In the obnoxious facility game, we design mechanisms that output a location of an undesirable facility based on the locations of players reported by themselves. The benefit of a player is defined to be the distance between her location and the facility. A player may try to manipulate the output of the mechanism by strategically misreporting her location. We wish to design a λ-group strategy-proof mechanism i.e., for every group of players, at least one player in the group cannot gain strictly more than λ times her primary benefit by having the entire group change their reports simultaneously. In this paper, we design a k-candidate λ-group strategy-proof mechanism for the obnoxious facility game in the metric defined by k half lines with a common endpoint such that each candidate is a point in each of the half-lines at the same distance to the common endpoint as other candidates. Then, we show that the benefit ratio of the mechanism is at most 1+2/(k-1)λ. Finally, we prove that the bound is nearly tight.

Recent research on graph drawing focuses on Right-Angle-Crossing (RAC) drawings of 1-plane graphs, where each edge is drawn as a straight line and two crossing edges only intersect at right angles. We give a transformation from a restricted case of the RAC drawing problem to a problem of finding a straight-line drawing of a maximal plane graph where some angles are required to be acute. For a restricted version of the latter problem, we show necessary and sufficient conditions for such a drawing to exist, and design an O(n2)-time algorithm that given an n-vertex plane graph produces a desired drawing of the graph or reports that none exists.

In this paper, we consider a problem of simultaneously optimizing a sequence of graphs and a route which exhaustively visits the vertices from each pair of successive graphs in the sequence. This type of problem arises from repetitive routing of grasp-and-delivery robots used in the production of printed circuit boards. The problem is formulated as follows. We are given a metric graph G*=(V*,E*), a set of m+1 disjoint subsets Ci ⊆ V* of vertices with |Ci|=n, i=0,1,...,m, and a starting vertex s ∈ C0. We seek to find a sequence π=(Ci1, Ci2, ..., Cim) of the subsets of vertices and a shortest walk P which visits all (m+1)n vertices in G* in such a way that after starting from s, the walk alternately visits the vertices in Cik-1 and Cik, for k=1,2,...,m (i0=0). Thus, P is a walk with m(2n-1) edges obtained by concatenating m alternating Hamiltonian paths between Cik-1 and Cik, k=1,2,...,m. In this paper, we show that an approximate sequence of subsets of vertices and an approximate walk with at most three times the optimal route length can be found in polynomial time.

The family of stable matching problems have been well-studied across a wide field of research areas, including economics, mathematics and computer science. In general, an instance of a stable matching problem is given by a set of participants who have expressed their preferences of each other, and asks to find a “stable” matching, that is, a pairing of the participants such that no unpaired participants prefer each other to their assigned partners. In the case of the Stable Roommates Problem (SR), it is known that given an even number n of participants, there might not exist a stable matching that pairs all of the participants, but there exist efficient algorithms to determine if this is possible or not, and if it is possible, produce such a matching. Common extensions of SR allow for the participants' preference lists to be incomplete, or include indifference. Allowing indifference in turn, gives rise to different possible definitions of stability, super, strong, and weak stability. While instances asking for super and strongly stable matching can be efficiently solved even if preference lists are incomplete, the case of weak stability is NP-complete. We examine a restricted case of indifference, introducing the concept of unranked entries. For this type of instances, we show that the problem of finding a weakly stable matching remains NP-complete even if each participant has a complete preference list with at most two unranked entries, or is herself unranked for up to three other participants. On the other hand, for instances where there are m acceptable pairs and there are in total k unranked entries in all of the participants' preference lists, we propose an O(2kn2)-time and polynomial space algorithm that finds a stable matching, or determines that none exists in the given instance.

Given an edge-weighted bipartite digraph G=(A,B;E), the Bipartite Traveling Salesman Problem (BTSP) asks to find the minimum cost of a Hamiltonian cycle of G, or determine that none exists. When |A|=|B|=n, the BTSP can be solved using polynomial space in O*(42nnlog n) time by using the divide-and-conquer algorithm of Gurevich and Shelah (SIAM Journal of Computation, 16(3), pp.486-502, 1987). We adapt their algorithm for the bipartite case, and show an improved time bound of O*(42n), saving the nlog n factor.