjoint cliques. A tessellation cover of G is a set of tessellations that covers its edge set E ( G ) . A total tessellation cover of G consists of a tessellation cover together with a compatible vertex coloring, such that the color of each vertex is different from the tessellation labels of the edges incident to the vertex. The total tessellation cover number T t ( G ) is the size of a minimum total tessellation cover of G . We present lower bounds T t ( G ) ≥ ω ( G ) and T t ( G ) ≥ s ( G ) + 1 , where ω ( G ) is the size of a maximum clique, and s ( G ) is the number of edges of a maximum induced star subgraph. A graph G is called a good total tessellable if T t ( G ) = ω ( G ) or T t ( G ) = s ( G ) + 1 . We study the complexity of the k - total tessellability problem, which aims to decide whether a given graph G has T t ( G ) ≤ k . We prove that k - total tessellability is in P for good total tessellable graphs. We establish the NP -completeness of the k - total tessellability when restricted to the following graph classes: bipartite graphs, line graphs of triangle-free graphs, universal graphs, ( 2 , 1 ) -chordal graphs and planar graphs.

Approximate Aggregate Utility Maximization Using Greedy Maximal Scheduling

In this paper, we study the problem of aggregate utility maximization in multihop wireless networks. Following standard approaches, we consider the dual of a convex optimization problem that can be decomposed into two sub-problems. One of the sub-problem is a scheduling problem that requires enumeration of all maximal independent sets in the network—a well-known NP-hard problem. To overcome this computational complexity, we resort to polynomial-time sub-optimal Greedy Maximal Scheduling (GMS). The majority of this paper is dedicated towards quantifying the sub-optimality that arises in the aggregate utility maximization problem due to GMS. We also provide some insights into factors affecting aggregate utility maximization by providing bounds on the maximum achievable aggregate utility.

On reconfigurability of target sets

We study the problem of deciding reconfigurability of target sets of a graph. Given a graph G with vertex thresholds τ, consider a dynamic process in which vertex v becomes activated once at least τ(v) of its neighbors are activated. A vertex set S is called a target set if all vertices of G would be activated when initially activating vertices of S. In the Target Set Reconfiguration problem, given two target sets X and Y of the same size, we are required to determine whether X can be transformed into Y by repeatedly swapping one vertex in the current set with another vertex not in the current set preserving every intermediate set as a target set. In this paper, we investigate the complexity of Target Set Reconfiguration in restricted cases. On the hardness side, we prove that Target Set Reconfiguration is PSPACE-complete on bipartite planar graphs of degree 3 and 4 and of threshold 2, bipartite 3-regular graphs and planar 3-regular graphs of threshold 1 and 2, and split graphs, which is in contrast to the fact that a special case called Vertex Cover Reconfiguration is in P for the last graph class. On the positive side, we present a polynomial-time algorithm for Target Set Reconfiguration on graphs of maximum degree 2 and trees. The latter result can be thought of as a generalization of that for Vertex Cover Reconfiguration.

A New Approach Based on Centrality Value in Solving the Minimum Vertex Cover Problem: Malatya Centrality Algorithm

The graph is a data structures and models that used to describe many real-world problems. Many engineering problems, such as safety and transportation, have a graph-like structure and are based on a similar model. Therefore, these problems can be solved using similar methods to the graph data model. Vertex cover problem that is used in modeling many problems is one of the important NP-complete problems in graph theory. Vertex-cover realization by using minimum number of vertex is called Minimum Vertex Cover Problem (MVCP). Since MVCP is an optimization problem, many algorithms and approaches have been proposed to solve this problem. In this article, Malatya algorithm, which offers an effective solution for the vertex-cover problem, is proposed. Malatya algorithm offers a polynomial approach to the vertex cover problem. In the proposed approach, MVCP consists of two steps, calculating the Malatya centrality value and selecting the covering nodes. In the first step, Malatya centrality values are calculated for the nodes in the graph. These values are calculated using Malatya algorithm. Malatya centrality value of each node in the graph consists of the sum of the ratios of the degree of the node to the degrees of the adjacent nodes. The second step is a node selection problem for the vertex cover. The node with the maximum Malatya centrality value is selected from the nodes in the graph and added to the solution set. Then this node and its coincident edges are removed from the graph. Malatya centrality values are calculated again for the new graph, and the node with the maximum Malatya centrality value is selected from these values, and the coincident edges to this node are removed from the graph. This process is continued until all the edges in the graph are covered. It is shown on the sample graph that the proposed Malatya algorithm provides an effective solution for MVCP. Successful test results and analyzes show the effectiveness of Malatya algorithm.

Approximation algorithms for some min–max and minimum stacker crane cover problems

Approximation algorithms for some min–max and minimum stacker crane cover problems

Parameterized algorithms for finding highly connected solution

To introduce our question and the parameterization, consider the classical Vertex Cover problem. In this problem, the input is a graph G on n vertices and a positive integer ℓ, and the goal is to find a vertex subset S of size at most ℓ such that G−S is an independent set. Further, we want that G[S] is highly connected. That is, G[S] should be n−k edge-connected. Clearly, the problem is NP-complete, as substituting k=n−1, we obtain the Connected Vertex Cover problem. A simple observation also shows that the problem admits an algorithm with running time nO(k). Since the problem is polynomial-time solvable for every fixed integer k, a natural parameter is the integer k. In all the problems we consider, the parameter is k, and the goal is to find a solution S of size at most ℓ, such that G[S] is n−k edge-connected and G−S satisfies a property. We show that this version of well-known problems such as Vertex Cover, Feedback Vertex Set, Odd Cycle Transversal and Multiway Cut admit an algorithm with running time f(k)⋅nO(1), that is, they are FPT with the parameter k. One of our main subroutines to obtain these algorithms is an FPT algorithm for n−k edge connected Steiner Subgraph, which could be of an independent interest. Finally, we also show that such an algorithm is not possible for Multicut.