Tobias Jacobs

The binary identification problem for weighted trees

The Binary Identification Problem for weighted trees asks for the minimum cost strategy (decision tree) for identifying a vertex in an edge weighted tree via testing edges. Each edge has assigned a different cost, to be paid for testing it. Testing an edge e reveals in which component of T-e lies the vertex to be identified. We give a complete characterization of the computational complexity of this problem with respect to both tree diameter and degree. In particular, we show that it is strongly NP-hard to compute a minimum cost decision tree for weighted trees of diameter at least 6, and for trees having degree three or more. For trees of diameter five or less, we give a polynomial time algorithm. Moreover, for the degree 2 case, we significantly improve the straightforward O(n^3) dynamic programming approach, and provide an O(n^2) time algorithm. Finally, this work contains the first approximate decision tree construction algorithm that breaks the barrier of factor logn.

On Eulerian extensions and their application to no-wait flowshop scheduling

We consider a variant of no-wait flowshop scheduling that is motivated by continuous casting in the multistage production process in steel manufacturing. The task is to find a feasible schedule with a minimum number of interruptions, i.e., continuous idle time intervals on the last production stage. Based on an interpretation as Eulerian Extension Problems, we fully settle the complexity status of any particular problem case: We give a very intuitive optimal algorithm for scheduling on two processing stages with one machine in the first stage, and we show that all other problem variants are strongly NP-hard. We also discuss alternative idle time related scheduling models and their justification in the considered steel manufacturing environment. Here, we derive constant factor approximations.

On Greedy Algorithms for Decision Trees

In the general search problem we want to identify a specific element using a set of allowed tests. The general goal is to minimize the number of tests performed, although different measures are used to capture this goal. In this work we introduce a novel greedy approach that achieves the best known approximation ratios simultaneously for many different variations of this identification problem. In addition to this flexibility, our algorithm admits much shorter and simpler analyses than previous greedy strategies. As a second contribution, we investigate the potential of greedy algorithms for the more restricted problem of identifying elements of partially ordered sets by comparison with other elements. We prove that the latter problem is as hard to approximate as the general identification problem. As a positive result, we show that a natural greedy strategy achieves an approximation ratio of 2 for tree-like posets, improving upon the previously best known 14-approximation for this problem.

Analytical aspects of tie breaking

This article describes an analytical framework for reasoning about the issue of tie breaking in algorithm theory. The core of this framework is the concept of robust algorithms. Such kind of algorithms have the convenient property that an arbitrary set of degenerate cases can be ignored without loss of generality during proofs of many important properties, e.g., runtime, space requirements, feasibility, competitive and approximation ratios. Here degeneracy is defined as a set of problem instances satisfying a certain property, and this property is independent from both the algorithm under consideration and the specific combinatorial problem structure. It turns out that robustness is related to the tie breaking policy of algorithms in two different ways. On the one hand, the set of inputs where a tie actually occurs is typically (but not always) a degenerate case. On the other hand, we show that for any algorithm there is a way to break ties such that it becomes robust. In particular, robustness is guaranteed by any tie breaking strategy that can be interpreted as symbolic perturbation. For many typical algorithms the implicit usage of symbolic perturbation can be easily verified and so the consideration of degenerate cases can be avoided during their analysis. The concept of robustness also gives a theoretical explanation of why tie breaking does often not matter at all.

On the Huffman and alphabetic tree problem with general cost functions

We address generalized versions of the Huffman and Alphabetic Tree Problem where the cost caused by each individual leaf i, instead of being linear, depends on its depth in the tree by an arbitrary function. The objective is to minimize either the total cost or the maximum cost among all leaves. We review and extend the known results in this direction and devise a number of new algorithms and hardness proofs.It turns out that the Dynamic Programming approach for the Alphabetic Tree Problem can be extended to arbitrary cost functions, resulting in a time O(n4) optimal algorithm using space O(n3). We identify classes of cost functions where the well-known trick to reduce the runtime by a factor of n via a “monotonicity” property can be applied.For the generalized Huffman Tree Problem we show that even the k-ary version can be solved by a generalized version of the Coin Collector Algorithm of Larmore and Hirschberg (in Proc. SODA’90, pp. 310–318, 1990) when the cost functions are nondecreasing and convex. Furthermore, we give an O(n2logn) algorithm for the worst case minimization variants of both the Huffman and Alphabetic Tree Problem with nondecreasing cost functions.Investigating the limits of computational tractability, we show that the Huffman Tree Problem in its full generality is inapproximable unless P = NP, no matter if the objective function is the sum of leaf costs or their maximum. The alphabetic version becomes NP-hard when the leaf costs are interdependent.

Binary identification problems for weighted trees

The Binary Identification Problem for weighted trees asks for the minimum cost strategy (decision tree) for identifying a node in an edge weighted tree via testing edges. Each edge has assigned a different cost, to be paid for testing it. Testing an edge e reveals in which component of T - e lies the vertex to be identified. We give a complete characterization of the computational complexity of this problem with respect to both tree diameter and degree. In particular, we show that it is strongly NP-hard to compute a minimum cost decision tree for weighted trees of diameter at least 6, and for trees having degree three or more. For trees of diameter five or less, we give a polynomial time algorithm. Moreover, for the degree 2 case, we significantly improve the straightforward O(n3) dynamic programming approach, and provide an O(n2) time algorithm. Finally, this work contains the first approximate decision tree construction algorithm that breaks the barrier of factor logn.