40 research outputs found
Randomized online computation with high probability guarantees
We study the relationship between the competitive ratio and the tail
distribution of randomized online minimization problems. To this end, we define
a broad class of online problems that includes some of the well-studied
problems like paging, k-server and metrical task systems on finite metrics, and
show that for these problems it is possible to obtain, given an algorithm with
constant expected competitive ratio, another algorithm that achieves the same
solution quality up to an arbitrarily small constant error a with high
probability; the "high probability" statement is in terms of the optimal cost.
Furthermore, we show that our assumptions are tight in the sense that removing
any of them allows for a counterexample to the theorem. In addition, there are
examples of other problems not covered by our definition, where similar high
probability results can be obtained.Comment: 20 pages, 2 figure
Advice Complexity of the Online Induced Subgraph Problem
Several well-studied graph problems aim to select a largest (or smallest)
induced subgraph with a given property of the input graph. Examples of such
problems include maximum independent set, maximum planar graph, and many
others. We consider these problems, where the vertices are presented online.
With each vertex, the online algorithm must decide whether to include it into
the constructed subgraph, based only on the subgraph induced by the vertices
presented so far. We study the properties that are common to all these problems
by investigating the generalized problem: for a hereditary property \pty, find
some maximal induced subgraph having \pty. We study this problem from the point
of view of advice complexity. Using a result from Boyar et al. [STACS 2015], we
give a tight trade-off relationship stating that for inputs of length n roughly
n/c bits of advice are both needed and sufficient to obtain a solution with
competitive ratio c, regardless of the choice of \pty, for any c (possibly a
function of n). Surprisingly, a similar result cannot be obtained for the
symmetric problem: for a given cohereditary property \pty, find a minimum
subgraph having \pty. We show that the advice complexity of this problem varies
significantly with the choice of \pty.
We also consider preemptive online model, where the decision of the algorithm
is not completely irreversible. In particular, the algorithm may discard some
vertices previously assigned to the constructed set, but discarded vertices
cannot be reinserted into the set again. We show that, for the maximum induced
subgraph problem, preemption cannot help much, giving a lower bound of
bits of advice needed to obtain competitive ratio ,
where is any increasing function bounded by \sqrt{n/log n}. We also give a
linear lower bound for c close to 1
On Semi-perfect 1-factorizations
The perfect 1-factorization conjecture by A. Kotzig [7] asserts the existence of a 1-factorization of a complete graph K2n in which any two 1-factors induce a Hamiltonian cycle. This conjecture is one of the prominent open problems in graph theory. Apart from its theoretical significance it has a number of applications, particularly in designing topologies for wireless communication. Recently, a weaker version of this conjecture has been proposed in [1] for the case of semi-perfect 1-factorizations. A semi-perfect 1-factorization is a decomposition of a graph G into distinct 1-factors F1,..., Fk such that F1 ∪ Fi forms a Hamiltonian cycle for any 1 < i ≤ k. We show that complete graphs K2n, hypercubes Q2n+1 and tori T2n×2n admit a semi-perfect 1-factorization
On Immunity and Catastrophic Indices of Graphs £
Immunity index of a graph is the least integer c1 such that each configuration of size c1 is immune. Catastrophic index of a graph is the least integer c2 such that each configuration of size c2 is catastrophic. This paper contains the first systematic study of immunity indices on a variety of interconnection networks and their distance from catastrophic indices
Randomized online computation with high probability guarantees
We study the relationship between the competitive ratio and the tail distribution of randomized online problems. To this end, we identify a broad class of online problems for which the existence of a randomized online algorithm with constant expected competitive ratio r implies the existence of a randomized online algorithm that has a competitive ratio of (1+ε)r with high probability, measured with respect to the optimal profit or cost, respectively. The class of problems includes some of the well-studied online problems such as paging, k-server, and metrical task systems on finite metric spaces.ISSN:0178-4617ISSN:1432-054