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 the Minimum Sum Edge Coloring problem we have to assign positive integers to the edges of a graph such that adjacent edges receive different integers and the sum of the assigned numbers is minimal. We show that the problem is (a) NP-hard for planar bipartite graphs with maximum degree 3, (b) NP-hard for 3-regular planar graphs, (c) NP-hard for partial 2-trees, and (d) APX-hard for bipartite ...

Much structural work on NP-complete sets has exploited SAT's d-self-reduci-bility. In this paper we exploit the additional fact that SAT is a d-cylinder to show that NP-complete sets are p-superterse unless P = NP. In fact, every set that is NP-hard under polynomial-time n o(1)-tt reductions is p-superterse unless P = NP. In particular no p-selective set is NP-hard under polynomial-time n o(1)-...

We study the sparse set conjecture for sets with low density. The sparse set conjecture states that P = NP if and only if there exists a sparse Turing hard set for NP. In this paper we study a weaker variant of the conjecture. We are interested in the consequences of NP having Tur-ing hard sets of density f (n), for (unbounded) functions f (n), that are sub-polynomial, for example log(n). We es...

For an optimization problem known to be NP-Hard, the dichotomy study investigates reduction instances determine line separating polynomial-time solvable vs NP-Hard (easy hard instances). In this paper, we investigate well-studied Hamiltonian cycle (HCYCLE), and present interesting result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete planar bipartite graphs with...

Cook's Theorem [Cormen, T.H., Leiserson, C.E., Rivest, R.L., 2001. Introduction to Algorithms, second ed., The MIT Press; Garey, M.R., Johnson, D.S., 1979. Computer and Intractability, Freeman, San Fransico, CA] is that if one algorithm for an NP-complete or an NP-hard problem will be developed, then other problems will be solved by means of reduction to that problem. Cook's Theorem has been de...