arXiv subjects

Randy Goebel

Publications and source records attributed to Randy Goebel.

40 records · Page 3Linked to original sources

arXiv2017

On rescheduling due to machine disruption while to minimize the total weighted completion time

We investigate a single machine rescheduling problem that arises from an unexpected machine unavailability, after the given set of jobs has already been scheduled to minimize the total weighted completion time. Such a disruption is represented as an unavailable time interval and is revealed to the production planner before any job is processed; the production planner wishes to reschedule the jobs to minimize the alteration to the originally planned schedule, which is measured as the maximum time deviation between the original and the new schedules for all the jobs. The objective function in this rescheduling problem is to minimize the sum of the total weighted completion time and the weighted maximum time deviation, under the constraint that the maximum time deviation is bounded above by a given value. That is, the maximum time deviation is taken both as a constraint and as part of the objective function. We present a pseudo-polynomial time exact algorithm and a fully polynomial time approximation scheme, the latter of which is the best possible given that the general problem is NP-hard.

cs.DS↗

arXiv2017

Approximation algorithms for the vertex happiness

We investigate the maximum happy vertices (MHV) problem and its complement, the minimum unhappy vertices (MUHV) problem. We first show that the MHV and MUHV problems are a special case of the supermodular and submodular multi-labeling (Sup-ML and Sub-ML) problems, respectively, by re-writing the objective functions as set functions. The convex relaxation on the Lovász extension, originally presented for the submodular multi-partitioning (Sub-MP) problem, can be extended for the Sub-ML problem, thereby proving that the Sub-ML (Sup-ML, respectively) can be approximated within a factor of $2 - \frac{2}{k}$ ($\frac{2}{k}$, respectively). These general results imply that the MHV and the MUHV problems can also be approximated within $\frac{2}{k}$ and $2 - \frac{2}{k}$, respectively, using the same approximation algorithms. For MHV, this $\frac{2}{k}$-approximation algorithm improves the previous best approximation ratio $\max \{\frac{1}{k}, \frac{1}{Δ+ 1}\}$, where $Δ$ is the maximum vertex degree of the input graph. We also show that an existing LP relaxation is the same as the concave relaxation on the Lovász extension for the Sup-ML problem; we then prove an upper bound of $\frac{2}{k}$ on the integrality gap of the LP relaxation. These suggest that the $\frac{2}{k}$-approximation algorithm is the best possible based on the LP relaxation. For MUHV, we formulate a novel LP relaxation and prove that it is the same as the convex relaxation on the Lovász extension for the Sub-ML problem; we then show a lower bound of $2 - \frac{2}{k}$ on the integrality gap of the LP relaxation. Similarly, these suggest that the $(2 - \frac{2}{k})$-approximation algorithm is the best possible based on the LP relaxation. Lastly, we prove that this $(2 - \frac{2}{k})$-approximation is optimal for the MUHV problem, assuming the Unique Games Conjecture.

cs.DS↗

arXiv2013

An approximation algorithm for the Bandpass-2 problem

The general Bandpass-$B$ problem is NP-hard and can be approximated by a reduction into the weighted $B$-set packing problem, with a worst case performance ratio of $O(B^2)$. When $B = 2$, a maximum weight matching gives a 2-approximation to the problem. In this paper, we call the Bandpass-2 problem simply the Bandpass problem. The Bandpass problem can be viewed as a variation of the maximum traveling salesman problem, in which the edge weights are dynamic rather than given at the front. We present a ${426}{227}$-approximation algorithm for the problem. Such an improved approximation is built on an intrinsic structural property proven for the optimal solution and several novel schemes to partition a $b$-matching into desired matchings.

cs.DS↗

arXiv2013

Integrating Probabilistic, Taxonomic and Causal Knowledge in Abductive Diagnosis

We propose an abductive diagnosis theory that integrates probabilistic, causal and taxonomic knowledge. Probabilistic knowledge allows us to select the most likely explanation; causal knowledge allows us to make reasonable independence assumptions; taxonomic knowledge allows causation to be modeled at different levels of detail, and allows observations be described in different levels of precision. Unlike most other approaches where a causal explanation is a hypothesis that one or more causative events occurred, we define an explanation of a set of observations to be an occurrence of a chain of causation events. These causation events constitute a scenario where all the observations are true. We show that the probabilities of the scenarios can be computed from the conditional probabilities of the causation events. Abductive reasoning is inherently complex even if only modest expressive power is allowed. However, our abduction algorithm is exponential only in the number of observations to be explained, and is polynomial in the size of the knowledge base. This contrasts with many other abduction procedures that are exponential in the size of the knowledge base.

cs.AI↗