Lasse Kliemann

Bipartite Graph Matchings in the Semi-streaming Model

We present an algorithm for finding a large matching in a bipartite graph in the semi-streaming model. In this model, the input graph G = (V, E) is represented as a stream of its edges in some arbitrary order, and storage of the algorithm is bounded by O(n , polylog n) bits, where n = |V|. For e> 0, our algorithm finds a \(\frac{1}{1+\epsilon}\)-approximation of a maximum-cardinality matching and uses \(O{({(\frac{1}{\epsilon})^8})}\) passes over the input stream. The only previously known algorithm with such arbitrarily good approximation – though for general graphs – required exponentially many \(\Omega({{(\frac{1}{\epsilon})^{\frac{1}{\epsilon}}}})\) passes (McGregor 2005).

Three‐dimensional reconstruction of seed implants by randomized rounding and visual evaluation

The development of efficient 3D seed reconstruction algorithms is an ongoing and vivid research topic. Since the 1980s many publications about seed assignment were published. In this paper a novel mathematical approach is described to solve the 3D assignment problem for the reconstruction of seeds with radiographs: we present a fast linear programming approach together with afterwards applying the so-called randomized rounding scheme to compute good (possibly partial) assignments. We apply a visualization software that allows user interaction to check the solution given by the algorithm and to augment partial assignments. The second step is justified as the randomized algorithm already returns optimal solutions is many cases, and in cases with partial assignments it fails to match only a very small number of seed images. Our algorithm transfers ideas from recent breakthrough research work on the design of efficient randomized algorithms in discrete optimization and computer science to the seed reconstruction problem.

The Price of Anarchy for Network Formation in an Adversary Model

We study network formation with n players and link cost α > 0. After the network is built, an adversary randomly deletes one link according to a certain probability distribution. Cost for player ν incorporates the expected number of players to which ν will become disconnected. We focus on unilateral link formation and Nash equilibrium . We show existence of Nash equilibria and a price of stability of 1 + ο (1) under moderate assumptions on the adversary and n ≥ 9. We prove bounds on the price of anarchy for two special adversaries: one removes a link chosen uniformly at random, while the other removes a link that causes a maximum number of player pairs to be separated. We show an Ο (1) bound on the price of anarchy for both adversaries, the constant being bounded by 15 + ο (1) and 9 + ο (1), respectively.

A New QEA Computing Near-Optimal Low-Discrepancy Colorings in the Hypergraph of Arithmetic Progressions

We present a new quantum-inspired evolutionary algorithm, the attractor population QEA (apQEA). Our benchmark problem is a classical and difficult problem from Combinatorics, namely finding low-discrepancy colorings in the hypergraph of arithmetic progressions on the first n integers, which is a massive hypergraph (e.g., with approx. 3.88 ×1011 hyperedges for n = 250 000). Its optimal low-discrepancy coloring bound \(\Theta(\sqrt[4]{n})\) is known and it has been a long-standing open problem to give practically and/or theoretically efficient algorithms. We show that apQEA outperforms known QEA approaches and the classical combinatorial algorithm (Sarkozy 1974) by a large margin. Regarding practicability, it is also far superior to the SDP-based polynomial-time algorithm of Bansal (2010), the latter being a breakthrough work from a theoretical point of view. Thus we give the first practical algorithm to construct optimal colorings in this hypergraph, up to a constant factor. We hope that our work will spur further applications of Algorithm Engineering to Combinatorics.

The Price of Anarchy in Bilateral Network Formation in an Adversary Model

We study the bilateral version of the adversary network formation game introduced by the author in 2010. In bilateral network formation, a link is formed only if both endpoints agree on it and then both have to pay the link cost

Brief announcement: the price of anarchy for distributed network formation in an adversary model

\alpha >0

Randomized algorithms for mixed matching and covering in hypergraphs in 3D seed reconstruction in brachytherapy

α>0 for it. In the adversary model, the cost of each player comprises this building cost plus the expected number of other players to which she will lose connection if one link is destroyed randomly according to a known probability distribution. Two adversaries are considered: one chooses the link to destroy uniformly at random from the set of all built links, the other instead concentrates the probability measure on the built links which cause a maximum number of player pairs to be separated when destroyed. Pairwise stability (PS) and pairwise Nash equilibrium (PNE) are used as equilibrium concepts. For the first adversary, we prove convexity of cost, hence PS and PNE coincide. The main result is an upper bound of 10 on the price of anarchy for this adversary, for link cost