Shopping in VR malls has been regarded as a paradigm shift for E-commerce, but most of the conventional VR shopping platforms are designed for a single user. In this paper, we envisage a scenario of VR group shopping, which brings major advantages over conventional group shopping in brick-and-mortar stores and Web shopping: 1) configure flexible display of items and partitioning of subgroups to address individual interests in the group, and 2) support social interactions in the subgroups to boost sales. Accordingly, we formulate the Social-aware VR Group-Item Configuration (SVGIC) problem to configure a set of displayed items for flexibly partitioned subgroups of users in VR group shopping. We prove SVGIC is APX-hard and also NP-hard to approximate within [EQUATION]. We design a 4-approximation algorithm based on the idea of Co-display Subgroup Formation (CSF) to configure proper items for display to different subgroups of friends. Experimental results on real VR datasets and a user study with hTC VIVE manifest that our algorithms outperform baseline approaches by at least 30.1% of solution quality.
We model the societal task of redistricting political districts as a partitioning problem: Given a set of n points in the plane, each belonging to one of two parties, and a parameter k, our goal is to compute a partition Π of the plane into regions so that each region contains roughly σ = n/k points. Π should satisfy a notion of "local" fairness, which is related to the notion of core, a well-studied concept in cooperative game theory. A region is associated with the majority party in that region, and a point is unhappy in Π if it belongs to the minority party. A group D of roughly σ contiguous points is called a deviating group with respect to Π if majority of points in D are unhappy in Π. The partition Π is locally fair if there is no deviating group with respect to Π.This paper focuses on a restricted case when points lie in 1D. The problem is non-trivial even in this case. We consider both adversarial and "beyond worst-case" settings for this problem. For the former, we characterize the input parameters for which a locally fair partition always exists; we also show that a locally fair partition may not exist for certain parameters. We then consider input models where there are "runs" of red and blue points. For such clustered inputs, we show that a locally fair partition may not exist for certain values of σ, but an approximate locally fair partition exists if we allow some regions to have smaller sizes. We finally present a polynomial-time algorithm for computing a locally fair partition if one exists.
Research on new queries for finding dense subgraphs and groups has been actively pursued due to their many applications, especially in social network analysis and graph mining. However, existing work faces two major weaknesses: i) incapability of supporting personalized neighborhood density, and ii) inability to find sparse groups. To tackle the above issues, we propose a new query, called Density-Customized Social Group Query (DCSGQ), that accommodates the need for personalized density by allowing individual users to flexibly configure their social tightness (and sparseness) for the target group. The proposed DCSGQ is general due to flexible in configuration of neighboring social density in queries. We prove the NP-hardness and inapproximability of DCSGQ, formulate an Integer Program (IP) as a baseline, and propose an efficient algorithm, FSGSel-RR, by relaxing the IP. We then propose a fixed-parameter tractable algorithm with a performance guarantee, named FSGSel-TD, and further combine it with FSGSel-RR into a hybrid approach, named FSGSel-Hybrid, in order to strike a good balance between solution quality and efficiency. Extensive experiments on multiple large real datasets demonstrate the superior solution quality and efficiency of our approaches over existing subgraph and group queries.
Shopping in VR malls has been regarded as a paradigm shift for E-commerce, but most of the conventional VR shopping platforms are designed for a single user. In this paper, we envisage a scenario of VR group shopping, which brings major advantages over conventional group shopping in brickand-mortar stores and Web shopping: 1) configure flexible display of items and partitioning of subgroups to address individual interests in the group, and 2) support social interactions in the subgroups to boost sales. Accordingly, we formulate the Social-aware VR Group-Item Configuration (SVGIC) problem to configure a set of displayed items for flexibly partitioned subgroups of users in VR group shopping. We prove SVGIC is APX-hard and also NP-hard to approximate within 32 31 − . We design an approximation algorithm based on the idea of Co-display Subgroup Formation (CSF) to configure proper items for display to different subgroups of friends. Experimental results on real VR datasets and a user study with hTC VIVE manifest that our algorithms outperform baseline approaches by at least 30.1% of solution quality.
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