Polynomial Time Approximation Schemes for Dense Instances of N P Hard Problems

Journal of Computer and System Sciences (Impact Factor: 1.14). 02/1999; 58(1):193-210. DOI: 10.1006/jcss.1998.1605
Source: DBLP


We present a unified framework for designing polynomial time approximation schemes (PTASs) for “dense” instances of many NP-hard optimization problems, including maximum cut, graph bisection, graph separation, minimumk-way cut with and without specified terminals, and maximum 3-satisfiability. By dense graphs we mean graphs with minimum degreeΩ(n), although our algorithms solve most of these problems so long as the average degree isΩ(n). Denseness for nongraph problems is defined similarly. The unified framework begins with the idea ofexhaustive sampling:picking a small random set of vertices, guessing where they go on the optimum solution, and then using their placement to determine the placement of everything else. The approach then develops into a PTAS for approximating certainsmoothinteger programs, where the objective function and the constraints are “dense” polynomials of constant degree.

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    • "For some special cases in terms of graph classes, values of k and optimal objective values, better approximations have been obtained for DkSP and HkSP. Arora et al. [3] "
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    ABSTRACT: Given a connected graph $G$ on $n$ vertices and a positive integer $k\le n$, a subgraph of $G$ on $k$ vertices is called a $k$-subgraph in $G$. We design combinatorial approximation algorithms for finding a connected $k$-subgraph in $G$ such that its density is at least a factor $\Omega(\max\{n^{-2/5},k^2/n^2\})$ of the density of the densest $k$-subgraph in $G$ (which is not necessarily connected). These particularly provide the first non-trivial approximations for the densest connected $k$-subgraph problem on general graphs.
    • "While the above reduction indicates that QKP and DkS have strong similarities in their approximability behavior, this analogy breaks down completely for dense graphs. It was shown by Arora et al. (1999) that there exists a PTAS for DkS on graphs with Ω(n 2 ) edges for k in Ω(n) (and DkS becomes trivial on complete graphs), while an instance of QKP can always be extended to a complete graph by adding edges with profit 0 without gaining anything w.r.t. approximability. "
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    ABSTRACT: We study the classical quadratic knapsack problem (QKP) on special graph classes. In this case the quadratic terms of the objective function are present only for certain pairs of knapsack items. These pairs are represented by the edges of a graph G=(V,E) whose vertices represent the knapsack items. We show that QKP permits an FPTAS on graphs of bounded treewidth and a PTAS on planar graphs and more generally on H-minor free graphs. The latter result is shown by adopting a technique of Demaine et al. (2005). We will also show strong NP-hardness of QKP on graphs that are 3-book embeddable, a natural graph class that is related to planar graphs. In addition we will argue that the problem might have a bad approximability behaviour on all graph classes containing large cliques (under certain complexity assumption used for showing hardness results for the densest k-subgraph problem).
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    • "For the general problem of finding the KHS, many approximate algorithms were introduced (e.g., [1] [2] [12]). For instance, in [1] the authors model the problem as a quadratic 0/1 program and apply random sampling and randomized rounding techniques resulting in a polynomial-time approximation scheme, and in [12], the algorithm is based on semi-definite programming relaxation. If K is very small, then the optimal solution can be found by enumeration. "

    SIGMOD 2013; 06/2013
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