Related papers: Phase transition for cutting-plane approach to ver…
The phase-transition behavior of the NP-hard vertex-cover (VC) combinatorial optimization problem is studied numerically by linear programming (LP) on ensembles of random graphs. As the basic Simplex (SX) algorithm suitable for such LPs may…
We review recent progress in the study of the vertex-cover problem (VC). VC belongs to the class of NP-complete graph theoretical problems, which plays a central role in theoretical computer science. On ensembles of random graphs, VC…
In this letter, we analytically describe the typical solution time needed by a backtracking algorithm to solve the vertex-cover problem on finite-connectivity random graphs. We find two different transitions: The first one is…
We study numerically the cluster structure of random ensembles of two NP-hard optimization problems originating in computational complexity, the vertex-cover problem and the number partitioning problem. We use branch-and-bound type…
In this letter we study the NP-complete vertex cover problem on finite connectivity random graphs. When the allowed size of the cover set is decreased, a discontinuous transition in solvability and typical-case complexity occurs. This…
In this paper, solution space organization of minimum vertex-cover problem is deeply investigated using the K\"{o}nig-Eg\'{e}rvary (KE) graph and theorem, in which a hierarchical decomposition mechanism named KE-layer structure of general…
The VertexCover problem is proven to be computationally hard in different ways: It is NP-complete to find an optimal solution and even NP-hard to find an approximation with reasonable factors. In contrast, recent experiments suggest that on…
To solve the combinatorial optimization problems especially the minimal Vertex-cover problem with high efficiency, is a significant task in theoretical computer science and many other subjects. Aiming at detecting the solution space of…
Covering all edges of a graph by a small number of vertices, this is the NP-complete Vertex Cover problem. It is among the most fundamental graph-algorithmic problems. Following a recent trend in studying temporal graphs (a sequence of…
The Traveling Salesperson problem asks for the shortest cyclic tour visiting a set of cities given their pairwise distances and belongs to the NP-hard complexity class, which means that with all known algorithms in the worst case instances…
Finding a minimum vertex cover in a network is a fundamental NP-complete graph problem. One way to deal with its computational hardness, is to trade the qualitative performance of an algorithm (allowing non-optimal outputs) for an improved…
Leaf-Removal process has been widely researched and applied in many mathematical and physical fields to help understand the complex systems, and a lot of problems including the minimal vertex-cover are deeply related to this process and the…
We study minimum vertex cover problems on random \alpha-uniform hypergraphs using two different approaches, a replica method in statistical mechanics of random systems and a leaf removal algorithm. It is found that there exists a phase…
Recently, there has been increasing interest and progress in improvising the approximation algorithm for well-known NP-Complete problems, particularly the approximation algorithm for the Vertex-Cover problem. Here we have proposed a…
A phase transition, like the one already found on Boolean satisfiability problem by Kirkpatrick and Selman, is found on max clique problem on ER graphs. Although number of the datapoints is limited, the transition seems to obey finite size…
We study the graph alignment problem over two independent Erd\H{o}s-R\'enyi graphs on $n$ vertices, with edge density $p$ falling into two regimes separated by the critical window around $p_c=\sqrt{\log n/n}$. Our result reveals an…
The distribution of the computational cost of linear-programming (LP) relaxation for vertex cover problems on Erdos-Renyi random graphs is evaluated by using the rare-event sampling method. As a large-deviation property, differences of the…
Many algorithms which exactly solve hard problems require branching on more or less complex structures in order to do their job. Those who design such algorithms often find themselves doing a meticulous analysis of numerous different cases…
We consider two independent Erd\H{o}s-R\'enyi random graphs, with possibly different parameters, and study two isomorphism problems, a graph embedding problem and a common subgraph problem. Under certain conditions on the graph parameters…
We consider the selective graph coloring problem, which is a generalization of the classical graph coloring problem. Given a graph together with a partition of its vertex set into clusters, we want to choose exactly one vertex per cluster…