58 research outputs found
Multicast Network Design Game on a Ring
In this paper we study quality measures of different solution concepts for
the multicast network design game on a ring topology. We recall from the
literature a lower bound of 4/3 and prove a matching upper bound for the price
of stability, which is the ratio of the social costs of a best Nash equilibrium
and of a general optimum. Therefore, we answer an open question posed by
Fanelli et al. in [12]. We prove an upper bound of 2 for the ratio of the costs
of a potential optimizer and of an optimum, provide a construction of a lower
bound, and give a computer-assisted argument that it reaches for any
precision. We then turn our attention to players arriving one by one and
playing myopically their best response. We provide matching lower and upper
bounds of 2 for the myopic sequential price of anarchy (achieved for a
worst-case order of the arrival of the players). We then initiate the study of
myopic sequential price of stability and for the multicast game on the ring we
construct a lower bound of 4/3, and provide an upper bound of 26/19. To the
end, we conjecture and argue that the right answer is 4/3.Comment: 12 pages, 4 figure
Minimizing the Cost of Team Exploration
A group of mobile agents is given a task to explore an edge-weighted graph
, i.e., every vertex of has to be visited by at least one agent. There
is no centralized unit to coordinate their actions, but they can freely
communicate with each other. The goal is to construct a deterministic strategy
which allows agents to complete their task optimally. In this paper we are
interested in a cost-optimal strategy, where the cost is understood as the
total distance traversed by agents coupled with the cost of invoking them. Two
graph classes are analyzed, rings and trees, in the off-line and on-line
setting, i.e., when a structure of a graph is known and not known to agents in
advance. We present algorithms that compute the optimal solutions for a given
ring and tree of order , in time units. For rings in the on-line
setting, we give the -competitive algorithm and prove the lower bound of
for the competitive ratio for any on-line strategy. For every strategy
for trees in the on-line setting, we prove the competitive ratio to be no less
than , which can be achieved by the algorithm.Comment: 25 pages, 4 figures, 5 pseudo-code
Tight bounds for online TSP on the line
We consider the online traveling salesperson problem (TSP), where requests appear online over time on the real line and need to be visited by a server initially located at the origin. We distinguish between closed and open online TSP, depending on whether the server eventually needs to return to the origin or not. While online TSP on the line is a very natural online problem that was introduced more than two decades ago, no tight competitive analysis was known to date. We settle this problem by providing tight bounds on the competitive ratios for both the closed and the open variant of the problem. In particular, for closed online TSP, we provide a 1.64-competitive algorithm,thus matching a known lower bound. For open online TSP, we give a new upper bound as well as a matching lower bound that establish the remarkable competitive ratio of 2.04. Additionally, we consider the online Dial-A-Ride problem on the line, where each request needs to be transported to a specified destination. We provide an improved non-preemptive lower bound of 1.75 for this setting, as well as an improved preemptive algorithm with competitive ratio 2.41.Finally, we generalize known and give new complexity results for the underlying offline problems. In particular, we give an algorithm with running time O(n2) for closed offline TSP on the line with release dates and show that both variants of offline Dial-A-Ride on the line are NP-hard for any capacity c≥2 of the server
Degree-constrained orientations of embedded graphs
Abstract. We investigate the problem of orienting the edges of an em-bedded graph in such a way that the in-degrees of both the nodes and faces meet given values. We show that the number of feasible solutions is bounded by 22g, where g is the genus of the embedding, and all so-lutions can be determined within time O(22g|E|2 + |E|3). In particular, for planar graphs the solution is unique if it exists, and in general the problem of finding a feasible orientation is fixed-parameter tractable in g. In sharp contrast to these results, we show that the problem becomes NP-complete even for a fixed genus if only upper and lower bounds on the in-degrees are specified instead of exact values.
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