We study graph search problems with imperfect detection. Both graph search and search with imperfect detection are well-studied subjects, but the natural combination is relatively new. In this setting, one is given an edge-weighted graph and a target is hidden at one of the vertices. One can walk through the graph to search for the hidden target and the goal is to find it as soon as possible. However, just visiting a vertex is not sufficient and a search needs to be done which takes a fixed amount of time. Each search attempt is only successful though with probability γ , given that the target is at the search location. For the hider, we consider the model where the target is hidden at random, and the adversarial model where the target is placed by an adversary. For the searcher, we consider both pathwise search and expanding search. For all cases, we obtain the first constant factor approximation guarantees.
We consider the buy-at-bulk facility location problem (BBFL), a problem combining the classic facility location problem with buy-at-bulk network design, which finds motivation in telecommunication networks. In it, we are given a graph with edge lengths, opening costs and demands for each vertex, and a monotone and subadditive capacity-cost function, and our task is to open facilities on a subset of the vertices and route the demand from each vertex to these facilities. The cost of a solution (which we want to minimize) is given by the opening costs of the chosen facilities, plus the cost on each edge, which is given by its length times the cost of providing enough capacity for the demands through the edge, given by the capacity-cost function. A common variant of the problem, the k-cable facility location problem (kCFL), considers the case where capacity is provided by buying copies of given cable types, each with a certain capacity and cost. We study BBFL on tree instances and show, for the unit-demand and splittable variants, that the problem admits a PTAS (a (1+ϵ ) -approximation for any ϵ > 0 ). We also consider kCFL in the new setting of cable-unsplittable demands, where the demand of a vertex cannot be split among multiple cables. We show that the problem is NP-hard to approximate to a factor better than 3/2 on stars, and then provide an algorithm for tree instances that outputs a solution with optimal cost, but which exceeds the capacity on each cable by a factor of 1+ϵ . As a consequence, we show that the problem has a 2-approximation algorithm on trees.
In this work, we study how to maintain a forest of arborescences of maximum arc cardinality under arc insertions while minimizing recourse—the total number of arcs changed in the maintained solution. This problem is the “arborescence version” of max cardinality matching. On the impossibility side, we observe that even in this insertion-only model, it is possible for m adversarial arc arrivals to necessarily incur (m · n) recourse, matching a trivial upper bound of O(m · n) . On the possibility side, we give an algorithm with expected recourse O(m ·log ^2 n) if all m arcs arrive uniformly at random.
We study online algorithms for maximum cardinality matchings with edge arrivals in graphs of low degree. Buchbinder, Segev, and Tkach showed that no online algorithm for maximum cardinality fractional matchings can achieve a competitive ratio larger than 4/(9-√(5))≈ 0.5914 even for graphs of maximum degree three. The negative result of Buchbinder et al. holds even when the graph is bipartite and edges are revealed according to vertex arrivals, i.e. once a vertex arrives, all edges are revealed that include the newly arrived vertex and one of the previously arrived vertices. In this work, we complement the negative result of Buchbinder et al. by providing an online algorithm for maximum cardinality fractional matchings with a competitive ratio at least 4/(9-√(5))≈ 0.5914 for graphs of maximum degree three. We also demonstrate that no online algorithm for maximum cardinality integral matchings can have the competitive guarantee 0.5807, establishing a gap between integral and fractional matchings for graphs of maximum degree three. Note that the work of Buchbinder et al. shows that for graphs of maximum degree two, there is no such gap between fractional and integral matchings, because for both of them the best achievable competitive ratio is 2/3. Also, our results demonstrate that for graphs of maximum degree three best possible competitive ratios for fractional matchings are the same in the vertex arrival and in the edge arrival models.
Given a set of geometric objects, A, in ℝ^d , a set of points P is a piercing set if every object in A includes a point in P. We consider the online version of the piercing set problem where recourse is allowed. In the traditional online setting, with irrevocable decisions, no algorithm can guarantee a competitive ratio better than (n) for piercing intervals, where n is the length of the input sequence. In this paper, we show a 2-competitive algorithm for the problem by allowing at most one recourse per input. We prove that to maintain an optimum piercing set for intervals, any online algorithm needs (n) amortised recourse. We also design an algorithm for piercing intervals which provides a trade-off between recourse and competitive ratio; the algorithm is (1+ϵ ) -competitive when allowed (1+2/ϵ ) recourse at each step. We extend our results to higher dimensions by obtaining a 2^d+1 -competitive algorithm with amortised two recourse for piercing axis-aligned hypercubes in ℝ^d .
