In this report, we will introduce an optimization problem called Maximum Density Set Cover Problem. We firstly prove it is a NP-hard problem and then we analyse its approximation algorithms. The performance guarantee of one of the approximation algorithms is θ( √ n). Experiment is conducted to verify the algorithms’ performance.

A common subproblem of DNF approximate counting and derandomizing RL is the discrepancy problem for combinatorial rectangles. We explicitly construct a poly(n)-size sample space that approximates the volume of any combinatorial rectangle in [n] to within o(1) error (improving on the constructions of [EGLNV92]). The construction extends the techniques of [LLSZ95] for the analogous hitting set pr...

For a polygonal region P with n vertices, a guard cover G is a set of points in P , such that any point in P can be seen from a point in G. In a colored guard cover, every element in a guard cover is assigned a color, such that no two guards with the same color have overlapping visibility regions. The Chromatic Art Gallery Problem (CAGP) asks for the minimum number of colors for which a colored...

We investigate the effect of certain natural connectivity constraints on the parameterized complexity of two fundamental graph covering problems, namely k-VERTEX COVER and k-EDGE COVER. Specifically, we impose the additional requirement that each connected component of a solution have at least t vertices (resp. edges from the solution), and call the problem t-TOTAL VERTEX COVER (resp. t-TOTAL E...

The multiple weighted hitting set problem is to find a subset of nodes in a hypergraph that hits every hyperedge in at least m nodes. We extend the problem to a notion of hypergraphs with so-called hypernodes and show that, for m = 2, it remains fixed-parameter tractable (FPT), parameterized by the number of hyperedges. This is accomplished by a nontrivial extension of the dynamic programming a...

In the double TSP with multiple stacks, a vehicle with several stacks performs a Hamiltonian circuit to pick up some items and stores them in its stacks. It then delivers every item by performing another Hamiltonian circuit while satisfying the last-in-first-out policy of its stacks. The consistency requirement ensuring that the pickup and delivery circuits can be performed by the vehicle is th...

We use a new approximation measure, the differential approximation ratio, to derive polynomial-time approximation algorithms for minimum set covering (for both weighted and unweighted cases), minimum graph coloring and bin-packing. We also propose differentialapproximation-ratio preserving reductions linking minimum coloring, minimum vertex covering by cliques, minimum edge covering by cliques ...

Abstract Consider the following variant of the set cover problem. We are given a universe U = {1, ..., n} and a collection of subsets C = {S1, ..., Sm} where Si ⊆ U . For every element u ∈ U we need to find a set φ (u) ∈ C such that u ∈ φ (u). Once we construct and fix the mapping φ : U 7→ C a subset X ⊆ U of the universe is revealed, and we need to cover all elements from X with exactly φ(X) :...

We study the complexity of the Hitting Set problem in set systems (hypergraphs) that avoid certain sub-structures. In particular, we characterize the classical and parameterized complexity of the problem when the Vapnik-Chervonenkis dimension (VC-dimension) of the input is small. VC-dimension is a natural measure of complexity of set systems. Several tractable instances of Hitting Set with a ge...

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