Clique-width is a graph parameter that measures in a certain sense the complexity of a graph. Hard graph problems (e.g., problems expressible in Monadic Second Order Logic with second-order quantification on vertex sets, that includes NP-hard problems) can be solved efficiently for graphs of certified small clique-width. It is widely believed that determining the clique-width of a graph is NP-h...

We prove that for every d ≥ 2, deciding if a pure, d-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d ≥ 2 and k ≥ 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable is NP-hard. For d ≥ 3, both problems remain NP-hard when restricted to...

In this course, we will be studying, as the title suggests, the approximability and inapproximability (limits of approximability) of different combinatorial optimization problems. All the problems we will be looking at will be ones that lack efficient algorithms and in particular will be NP-hard problems. The last two-three decades has seen remarkable progress in approximation algorithms for se...

This paper considers a number of NP-complete problems, and provides faster algorithms for solving them. The solutions are based on a recursive partitioning of the problem domain, and careful elimination of some of the branches along the search without actually checking them. The time complexity of the proposed algorithms is of the form O(2 ) for constant 0¡ ¡ 1, where n is the output size of th...

Digital circuits with feedback loops can solve some instances of NP-hard problems by relaxation: the circuit will either oscillate or settle down to a stable state that represents a solution to the problem instance. This approach differs from using hardware accelerators to speed up the execution of deterministic algorithms, as it exploits stabilisation properties of circuits with feedback, and ...

Algorithms for NP-hard Optimization Problems and Cluster Analysis by Nan Li The set cover problem, weighted set cover problem, minimum dominating set problem and minimum weighted dominating set problem are all classical NP-hard optimization problems of great importance in both theory and real applications. Since the exact algorithms, which require exhaustive exploration of exponentially many op...

The paper describes a novel algorithm, inspired by the phenomenon of wisdom of crowds, for solving instances of NP-hard problems. The proposed approach achieves superior performance compared to the genetic algorithm-based approach and requires modest computational resources. On average, a 6%–9% improvement in quality of solutions has been observed.

NP-hard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in<2 (more generally, in<d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (distance between nodes (x1, y1) and (x2, y2) is ((x1−x2)+(y1−y2))) then the problem is called Euclidean...