80,683 research outputs found
Multiflow Transmission in Delay Constrained Cooperative Wireless Networks
This paper considers the problem of energy-efficient transmission in
multi-flow multihop cooperative wireless networks. Although the performance
gains of cooperative approaches are well known, the combinatorial nature of
these schemes makes it difficult to design efficient polynomial-time algorithms
for joint routing, scheduling and power control. This becomes more so when
there is more than one flow in the network. It has been conjectured by many
authors, in the literature, that the multiflow problem in cooperative networks
is an NP-hard problem. In this paper, we formulate the problem, as a
combinatorial optimization problem, for a general setting of -flows, and
formally prove that the problem is not only NP-hard but it is
inapproxmiable. To our knowledge*, these results provide
the first such inapproxmiablity proof in the context of multiflow cooperative
wireless networks. We further prove that for a special case of k = 1 the
solution is a simple path, and devise a polynomial time algorithm for jointly
optimizing routing, scheduling and power control. We then use this algorithm to
establish analytical upper and lower bounds for the optimal performance for the
general case of flows. Furthermore, we propose a polynomial time heuristic
for calculating the solution for the general case and evaluate the performance
of this heuristic under different channel conditions and against the analytical
upper and lower bounds.Comment: 9 pages, 5 figure
Asymmetric Traveling Salesman Path and Directed Latency Problems
We study integrality gaps and approximability of two closely related problems
on directed graphs. Given a set V of n nodes in an underlying asymmetric metric
and two specified nodes s and t, both problems ask to find an s-t path visiting
all other nodes. In the asymmetric traveling salesman path problem (ATSPP), the
objective is to minimize the total cost of this path. In the directed latency
problem, the objective is to minimize the sum of distances on this path from s
to each node. Both of these problems are NP-hard. The best known approximation
algorithms for ATSPP had ratio O(log n) until the very recent result that
improves it to O(log n/ log log n). However, only a bound of O(sqrt(n)) for the
integrality gap of its linear programming relaxation has been known. For
directed latency, the best previously known approximation algorithm has a
guarantee of O(n^(1/2+eps)), for any constant eps > 0. We present a new
algorithm for the ATSPP problem that has an approximation ratio of O(log n),
but whose analysis also bounds the integrality gap of the standard LP
relaxation of ATSPP by the same factor. This solves an open problem posed by
Chekuri and Pal [2007]. We then pursue a deeper study of this linear program
and its variations, which leads to an algorithm for the k-person ATSPP (where k
s-t paths of minimum total length are sought) and an O(log n)-approximation for
the directed latency problem
A Combinatorial, Strongly Polynomial-Time Algorithm for Minimizing Submodular Functions
This paper presents the first combinatorial polynomial-time algorithm for
minimizing submodular set functions, answering an open question posed in 1981
by Grotschel, Lovasz, and Schrijver. The algorithm employs a scaling scheme
that uses a flow in the complete directed graph on the underlying set with each
arc capacity equal to the scaled parameter. The resulting algorithm runs in
time bounded by a polynomial in the size of the underlying set and the largest
length of the function value. The paper also presents a strongly
polynomial-time version that runs in time bounded by a polynomial in the size
of the underlying set independent of the function value.Comment: 17 page
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