When training complex neural networks, memory usage can be an important bottleneck.

Neural Information Processing Systems http://nips.cc/

A preference profile is single-peaked on a tree if the candidate set can be equipped with a tree structure so that the preferences of each voter are decreasing from their top candidate along all paths in the tree. This notion was introduced by Demange (1982), and subsequently Trick (1989b) described an efficient algorithm for deciding if a given profile is single-peaked on a tree. We study the complexity of multiwinner elections under several variants of the Chamberlin–Courant rule for preferences single-peaked on trees. We show that in this setting the egalitarian version of this rule admits a polynomial-time winner determination algorithm. For the utilitarian version, we prove that winner determination remains NP-hard for the Borda scoring function; indeed, this hardness results extends to a large family of scoring functions. However, a winning committee can be found in polynomial time if either the number of leaves or the number of internal vertices of the underlying tree is bounded by a constant. To benefit from these positive results, we need a procedure that can determine whether a given profile is single-peaked on a tree that has additional desirable properties (such as, e.g., a small number of leaves). To address this challenge, we develop a structural approach that enables us to compactly represent all trees with respect to which a given profile is single-peaked. We show how to use this representation to efficiently find the best tree for a given profile for use with our winner determination algorithms: Given a profile, we can efficiently find a tree with the minimum number of leaves, or a tree with the minimum number of internal vertices among trees on which the profile is single-peaked. We then explore the power and limitations of this framework: we develop polynomial-time algorithms to find trees with the smallest maximum degree, diameter, or pathwidth, but show that it is NP-hard to check whether a given profile is single-peaked on a tree that is isomorphic to a given tree, or on a regular tree.

In its most traditional setting, the main concern of optimization theory is the search for optimal solutions for instances of a given computational problem. A recent trend of research in artificial intelligence, called solution diversity, has focused on the development of notions of optimality that may be more appropriate in settings where subjectivity is essential. The idea is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is most appropriate to the context at hand. It also displays the richness of the solution space. When combined with techniques from parameterized complexity theory, the paradigm of diversity of solutions offers a powerful algorithmic framework to address problems of practical relevance. In this work, we investigate the impact of this combination in the field of Kemeny Rank Aggregation, a well-studied class of problems lying in the intersection of order theory and social choice theory and also in the field of order theory itself. In particular, we show that the Kemeny Rank Aggregation problem is fixed-parameter tractable with respect to natural parameters providing natural formalizations of the notions of diversity and of the notion of a sufficiently good solution. Our main results work both when considering the traditional setting of aggregation over linearly ordered votes, and in the more general setting where votes are partially ordered.

The problem of deciding the validity (QSAT) of quantified Boolean formulas (QBF) is a vivid research area in both theory and practice. In the field of parameterized algorithmics, the well-studied graph measure treewidth turned out to be a successful parameter. A well-known result by Chen in parameterized complexity is that QSAT when parameterized by the treewidth of the primal graph of the input formula together with the quantifier depth of the formula is fixed-parameter tractable. More precisely, the runtime of such an algorithm is polynomial in the formula size and exponential in the treewidth, where the exponential function in the treewidth is a tower, whose height is the quantifier depth. A natural question is whether one can significantly improve these results and decrease the tower while assuming the Exponential Time Hypothesis (ETH). In the last years, there has been a growing interest in the quest of establishing lower bounds under ETH, showing mostly problem-specific lower bounds up to the third level of the polynomial hierarchy. Still, an important question is to settle this as general as possible and to cover the whole polynomial hierarchy. In this work, we show lower bounds based on the ETH for arbitrary QBFs parameterized by treewidth (and quantifier depth). More formally, we establish lower bounds for QSAT and treewidth, namely, that under ETH there cannot be an algorithm that solves QSAT of quantifier depth i in runtime significantly better than i-fold exponential in the treewidth and polynomial in the input size. In doing so, we provide a versatile reduction technique to compress treewidth that encodes the essence of dynamic programming on arbitrary tree decompositions. Further, we describe a general methodology for a more fine-grained analysis of problems parameterized by treewidth that are at higher levels of the polynomial hierarchy.

The Partner Units Problem is a specific type of configuration problem with important applications in the area of surveillance and security. In this work we show that a special case of the problem, that is of great interest to our partners in industry, can directly be tackled via a structural problem decompostion method. Combining these theoretical insights with general purpose AI techniques such as constraint satisfaction and SAT solving proves to be particularly effective in practice.