Measure preserving homomorphisms and independent sets in tensor graph powers

In this note, we study the behavior of independent sets of maximum probability measure in tensor graph powers. To do this, we introduce an upper bound using measure preserving homomorphisms. This work extends some previous results about independence ratios of tensor graph powers.Comment: 5 page

Approximation Algorithms for Minimum-Load k-Facility Location

We consider a facility-location problem that abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. Formally, we consider the minimum-load k-facility location (MLkFL) problem, which is defined as follows. We have a set F of facilities, a set C of clients, and an integer k > 0. Assigning client j to a facility f incurs a connection cost d(f, j). The goal is to open a set F\u27 of k facilities, and assign each client j to a facility f(j) in F\u27 so as to minimize maximum, over all facilities in F\u27, of the sum of distances of clients j assigned to F\u27 to F\u27. We call this sum the load of facility f. This problem was studied under the name of min-max star cover in [6, 2], who (among other results) gave bicriteria approximation algorithms for MLkFL for when F = C. MLkFL is rather poorly understood, and only an O(k)-approximation is currently known for MLkFL, even for line metrics. Our main result is the first polynomial time approximation scheme (PTAS) for MLkFL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that MLkFL is strongly NP-hard on line metrics. We also devise a quasi-PTAS for MLkFL on tree metrics. MLkFL turns out to be surprisingly challenging even on line metrics, and resilient to attack by the variety of techniques that have been successfully applied to facility-location problems. For instance, we show that: (a) even a configuration-style LP-relaxation has a bad integrality gap; and (b) a multi-swap k-median style local-search heuristic has a bad locality gap. Thus, we need to devise various novel techniques to attack MLkFL. Our PTAS for line metrics consists of two main ingredients. First, we prove that there always exists a near-optimal solution possessing some nice structural properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP) encoding the problem, and argue that a MILP-solution minimizing a certain potential function possesses the desired structure, and then use a rounding algorithm for the generalized-assignment problem to "transfer" this structure to the rounded integer solution. Complementing this, we show that these structural properties enable one to find such a structured solution via dynamic programming

USING MACHINE TRANSLATION FOR QUERY REWRITE GENERATION

A query rewrite system can be used to improve query understanding of search engines by generating multiple reformulations of the same query using machine translation. The query rewrite system receives an initial query from a user. It then retrieves known related queries to the initial query from a database. Subsequently, the system generates variant queries from the query and the related queries by passing them through a monolingual machine translator. Thereafter, the system ranks the variant queries according to predefined parameters and transmits one or more of the ranked variant queries to a search engine for further processing

A New Learning Algorithm for the MAXQ Hierarchical Reinforcement Learning Method

One of the most effective methods in hierarchical reinforcement learning is MAXQ method introduced in [1]. Although this method is shown to be effective in many applications, it is computationally expensive in applications with deep hierarchy [2], which makes it impractical for use in such applications. In this paper, we propose a new learning algorithm for MAXQ method to address the open problem of reducing its computational complexity. This new algorithm, which is an improved version of MAXQ-Q learning algorithm [2], learns value functions instead of computing them with a complete search of all paths thorough the MAXQ graph. We use the new learning algorithm to solve some instances of the simple Taxi Domain Problem. In this domain, our experimental results show that the new learning algorithm always converges to optimal policy, its convergence behavior is similar to MAXQ-Q learning algorithm, and as it is expected, its overall running time is less than MAXQ-Q learning algorithm