Parameterized Complexity of Eulerian Deletion Problems

Marek Cygan 0 1 2 Dniel Marx 0 1 2 Marcin Pilipczuk 0 1 2 Micha Pilipczuk 0 1 2 Ildik Schlotter 0 1 2 0 M. Pilipczuk ( ) Department of Informatics, University of Bergen , Postboks 7803, 5020 Bergen, Norway 1 D. Marx Institut fr Informatik, Humboldt-Universitt zu Berlin , Unter den Linden 6, 10099 Berlin, Germany 2 I. Schlotter Department of Computer Science and Information Theory, Budapest University of Technology and Economics , Magyar tudsok krtja 2, 1117 Budapest, Hungary We study a family of problems where the goal is to make a graph Eulerian, i.e., connected and with all the vertices having even degrees, by a minimum number of deletions. We completely classify the parameterized complexity of various versions: undirected or directed graphs, vertex or edge deletions, with or without the requirement of connectivity, etc. The collection of results shows an interesting contrast: while the node-deletion variants remain intractable, i.e., W[1]-hard for all the studied cases, edge-deletion problems are either fixed-parameter tractable or polynomial-time solvable. Of particular interest is a randomized FPT algorithm for making an undirected graph Eulerian by deleting the minimum number of edges, based on a novel application of the color coding technique. For versions that remain NP-complete but fixed-parameter tractable we consider also possibilities of 1 Introduction An undirected graph is Eulerian if it is connected and every vertex has even degree; a directed graph is Eulerian if it is strongly connected and every vertex is balanced (i.e., the indegree equals the outdegree). The class of Eulerian graphs is a well-studied and classical notion in the graph theory. We investigate several algorithmic problems related to the question of how to make a graph Eulerian. We focus on deletion problems, where either vertices or edges can be deleted from the input graph to make it Eulerian, using as few deletions as possible. What makes these problems interesting is the interplay of two different type of constraints: each vertex locally prescribes the constraint that it has to be even/balanced, while retaining connectivity is a global requirement. For comparison, we also investigate the variant of the problem where we have only the local constraints (i.e., the task is to delete the minimum number of edges or nodes to make every vertex even/balanced). As many of the studied problems turn out to be NP-hard, we apply the framework of parameterized complexity to get a more detailed insight. The investigation of these problems was initiated by Cai and Yang [9] who presented parameterized results for some cases. We complement their work by answering several open questions raised in [9]. Another motivation for our work comes from an observation of Cechlrov and Schlotter [10]: computing the deficiency for a certain type of housing market is equivalent to finding the minimum number of arcs whose deletion makes every strongly connected component of the graph balanced. While we are not able to determine the parameterized complexity of this problem, our results shed light on the complexity of several related ones. Related Work Subgraph problems have been widely studied in the literature. To name a few examples, Lewis and Yannakakis [21] investigated the complexity of the node-deletion problem for hereditary properties, Alon et al. [2] examined edgedeletion problems for monotone properties, while Natanzon et al. [28] and Burzyn et al. [6] studied the classical complexity of edge modification problems for various graph classes. Subgraph problems have also been looked at from the parameterized perspective. The most extensively studied variants are the node-deletion problems for hereditary properties: the results by Cai [8], and Khot and Raman [18], yield a complete characterization of the fixed-parameter tractable cases. Apart from hereditary properties, FPT algorithms are known for node-deletion problems where the task is to obtain a regular graph [26], a chordal graph [23], a grid [12], etc. Parameterized hardness results have been obtained in numerous cases as well [22, 24]. Recently, researchers focused on the issue of kernelization, yielding both positive [4, 17, 29] and negative results [20]. There is much less known about directed graphs. Raman and Sikdar [32] investigated the parameterized complexity of hereditary node-deletion problems in digraphs, while Raman and Saurabh [31] examined feedback set problems in tournaments. The FPT algorithm by Chen et al. for finding a feedback vertex set in a directed graph [11] resolved a long-standing open question. Work related to the class of Eulerian graphs mainly concentrated on the extension problem, where the task is to add a minimum number of edges or arcs in order to make the given graph Eulerian. FPT algorithms were given for various settings by Dorn et al. [13] and by Sorge [33]. Eulerian deletion problems were studied by Cai and Yang [9]. Our Contribution To settle the classical complexity of the examined problems, first we observe (Theorems 1 and 2) that classical results imply polynomial-time algorithms for the edge-deletion problems where the task is to make the given graph even/balanced: in the undirected case, this is essentially a T -join problem, while the directed case can be reduced to a flow problem. These observations answer a question raised by Cai and Yang [9], who observed that the analogous node-deletion problems are NP-hard. Moreover, the aforementioned algorithms are used as subroutines in our FPT results. By contrast to the polynomial time algorithms, we show that the seemingly similar edge- (or arc-) deletion problems where we aim for an Eulerian graph are NP-hard, even in the extremely restricted case when the input is a cubic planar graph and the number of deletions can be arbitrary (Theorem 3). We investigate both the undirected and the directed cases of Eulerian edge-deletion problem thoroughly from the parameterized point of view: we present a fixed-parameter tractable algorithm for both cases where the parameter is the number of deletions allowed (Theorem 4), and prove that these problems do not admit a polynomial-size kernel unless NP coNP/poly (Theorem 5), which is known to imply a collapse of the polynomial hierarchy to its third level [7, 34]. The FPT results use a novel argument that might be of independent interest. Intuitively, we need to find a solution S to a T -join problem and a witness (disjoint from S) certifying that the graph remains connected after the removal of S. Using a random coloring, we partition the edges into two types: each edge can contribute either to the solution or to the witness of the solution. This partition ensures that the solution and the witness are disjoint. While the use of random colorings is a standard technique for finding a solution consisting of disjoint objects [3], we use this technique to separate the solution from its proof of feasibility. The undirected (...truncated)