Consider a digraph where the vertices represent people and an arc (i, j) represents the possibility of i giving a gift to j. The basic question we pose is whether there is an anonymity-preserving “gift assignment” such that each person makes and receives exactly one gift, and such that no person i can infer the remaining gift assignments from the fact that i is assigned to give a gift to j. We ...

The Set Cover problem belongs to a group of hard problems which are neither approximable in polynomial time (at least with a constant factor) nor fixed parameter tractable, under widely believed complexity assumptions. In recent years, many researchers design exact exponential-time algorithms for problems of that kind. The goal is getting the time complexity still of order O(c), but with the co...

In the hitting set problem one is given m subsets of a nite set N and one has to nd an X N of minimum cardinality that hits intersects all of them The problem is NP hard It is not known whether there exists a polynomial time approximation algorithm for the hitting set problem with a nite performance ratio Special cases of the hitting set problem are described for which nite performance ratios a...

In this paper we study problems where one has to prove that certain graph parameter attains its maximum at the star and its minimum at the path among the trees on a fixed number of vertices. We give many applications of the socalled generalized tree shift which seems to be a powerful tool to attack the problems of the above mentioned kind. We show that the generalized tree shift increases the l...

In the Vertex Cover Reconfiguration (VCR) problem, given graph G = (V,E), positive integers k and `, and two vertex covers S and T of G of size at most k, we determine whether S can be transformed into T by a sequence of at most ` vertex additions or removals such that each operation results in a vertex cover of size at most k. Motivated by recent results establishing the W[1]-hardness of VCR w...

We improve on the classical Nemhauser-Trotter Theorem, which is a key tool for the Minimum (Weighted) Vertex Cover problem in the design of both, approximation algorithms and exact fixedparameter algorithms. Namely, we provide in polynomial time for a graph G with vertex weights w : V → 〈0,∞) a partition of V into three subsets V0, V1, V 1 2 , with no edges between V0 and V 1 2 or within V0, su...

We revisit the traveling salesman problem with neighborhoods (TSPN) and present the first constant-ratio approximation for disks in the plane: Given a set of n disks in the plane, a TSP tour whose length is at most O(1) times the optimal can be computed in time that is polynomial in n. Our result is the first constant-ratio approximation for a class of planar convex bodies of arbitrary size and...

Given an instance of a geometric set cover problem on a set of points X and a set of objects R, the dual is a geometric hitting set problem on a set of points P and a set of objects Q, where there exists a one-to-one mapping from each xj ∈ X to a dual object Qj ∈ Q and for each Ri ∈ R to a dual point in pi ∈ P , so that a dual point pi is contained in a dual object Qj if and only if the corresp...

We describe a polynomial time O(k log log OPTk(P ))approximation algorithm for the k-guarding problem of finding a minimum number, OPTk(P ), of vertex guards of an n-vertex simple polygon P so that for every point p ∈ P , the number of guards that see p is at least the minimum of k and the number of vertices that see p. Our approach finds O ( k ε log log 1 ε ) size (k, ε)-nets for instances of ...

abstract. suppose g is an nvertex and medge simple graph with edge set e(g). an integervalued function f: e(g) → z is called a flow. tutte was introduced the flow polynomial f(g, λ) as a polynomial in an indeterminate λ with integer coefficients by f(g,λ) in this paper the flow polynomial of some dendrimers are computed.