35 research outputs found
Approximating the Regular Graphic TSP in near linear time
We present a randomized approximation algorithm for computing traveling
salesperson tours in undirected regular graphs. Given an -vertex,
-regular graph, the algorithm computes a tour of length at most
, with high probability, in time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS
2012) for the same problem, in terms of both approximation factor, and running
time. The key ingredient of our algorithm is a technique that uses
edge-coloring algorithms to sample a cycle cover with cycles with
high probability, in near linear time.
Additionally, we also give a deterministic
factor approximation algorithm
running in time .Comment: 12 page
Metrical Service Systems with Multiple Servers
We study the problem of metrical service systems with multiple servers
(MSSMS), which generalizes two well-known problems -- the -server problem,
and metrical service systems. The MSSMS problem is to service requests, each of
which is an -point subset of a metric space, using servers, with the
objective of minimizing the total distance traveled by the servers.
Feuerstein initiated a study of this problem by proving upper and lower
bounds on the deterministic competitive ratio for uniform metric spaces. We
improve Feuerstein's analysis of the upper bound and prove that his algorithm
achieves a competitive ratio of . In the randomized
online setting, for uniform metric spaces, we give an algorithm which achieves
a competitive ratio , beating the deterministic lower
bound of . We prove that any randomized algorithm for
MSSMS on uniform metric spaces must be -competitive. We then
prove an improved lower bound of on
the competitive ratio of any deterministic algorithm for -MSSMS, on
general metric spaces. In the offline setting, we give a pseudo-approximation
algorithm for -MSSMS on general metric spaces, which achieves an
approximation ratio of using servers. We also prove a matching
hardness result, that a pseudo-approximation with less than servers is
unlikely, even for uniform metric spaces. For general metric spaces, we
highlight the limitations of a few popular techniques, that have been used in
algorithm design for the -server problem and metrical service systems.Comment: 18 pages; accepted for publication at COCOON 201
On Randomized Memoryless Algorithms for the Weighted -server Problem
The weighted -server problem is a generalization of the -server problem
in which the cost of moving a server of weight through a distance
is . The weighted server problem on uniform spaces models
caching where caches have different write costs. We prove tight bounds on the
performance of randomized memoryless algorithms for this problem on uniform
metric spaces. We prove that there is an -competitive memoryless
algorithm for this problem, where ;
. On the other hand we also prove that no randomized memoryless
algorithm can have competitive ratio better than .
To prove the upper bound of we develop a framework to bound from
above the competitive ratio of any randomized memoryless algorithm for this
problem. The key technical contribution is a method for working with potential
functions defined implicitly as the solution of a linear system. The result is
robust in the sense that a small change in the probabilities used by the
algorithm results in a small change in the upper bound on the competitive
ratio. The above result has two important implications. Firstly this yields an
-competitive memoryless algorithm for the weighted -server problem
on uniform spaces. This is the first competitive algorithm for which is
memoryless. Secondly, this helps us prove that the Harmonic algorithm, which
chooses probabilities in inverse proportion to weights, has a competitive ratio
of .Comment: Published at the 54th Annual IEEE Symposium on Foundations of
Computer Science (FOCS 2013
On 2-coloring certain k-uniform hypergraphs
AbstractWe give some sufficient conditions for the existence of a 2-coloring for k-uniform hypergraphs. When the number of edges equals the size of the base set of the hypergraph, these conditions are based on the permanent of the incidence matrix
Depth-3 arithmetic circuits for S<SUB>n</SUB><SUP>2</SUP>(X) and extensions of the Graham-Pollack theorem
We consider the problem of computing the second elementary symmetric polynomial S2n (X) = Δ ∑ 1≤ i< j≤n XiXj using depth-three arithmetic circuits of the form ∑ri=1 Πsij=1 Lij(X), where each Lij is a linear form in X1, . . . ,Xn. We consider this problem over several fields and determine exactly the number of multiplication gates required. The lower bounds are proved for inhomogeneous circuits where the Lij's are allowed to have constants; the upper bounds are proved in the homogeneous model. For reals and rationals, the number of multiplication gates required is exactly n-1; in most other cases, it is [n/2]. This problem is related to the Graham-Pollack theorem in algebraic graph theory. In particular, our results answer the following question of Babai and Frankl: what is the minimum number of complete bipartite graphs required to cover each edge of a complete graph an odd number of times? We show that for infinitely many n, the answer is [n/2]