The 2-Attractor Problem is NP-Complete

A $k$-attractor is a combinatorial object unifying dictionary-based compression. It allows to compare the repetitiveness measures of different dictionary compressors such as Lempel-Ziv 77, the Burrows-Wheeler transform, straight line programs and macro schemes. For a string $T \in \Sigma^n$, the $k$-attractor is defined as a set of positions $\Gamma \subseteq [1,n]$, such that every distinct substring of length at most $k$ is covered by at least one of the selected positions. Thus, if a substring occurs multiple times in $T$, one position suffices to cover it. A 1-attractor is easily computed in linear time, while Kempa and Prezza [STOC 2018] have shown that for $k \geq 3$, it is NP-complete to compute the smallest $k$-attractor by a reduction from $k$-set cover. The main result of this paper answers the open question for the complexity of the 2-attractor problem, showing that the problem remains NP-complete. Kempa and Prezza's proof for $k \geq 3$ also reduces the 2-attractor problem to the 2-set cover problem, which is equivalent to edge cover, but that does not fully capture the complexity of the 2-attractor problem. For this reason, we extend edge cover by a color function on the edges, yielding the colorful edge cover problem. Any edge cover must then satisfy the additional constraint that each color is represented. This extension raises the complexity such that colorful edge cover becomes NP-complete while also more precisely modeling the 2-attractor problem. We obtain a reduction showing $k$-attractor to be NP-complete for any $k \geq 2$

The Complexity of Online Graph Games

Online computation is a concept to model uncertainty where not all information on a problem instance is known in advance. An online algorithm receives requests which reveal the instance piecewise and has to respond with irrevocable decisions. Often, an adversary is assumed that constructs the instance knowing the deterministic behavior of the algorithm. From a game theoretical point of view, the adversary and the online algorithm are players in a two-player game. By applying this view on combinatorial graph problems, especially on problems where the solution is a subset of the vertices, we analyze their complexity. For this, we introduce a framework based on gadget reductions from 3-Satisfiability and extend it to an online setting where the graph is a priori known by a map. This is done by identifying a set of rules for the reductions and providing schemes for gadgets. The extension of the framework to the online setting enable reductions from TQBF. We provide example reductions to the well-known problems Vertex Cover, Independent Set and Dominating Set and prove that they are PSPACE-complete. Thus, this paper establishes that the online version with a map of NP-complete graph problems form a large class of PSPACE-complete problems

The Complexity of Graph Exploration Games

The graph exploration problem asks a searcher to explore an unknown graph. This problem can be interpreted as the online version of the Traveling Salesman Problem. The treasure hunt problem is the corresponding online version of the shortest s-t-path problem. It asks the searcher to find a specific vertex in an unknown graph at which a treasure is hidden. Recently, the analysis of the impact of a priori knowledge is of interest. In graph problems, one form of a priori knowledge is a map of the graph. We survey the graph exploration and treasure hunt problem with an unlabeled map, which is an isomorphic copy of the graph, that is provided to the searcher. We formulate decision variants of both problems by interpreting the online problems as a game between the online algorithm (the searcher) and the adversary. The map, however, is not controllable by the adversary. The question is, whether the searcher is able to explore the graph fully or find the treasure for all possible decisions of the adversary. We prove the PSPACE-completeness of these games, whereby we analyze the variations which ask for the mere existence of a tour through the graph or path to the treasure and the variations that include costs. Additionally, we analyze the complexity of related problems that ask for a tour in the graph or a s-t path

Online matching in regular bipartite graphs

In an online problem, the input is revealed one piece at a time. In every time step, the online algorithm has to produce a part of the output, based on the partial knowledge of the input. Such decisions are irrevocable, and thus online algorithms usually lead to nonoptimal solutions. The impact of the partial knowledge depends strongly on the problem. If the algorithm is allowed to read binary information about the future, the amount of bits read that allow the algorithm to solve the problem optimally is the socalled advice complexity. The quality of an online algorithm is measured by its competitive ratio, which compares its performance to that of an optimal offline algorithm. In this paper we study online bipartite matchings focusing on the particular case of bipartite matchings in regular graphs. We give tight upper and lower bounds on the competitive ratio of the online deterministic bipartite matching problem. The competitive ratio turns out to be asymptotically equal to the known randomized competitive ratio. Afterwards, we present an upper and lower bound for the advice complexity of the online deterministic bipartite matching problem.Postprint (author's final draft

The Complexity of Packing Edge-Disjoint Paths

We introduce and study the complexity of Path Packing. Given a graph G and a list of paths, the task is to embed the paths edge-disjoint in G. This generalizes the well known Hamiltonian-Path problem. Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least three. Surprisingly the latter is an almost tight result by runtime and parameterization. We show an ETH lower bound almost matching our runtime. Moreover, if two of the three values are constant and one is unbounded the problem becomes NP-hard. Further, we study restrictions to the given list of paths. On the positive side, we present an FPT-algorithm parameterized by the sum of the lengths of the paths. Packing paths of length two is polynomial time solvable, while packing paths of length three is NP-hard. Finally, even the spacial case Exact Path Packing where the paths have to cover every edge in G exactly once is already NP-hard for two paths on 4-regular graphs

Exploring sparse graphs with advice

Graph exploration is a theoretical model of the crucial task of moving an agent through an unknown environment. Here, an algorithm has to guide an explorer through a network with n vertices and m edges, visiting every vertex at least once. We consider the fixed-graph scenario by Kalyanasundaram and Pruhs (ICALP, 1993), where the explorer sees all vertices reachable in one step, their unique names and their distance from the current position. The algorithm only learns the structure of the graph during computation. Therefore, we are interested in the amount of crucial a-priori information (the advice complexity) needed to solve the problem optimally. We look at graph exploration on directed graphs and focus on cyclic solutions. It is known that O(n log n) bits of advice are necessary and sufficient to compute an optimal solution for general graphs. We present algorithms with O(m) advice, thus improving the bound for sparse graphs.ISSN:0890-540

Online matching in regular bipartite graphs

In an online problem, the input is revealed one piece at a time. In every time step, the online algorithm has to produce a part of the output, based on the partial knowledge of the input. Such decisions are irrevocable, and thus online algorithms usually lead to nonoptimal solutions. The impact of the partial knowledge depends strongly on the problem. If the algorithm is allowed to read binary information about the future, the amount of bits read that allow the algorithm to solve the problem optimally is the socalled advice complexity. The quality of an online algorithm is measured by its competitive ratio, which compares its performance to that of an optimal offline algorithm. In this paper we study online bipartite matchings focusing on the particular case of bipartite matchings in regular graphs. We give tight upper and lower bounds on the competitive ratio of the online deterministic bipartite matching problem. The competitive ratio turns out to be asymptotically equal to the known randomized competitive ratio. Afterwards, we present an upper and lower bound for the advice complexity of the online deterministic bipartite matching problem