In this paper, we study the two-facility location game with optional preference where the acceptable set of facilities for each agent could be different and an agent's cost is his distance to the closest facility within his acceptable set. The objective is to minimize the total cost of all agents while achieving strategyproofness. For general metrics, we design a deterministic strategyproof mechanism for the problem with approximation ratio of 1+2alpha, where alpha is the approximation ratio of the optimization version. In particular, for the setting on a line, we improve the earlier best ratio of n/2+1 to a ratio of 2.75.

The Elateridae family (click beetles) represents a highly diverse lineage that possesses a specialized clicking mechanism to startle predators. At present, however, phylogenetic relationships, especially among recognized subfamilies, remain contentious. Mitochondrial genomes (mitogenomes) can help resolve previously intractable phylogenetic relationships using morphological or limited molecular data. Here, we report the complete mitogenome of Pectocera sp. (Elateridae: Dendrometrinae: Oxynopterini), which was 15,962 bp in length and showed a typical gene number and order as most beetle mitogenomes, including 13 protein-coding genes (PCGs), 22 transfer RNA genes, 2 ribosomal RNA genes, and 1 noncoding control region (ATrich region). Comparative genomic analyses showed a high degree of feature similarity among Pectocera sp. and other click beetles. Evolutionary analysis of all PCGs based on the nonsynonymous/synonymous substitution rate ratio (ω) indicated that cox1 and atp8 exhibited the lowest and highest evolutionary rates, respectively, and that the evolutionary rates of all PCGs, except for cox3, nad2, and nad3, were lower than the average ω of click beetles.Phylogenetic analyses based on concatenated and coalescent approaches indicated that Pectocera sp. was sister to

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