Proof Compression and NP Versus PSPACE II: Addendum

In [3] we proved the conjecture NP = PSPACE by advanced proof theoretic methods that combined Hudelmaier's cut-free sequent calculus for minimal logic (HSC) [5] with the horizontal compressing in the corresponding minimal Prawitz-style natural deduction (ND) [6]. In this Addendum we show how to prove a weaker result NP = coNP without referring to HSC. The underlying idea (due to the second author) is to omit full minimal logic and compress only \naive" normal tree-like ND refutations of the existence of Hamiltonian cycles in given non-Hamiltonian graphs, since the Hamiltonian graph problem in NP-complete. Thus, loosely speaking, the proof of NP = coNP can be obtained by HSC-elimination from our proof of NP = PSPACE [3].

Numerical experiments with LP formulations of the maximum clique problem

The maximum clique problems calls for determining the size of the largest clique in a given graph. This graph problem affords a number of zero-one linear programming formulations. In this case study we deal with some of these formulations. We consider ways for tightening the formulations. We carry out numerical experiments to see the improvements the tightened formulations provide.

Exact Weight Perfect Matching of Bipartite Graph Problem Simplified

The study of perebor dates back to the Soviet-era mathematics, especially in the 1980s [1]. Post-Soviet mathematicians has been working on many problems in combinatorial optimization. One of them is Exact Weight Perfect Matching of Bipartite Graph (EWPM). This particular problem has been thoroughly considered by [2], [3], [4]. In this note, we give a simpler proof about the solvability of EWPM.

An Average Case NP-complete Graph Colouring Problem

NP-complete problems should be hard on some instances but those may be extremely rare. On generic instances many such problems, especially related to random graphs, have been proved to be easy. We show the intractability of random instances of a graph colouring problem: this graph problem is hard on average unless all NP problems under all samplable (i.e. generatable in polynomial time) distributions are easy. Worst case reductions use special gadgets and typically map instances into a negligible fraction of possible outputs. Ours must output nearly random graphs and avoid any super-polynomial distortion of probabilities. This poses significant technical difficulties.