In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

[Q]: The theory of actual causality, defined by Halpern and Pearl, and its quantitative measure – the degree of responsibility – was shown to be extremely useful in various areas of computer science due to a good match between the results it produces and our intuition. In this paper, I describe the applications of causality to formal verification, namely, explanation of counter-examples, refinement of coverage metrics, and symbolic trajectory evaluation. I also briefly discuss recent applications of causality to legal reasoning.
[A]: Causality and Responsibility for Formal Verification and Beyond


[Q]: Clustering is a useful data exploratory method with its wide applicability in multiple fields. However, data clustering greatly relies on initialization of cluster centers that can result in large intra-cluster variance and dead centers, therefore leading to sub-optimal solutions. This paper proposes a novel variance based version of the conventional Moving K-Means (MKM) algorithm called Variance Based Moving K-Means (VMKM) that can partition data into optimal homogeneous clusters, irrespective of cluster initialization. The algorithm utilizes a novel distance metric and a unique data element selection criteria to transfer the selected elements between clusters to achieve low intra-cluster variance and subsequently avoid dead centers. Quantitative and qualitative comparison with various clustering techniques is performed on four datasets selected from image processing, bioinformatics, remote sensing and the stock market respectively. An extensive analysis highlights the superior performance of the proposed method over other techniques.
[A]: Variance Based Moving K-Means Algorithm


[Q]: In some domestic professional sports leagues, the home stadiums are located in cities connected by a common train line running in one direction. For these instances, we can incorporate this geographical information to determine optimal or nearly-optimal solutions to the n-team Traveling Tournament Problem (TTP), an NP-hard sports scheduling problem whose solution is a double round-robin tournament schedule that minimizes the sum total of distances traveled by all n teams. We introduce the Linear Distance Traveling Tournament Problem (LD-TTP), and solve it for n = 4 and n = 6, generating the complete set of possible solutions through elementary combinatorial techniques. For larger n, we propose a novel “expander construction” that generates an approximate solution to the LD-TTP. For n ≡ 4 (mod 6), we show that our expander construction produces a feasible double round-robin tournament schedule whose total distance is guaranteed to be no worse than 4 3 times the optimal solution, regardless of where the n teams are located. This 4 3 -approximation for the LD-TTP is stronger than the currently best-known ratio of 5 3 + for the general TTP. We conclude the paper by applying this linear distance relaxation to general (nonlinear) n-team TTP instances, where we develop fast approximate solutions by simply “assuming” the n teams lie on a straight line and solving the modified problem. We show that this technique surprisingly generates the distance-optimal tournament on all benchmark sets on 6 teams, as well as close-to-optimal schedules for larger n, even when the teams are located around a circle or positioned in three-dimensional space.
[A]:
Generating Approximate Solutions to the Traveling Tournament Problem using a Linear Distance Relaxation