Tobias Mömke

On the Advice Complexity of Online Problems

In this paper, we investigate to what extent the solution quality of online algorithms can be improved by allowing the algorithm to extract a given amount of information about the input. We consider the recently introduced notion of advice complexity where the algorithm, in addition to being fed the requests one by one, has access to a tape of advice bits that were computed by some oracle function from the complete input. The advice complexity is the number of advice bits read. We introduce an improved model of advice complexity and investigate the connections of advice complexity to the competitive ratio of both deterministic and randomized online algorithms using the paging problem, job shop scheduling, and the routing problem on a line as sample problems. We provide both upper and lower bounds on the advice complexity of all three problems. Our results for all of these problems show that very small advice (only three bits in the case of paging) already suffices to significantly improve over the best deterministic algorithm. Moreover, to achieve the same competitive ratio as any randomized online algorithm, a logarithmic number of advice bits is sufficient. On the other hand, to obtain optimality, much larger advice is necessary.

On the hardness of reoptimization

We consider the following reoptimization scenario: Given an instance of an optimization problem together with an optimal solution, we want to find a high-quality solution for a locally modified instance. The naturally arising question is whether the knowledge of an optimal solution to the unaltered instance can help in solving the locally modified instance. In this paper, we survey some partial answers to this questions: Using some variants of the traveling salesman problem and the Steiner tree problem as examples, we show that the answer to this question depends on the considered problem and the type of local modification and can be totally different: For instance, for some reoptimization variant of the metric TSP, we get a 1.4-approximation improving on the best known approximation ratio of 1.5 for the classical metric TSP. For the Steiner tree problem on graphs with bounded cost function, which is APX-hard in its classical formulation, we even obtain a PTAS for the reoptimization variant. On the other hand, for a variant of TSP, where some vertices have to be visited before a prescribed deadline, we are able to show that the reoptimization problem is exactly as hard to approximate as the original problem.

Reoptimization of Steiner Trees

In this paper we study the problem of finding a minimum Steiner Tree given a minimum Steiner Tree for similar problem instance. We consider scenarios of altering an instance by locally changing the terminal set or the weight of an edge. For all modification scenarios we provide approximation algorithms that improve best currently known corresponding approximation ratios.

Reoptimization of Steiner trees

Given an instance of the Steiner tree problem together with an optimal solution, we consider the scenario where this instance is modified locally by adding one of the vertices to the terminal set or removing one vertex from it. In this paper, we investigate the problem of whether the knowledge of an optimal solution to the unaltered instance can help in solving the locally modified instance. Our results are as follows: (i) We prove that these reoptimization variants of the Steiner tree problem are NP-hard, even if edge costs are restricted to values from {1,2}. (ii) We design 1.5-approximation algorithms for both variants of local modifications. This is an improvement over the currently best known approximation algorithm for the classical Steiner tree problem which achieves an approximation ratio of 1+ln(3)/2?1.55. (iii) We present a PTAS for the subproblem in which the edge costs are natural numbers {1,?,k} for some constant k.

On the Advice Complexity of the Set Cover Problem

Recently, a new approach to get a deeper understanding of online computation has been introduced: the study of the advice complexity of online problems. The idea is to measure the information that online algorithms need to be supplied with to compute high-quality solutions and to overcome the drawback of not knowing future input requests. In this paper, we study the advice complexity of an online version of the well-known set cover problem introduced by Alon et al.: for a ground set of size n and a set family of m subsets of the ground set, we obtain bounds in both n and m. We prove that a linear number of advice bits is both sufficient and necessary to perform optimally. Furthermore, for any constant c, we prove that n − c bits are enough to construct a c-competitive online algorithm and this bound is tight up to a constant factor (only depending on c). Moreover, we show that a linear number of advice bits is both necessary and sufficient to be optimal with respect to m, as well. We further show lower and upper bounds for achieving c-competitiveness also in m.

Size complexity of rotating and sweeping automata

We examine the succinctness of one-way, rotating, sweeping, and two-way deterministic finite automata (1DFAs, RDFAs, SDFAs, 2DFAs) and their nondeterministic and randomized counterparts. Here, a SDFA is a 2DFA whose head can change direction only on the end-markers and a RDFA is a SDFA whose head is reset to the left end of the input every time the right end-marker is read. We study the size complexity classes defined by these automata, i.e., the classes of problems solvable by small automata of certain type. For any pair of classes of one-way, rotating, and sweeping deterministic (1D, RD, SD), self-verifying (1@D, R@D, S@D) and nondeterministic (1N, RN, SN) automata, as well as for their complements and reversals, we show that they are equal, incomparable, or one is strictly included in the other. The provided map of the complexity classes has interesting implications on the power of randomization for finite automata. Among other results, it implies that Las Vegas sweeping automata can be exponentially more succinct than SDFAs. We introduce a list of language operators and study the corresponding closure properties of the size complexity classes defined by these automata as well. Our conclusions reveal also the logical structure of certain proofs of known separations among the complexity classes and allow us to systematically construct alternative witnesses of these separations.

Removing and Adding Edges for the Traveling Salesman Problem

We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graph-TSP), we show that the approach gives a 1.461-approximation algorithm with respect to the Held-Karp lower bound. For graph-TSP restricted either to half-integral solutions to the Held-Karp relaxation or to a class of graphs that contains subcubic and claw-free graphs, we show that the integrality gap of the Held-Karp relaxation matches the conjectured ratio 4/3. The framework also allows for generalizations in a natural way and leads to analogous results for the s, t-path traveling salesman problem on graphic metrics where the start and end vertices are prespecified.

New approximation schemes for unsplittable flow on a path

We study the unsplittable flow on a path problem which has received a lot of attention in the research community recently. Given is a path with capacities on its edges and a set of tasks where each task is characterized by a source and a sink vertex, a demand, and a profit. The goal is to find a subset of the tasks of maximum total profit such that all task demands from this subset can be routed simultaneously without violating the capacity constraints. The best known approximation results are a quasi-polynomial time-approximation scheme if the task demands are in a quasi-polynomial range [Bansal et al., STOC 2006] and a polynomial time (2 + e)-approximation algorithm [Anagnostopoulos et al., SODA 2014]. Finding a PTAS for it has remained an important open question. In this paper we make progress towards this goal. When the task densities---defined as the ratio of a tasks profit and demand---lie in a constant range, we obtain a PTAS. We also improve the QPTAS of Bansal et al. by removing the assumption that the demands need to lie in a quasi-polynomial range. Our third result is a PTAS for the case where we are allowed to shorten the paths of the tasks by at most an e-fraction. This is particularly motivated by bandwidth allocation and scheduling applications of our problem if we are allowed to slightly increase the speed of the underlying transmission link/machine. Each of these results critically uses a sparsification lemma which we believe could be of independent interest. The lemma shows that in any (optimal) solution there exists an O(e)-fraction (measured by weight) of its tasks whose removal creates, on each edge, a slack which is at least as large as the (1/e)th largest demand using that edge. This slack can then be used to allow slight errors when estimating or rounding quantities arising in the computation.

Reoptimization of the Shortest Common Superstring Problem

A reoptimization problem describes the following scenario: Given an instance of an optimization problem together with an optimal solution for it, we want to find a good solution for a locally modified instance. In this paper, we deal with reoptimization variants of the shortest common superstring problem where the local modifications consist of adding or removing a single string. We show NP-hardness of these reoptimization problems and design several approximation algorithms for them.

An exponential gap between Las Vegas and deterministic sweeping finite automata

A two-way finite automaton is sweeping if its input head can change direction only on the end-markers. For each n ≥ 2, we exhibit a problem that can be solved by a O(n2)-state sweeping LasVegas automaton, but needs 2Ω(n) states on every sweeping deterministic automaton.