Concepts inFPTAS for #BIS with Degree Bounds on One Side
Polynomial-time approximation scheme
In computer science, a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems). A PTAS is an algorithm which takes an instance of an optimization problem and a parameter ¿ > 0 and, in polynomial time, produces a solution that is within a factor 1 + ¿ of being optimal (or 1 - ¿ for maximization problems).
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Degree (graph theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges incident to the vertex, with loops counted twice. The degree of a vertex is denoted The maximum degree of a graph G, denoted by ¿(G), and the minimum degree of a graph, denoted by ¿(G), are the maximum and minimum degree of its vertices. In the graph on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, all degrees are the same, and so we can speak of the degree of the graph.
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Independent set (graph theory)
In graph theory, an independent set or stable set is a set of vertices in a graph, no two of which are adjacent. That is, it is a set I of vertices such that for every two vertices in I, there is no edge connecting the two. Equivalently, each edge in the graph has at most one endpoint in I. The size of an independent set is the number of vertices it contains. A maximal independent set is an independent set such that adding any other vertex to the set forces the set to contain an edge.
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Bipartite graph
In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint sets U and V such that every edge connects a vertex in U to one in V; that is, U and V are independent sets. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles.
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P (complexity)
In computational complexity theory, P, also known as PTIME or DTIME(n), is one of the most fundamental complexity classes. It contains all decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time.
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Counting
Counting is the action of finding the number of elements of a finite set of objects.
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