Reducing the NP-problems to the convex/linear analysis on the Birkhoff polytope. Introduction Since the classical works of J. Edmonds [2, and others], linear modeling became a common technique in combinatorial optimization [8, 9, 13, 14, 15, 16, and others]. Often, the linear models are expressed with some constrains on the incidence vector. The major benefit of this approach is the symmetry of...

We study the effects of faulty data on NP-hard sets. We consider hard sets for several polynomial time reductions, add corrupt data and then analyze whether the resulting sets are still hard for NP. We explain that our results are related to a weakened deterministic variant of the notion of program self-correction by Blum, Luby, and Rubinfeld. Among other results, we use the Left-Set technique ...

The aim of edge modification problems is to change the edge set of a given graph as little as possible in order to satisfy a certain property. Edge modification problems in graphs have a lot of applications in different areas, and many polynomial-time algorithms and NP-completeness proofs for this kind of problems are known. In this work we prove new NP-completeness results for these problems i...

Abstract This paper describes the formal verification of NP-hardness reduction functions two key problems relevant in algebraic lattice theory: closest vector problem and shortest problem, both infinity norm. The formalization uncovered a number with existing proofs literature. how these were corrected formalization. work was carried out proof assistant Isabelle.

In this paper we define the exact k-coverage problem, and study it for the special cases of intervals and circular-arcs. Given a set system consisting of a ground set of n points with integer demands {d0, . . . , dn−1} and integer rewards, subsets of points, and an integer k, select up to k subsets such that the sum of rewards of the covered points is maximized, where point i is covered if exac...

Geometric covering is a well-studied topic in computational geometry. We study three covering problems: Disjoint Unit-Disk Cover, Depth-(≤ K) Packing and Red-Blue UnitSquare Cover. In the Disjoint Unit-Disk Cover problem, we are given a point set and want to cover the maximum number of points using disjoint unit disks. We prove that the problem is NP-complete and give a polynomial-time approxim...

For a given graph G, the Separator Problem asks whether a vertex or edge set of small cardinality (or weight) exists whose removal partitions G into two disjoint graphs of approximately equal sizes. Called the Vertex Separator Problem when the removed set is a vertex set, and the Edge Separator Problem when it is an edge set, both problems are NP-complete for general unweighted graphs [6]. Desp...

Introduction The vast majority of scientific community believes that P!=NP, with countless supporting arguments. The number of people who believe otherwise probably amounts to as few as those opposing the 2 Law of Thermodynamics. But isn’t nature elegant enough, not to resource to brute-force search? In this article, a novel concept of dimensionality is presented, which may lead to a more effic...