Efficient approximation by “low-complexity” exponential algorithms

This paper proposes a way to bring together two seemingly “foreign” domains that are the polynomial approximation and the exact computation for NP-hard problems. We show how one can match ideas from both areas in order to design approximation algorithms achieving ratios unachievable in polynomial time (unless a very unlikely complexity conjecture is confirmed) with worst-case complexity much lower (though super-polynomial) than that of an exact computation. We then apply these ideas to two famous combinatorial optimization problems, namely, the max independent set and the min vertex cover, as well as to some other problems mainly linked to max independent set by simple approximation preserving reductions.

[1]  Vijay V. Vazirani,et al.  Approximation Algorithms , 2001, Springer Berlin Heidelberg.

[2]  Giorgio Gambosi,et al.  Complexity and approximation: combinatorial optimization problems and their approximability properties , 1999 .

[3]  J. Håstad.  Clique is hard to approximate withinn1−ε , 1999 .

[4]  J. Håstad.  Clique is hard to approximate within n 1-C , 1996 .

[5]  Subhash Khot,et al.  Vertex cover might be hard to approximate to within 2-/spl epsiv/ , 2003, 18th IEEE Annual Conference on Computational Complexity, 2003. Proceedings..

[6]  Piotr Berman,et al.  On the Approximation Properties of Independent Set Problem in Degree 3 Graphs , 1999, WADS.

[7]  Gerhard J. Woeginger,et al.  Exact Algorithms for NP-Hard Problems: A Survey , 2001, Combinatorial Optimization.

[8]  Liming Cai,et al.  Fixed-Parameter Approximation: Conceptual Framework and Approximability Results , 2010, Algorithmica.

[9]  Michael R. Fellows,et al.  Parameterized Approximation Problems , 2006, IWPEC.

[10]  Dorit S. Hochba,et al.  Approximation Algorithms for NP-Hard Problems , 1997, SIGA.

[11]  David S. Johnson,et al.  Approximation algorithms for combinatorial problems , 1973, STOC.

[12]  Yijia Chen,et al.  On Parameterized Approximability , 2006, IWPEC.

[13]  Weijia Jia,et al.  Vertex Cover: Further Observations and Further Improvements , 1999, J. Algorithms.

[14]  Leslie E. Trotter,et al.  Vertex packings: Structural properties and algorithms , 1975, Math. Program..

[15]  A. Bonato,et al.  Graphs and Hypergraphs , 2022 .

[16]  Fabrizio Grandoni,et al.  Measure and Conquer: Domination - A Case Study , 2005, ICALP.

[17]  Hans Ulrich Simon,et al.  On Approximate Solutions for Combinatorial Optimization Problems , 1990, SIAM J. Discret. Math..

[18]  Martin Fürer,et al.  A Faster Algorithm for Finding Maximum Independent Sets in Sparse Graphs , 2006, LATIN.

[19]  Ge Xia,et al.  Labeled Search Trees and Amortized Analysis: Improved Upper Bounds for NP-Hard Problems , 2003, Algorithmica.

[20]  Weijia Jia,et al.  Vertex Cover: Further Observations and Further Improvements , 2001, J. Algorithms.

[21]  Robert E. Tarjan,et al.  Finding a Maximum Independent Set , 1976, SIAM J. Comput..

[22]  Carsten Lund,et al.  Proof verification and the intractability of approximation problems , 1992, FOCS 1992.

[23]  Richard Beigel,et al.  Finding maximum independent sets in sparse and general graphs , 1999, SODA '99.