7,215 research outputs found
Fast Convex Decomposition for Truthful Social Welfare Approximation
Approximating the optimal social welfare while preserving truthfulness is a
well studied problem in algorithmic mechanism design. Assuming that the social
welfare of a given mechanism design problem can be optimized by an integer
program whose integrality gap is at most , Lavi and Swamy~\cite{Lavi11}
propose a general approach to designing a randomized -approximation
mechanism which is truthful in expectation. Their method is based on
decomposing an optimal solution for the relaxed linear program into a convex
combination of integer solutions. Unfortunately, Lavi and Swamy's decomposition
technique relies heavily on the ellipsoid method, which is notorious for its
poor practical performance. To overcome this problem, we present an alternative
decomposition technique which yields an approximation
and only requires a quadratic number of calls to an integrality gap verifier
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
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