2,287 research outputs found
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
Metric dimension for random graphs
The metric dimension of a graph is the minimum number of vertices in a
subset of the vertex set of such that all other vertices are uniquely
determined by their distances to the vertices in . In this paper we
investigate the metric dimension of the random graph for a wide range
of probabilities
Polynomial Kernels for Weighted Problems
Kernelization is a formalization of efficient preprocessing for NP-hard
problems using the framework of parameterized complexity. Among open problems
in kernelization it has been asked many times whether there are deterministic
polynomial kernelizations for Subset Sum and Knapsack when parameterized by the
number of items.
We answer both questions affirmatively by using an algorithm for compressing
numbers due to Frank and Tardos (Combinatorica 1987). This result had been
first used by Marx and V\'egh (ICALP 2013) in the context of kernelization. We
further illustrate its applicability by giving polynomial kernels also for
weighted versions of several well-studied parameterized problems. Furthermore,
when parameterized by the different item sizes we obtain a polynomial
kernelization for Subset Sum and an exponential kernelization for Knapsack.
Finally, we also obtain kernelization results for polynomial integer programs
The distributions of functions related to parametric integer optimization
We consider the asymptotic distribution of the IP sparsity function, which
measures the minimal support of optimal IP solutions, and the IP to LP distance
function, which measures the distance between optimal IP and LP solutions. We
create a framework for studying the asymptotic distribution of general
functions related to integer optimization. There has been a significant amount
of research focused around the extreme values that these functions can attain,
however less is known about their typical values. Each of these functions is
defined for a fixed constraint matrix and objective vector while the right hand
sides are treated as input. We show that the typical values of these functions
are smaller than the known worst case bounds by providing a spectrum of
probability-like results that govern their overall asymptotic distributions.Comment: Accepted for journal publicatio
On Colorful Bin Packing Games
We consider colorful bin packing games in which selfish players control a set
of items which are to be packed into a minimum number of unit capacity bins.
Each item has one of colors and cannot be packed next to an item of
the same color. All bins have the same unitary cost which is shared among the
items it contains, so that players are interested in selecting a bin of minimum
shared cost. We adopt two standard cost sharing functions: the egalitarian cost
function which equally shares the cost of a bin among the items it contains,
and the proportional cost function which shares the cost of a bin among the
items it contains proportionally to their sizes. Although, under both cost
functions, colorful bin packing games do not converge in general to a (pure)
Nash equilibrium, we show that Nash equilibria are guaranteed to exist and we
design an algorithm for computing a Nash equilibrium whose running time is
polynomial under the egalitarian cost function and pseudo-polynomial for a
constant number of colors under the proportional one. We also provide a
complete characterization of the efficiency of Nash equilibria under both cost
functions for general games, by showing that the prices of anarchy and
stability are unbounded when while they are equal to 3 for black and
white games, where . We finally focus on games with uniform sizes (i.e.,
all items have the same size) for which the two cost functions coincide. We
show again a tight characterization of the efficiency of Nash equilibria and
design an algorithm which returns Nash equilibria with best achievable
performance
- …