Carsten Moldenhauer

Optimal solutions for moving target search

Moving target search or the game of cops and robbers has been given much attention during the last two decades. It is known that optimal solutions, given a n-cop-win graph, are computable in polynomial time in the size of the input graph. However, a first practical polytime algorithm was only given recently by Hahn et al. [3]. All other known approaches are learning and anytime algorithms that try to approximate the optimal solution. In this work we present four algorithms: an adaptation of Two-Agent IDA*, Proof-Number Search, alpha-beta, and Reverse Minimax A*, a new algorithm. We show how these techniques can be applied to compute optimal moving target search solutions and give benchmarks on their performance for the one cop one robber problem.

Primal-dual approximation algorithms for Node-Weighted Steiner Forest on planar graphs

Node-Weighted Steiner Forest is the following problem: Given an undirected graph, a set of pairs of terminal vertices, a weight function on the vertices, find a minimum weight set of vertices that includes and connects each pair of terminals. We consider the restriction to planar graphs where the problem remains NP-complete. Demaine et al. showed that the generic primal-dual algorithm of Goemans and Williamson is a 6-approximation on planar graphs. We present (1) two different analyses to prove an approximation factor of 3, (2) show that our analysis is best possible for the chosen proof strategy, and (3) generalize this result to feedback problems on planar graphs. We give a simple proof for the first result using contraction techniques and following a standard proof strategy for the generic primal-dual algorithm. Given this proof strategy our analysis is best possible which implies that proving a better upper bound for this algorithm, if possible, would require different proof methods. Then, we give a reduction on planar graphs of Feedback Vertex Set to Node-Weighted Steiner Tree, and Subset Feedback Vertex Set to Node-Weighted Steiner Forest. This generalizes our result to the feedback problems studied by Goemans and Williamson. For the opposite direction, we show how our constructions can be combined with the proof idea for the feedback problems to yield an alternative proof of the same approximation guarantee for Node-Weighted Steiner Forest.

Solving the stable set problem in terms of the odd cycle packing number

The classic stable set problem asks to find a maximum cardinality set of pairwise non-adjacent vertices in an undirected graph G. This problem is NP-hard to approximate with factor n^{1-epsilon} for any constant epsilon>0 [Hastad/Acta Mathematica/1996; Zuckerman/STOC/2006], where n is the number of vertices, and therefore there is no hope for good approximations in the general case. We study the stable set problem when restricted to graphs with bounded odd cycle packing number ocp(G), possibly by a function of n. This is the largest number of vertex-disjoint odd cycles in G. Equivalently, it is the logarithm of the largest absolute value of a sub-determinant of the edge-node incidence matrix A_G of G. Hence, if A_G is totally unimodular, then ocp(G)=0. Therefore, ocp(G) is a natural distance measure of A_G to the set of totally unimodular matrices on a scale from 1 to n/3. When ocp(G)=0, the graph is bipartite and it is well known that stable set can be solved in polynomial time. Our results imply that the odd cycle packing number indeed strongly influences the approximability of stable set. More precisely, we obtain a polynomial-time approximation scheme for graphs with ocp(G)=o(n/log(n)), and an alpha-approximation algorithm for any graph where alpha smoothly increases from a constant to n as ocp(G) grows from O(n/log(n)) to n/3. On the hardness side, we show that, assuming the exponential-time hypothesis, stable set cannot be solved in polynomial time if ocp(G)=Omega(log^{1+epsilon}(n)) for some epsilon>0. Finally, we generalize a theorem by Gyori et al. [Gyori et al./Discrete Mathematics/1997] and show that graphs without odd cycles of small weight can be made bipartite by removing a small number of vertices. This allows us to extend some of our above results to the weighted stable set problem.

Approximation Algorithms for Node-Weighted Prize-Collecting Steiner Tree Problems on Planar Graphs.

We study the prize-collecting version of the Node-weighted Steiner Tree problem (NWPCST) restricted to planar graphs. We give a new primal-dual Lagrangian-multiplier-preserving (LMP) 3-approximation algorithm for planar NWPCST. We then show a (

Balanced Interval Coloring

2.88 + \epsilon

Evaluating strategies for running from the cops

)-approximation which establishes a new best approximation guarantee for planar NWPCST. This is done by combining our LMP algorithm with a threshold rounding technique and utilizing the 2.4-approximation of Berman and Yaroslavtsev for the version without penalties. We also give a primal-dual 4-approximation algorithm for the more general forest version using techniques introduced by Hajiaghay and Jain.

Single-frontier bidirectional search

We consider the discrepancy problem of coloring n intervals with k colors such that at each point on the line, the maximal difference between the number of intervals of any two colors is minimal. Somewhat surprisingly, a coloring with maximal difference at most one always exists. Furthermore, we give an algorithm with running time O(n logn +kn logk) for its construction. This is in particular interesting because many known results for discrepancy problems are non-constructive. This problem naturally models a load balancing scenario, where n tasks with given start- and endtimes have to be distributed among k servers. Our results imply that this can be done ideally balanced. When generalizing to d-dimensional boxes (instead of intervals), a solution with difference at most one is not always possible. We show that for anyd 2 and anyk 2 it is NP-complete to decide if such a solution exists, which implies also NP-hardness of the respective minimization problem. In an online scenario, where intervals arrive over time and the color has to be decided upon arrival, the maximal difference in the size of color classes can become arbitrarily high for any online algorithm. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems—Sequencing and scheduling