Computer Science

Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity, DNF sparsification, randomness extractors, and recent advances on the Erdős-Rado sunflower conjecture. The recent breakthrough of Alweiss, Lovett, Wu and Zhang gives an improved bound on the maximum size of a w -set system that excludes a robust sunflower. In this paper, we use this result to obtain an exp( n 1/2−o(1) ) lower bound on the monotone circuit size of an explicit n -variate monotone function, improving the previous best known exp( n 1/3−o(1) ) due to Andreev and Harnik and Raz. We also show an exp(Ω(n)) lower bound on the monotone arithmetic circuit size of a related polynomial. Finally, we introduce a notion of robust clique-sunflowers and use this to prove an n Ω(k) lower bound on the monotone circuit size of the CLIQUE function for all k≤ n 1/3−o(1) , strengthening the bound of Alon and Boppana.

Generalized circuits are an important tool in the study of the computational complexity of equilibrium approximation problems. However, in this paper, we reveal that they have a conceptual flaw, namely that the solution concept is not monotonic. By this we mean that if ε< ε ′ , then an ε -approximate solution for a certain generalized circuit is not necessarily also an ε ′ -approximate solution. The reason for this non-monotonicity is the way Boolean operations are modeled. We illustrate that non-monotonicity creates subtle technical issues in prior work that require intricate additional arguments to circumvent. To eliminate this problem, we show that the Boolean gates are a redundant feature: one can simulate stronger, monotonic versions of the Boolean gates using the other gate types. Arguing at the level of these stronger Boolean gates eliminates all of the aforementioned issues in a natural way. We hope that our results will enable new studies of sub-classes of generalized circuits and enabler simpler and more natural reductions from generalized circuits to other equilibrium search problems.

Since the early days of research in algorithms and complexity, the computation of stable matchings is a core topic. While in the classic setting the goal is to match up two agents (either from different "gender" (this is Stable Marriage) or "unrestricted" (this is Stable Roommates)), Knuth [1976] triggered the study of three- or multidimensional cases. Here, we focus on the study of Multidimensional Stable Roommates, known to be NP-hard since the early 1990's. Many NP-hardness results, however, rely on very general input instances that do not occur in at least some of the specific application scenarios. With the quest for identifying islands of tractability for Multidimensional Stable Roommates, we study the case of master lists. Here, as natural in applications where agents express their preferences based on "objective" scores, one roughly speaking assumes that all agent preferences are "derived from" a central master list, implying that the individual agent preferences shall be similar. Master lists have been frequently studied in the two-dimensional (classic) stable matching case, but seemingly almost never for the multidimensional case. This work, also relying on methods from parameterized algorithm design and complexity analysis, performs a first systematic study of Multidimensional Stable Roommates under the assumption of master lists.

We suggest a generalization of Karchmer-Wigderson communication games to the multiparty setting. Our generalization turns out to be tightly connected to circuits consisting of threshold gates. This allows us to obtain new explicit constructions of such circuits for several functions. In particular, we provide an explicit (polynomial-time computable) log-depth monotone formula for Majority function, consisting only of 3-bit majority gates and variables. This resolves a conjecture of Cohen et al. (CRYPTO 2013).

Electing a single committee of a small size is a classical and well-understood voting situation. Being interested in a sequence of committees, we introduce and study two time-dependent multistage models based on simple Plurality voting. Therein, we are given a sequence of voting profiles (stages) over the same set of agents and candidates, and our task is to find a small committee for each stage of high score. In the conservative model we additionally require that any two consecutive committees have a small symmetric difference. Analogously, in the revolutionary model we require large symmetric differences. We prove both models to be NP-hard even for a constant number of agents, and, based on this, initiate a parameterized complexity analysis for the most natural parameters and combinations thereof. Among other results, we prove both models to be in XP yet W[1]-hard regarding the number of stages, and that being revolutionary seems to be "easier" than being conservative: If the (upper- resp. lower-) bound on the size of symmetric differences is constant, the conservative model remains NP-hard while the revolutionary model becomes polynomial-time solvable.

Covering all edges of a graph by a small number of vertices, this is the NP-complete Vertex Cover problem. It is among the most fundamental graph-algorithmic problems. Following a recent trend in studying temporal graphs (a sequence of graphs, so-called layers, over the same vertex set but, over time, changing edge sets), we initiate the study of Multistage Vertex Cover. Herein, given a temporal graph, the goal is to find for each layer of the temporal graph a small vertex cover and to guarantee that two vertex cover sets of every two consecutive layers differ not too much (specified by a given parameter). We show that, different from classic Vertex Cover and some other dynamic or temporal variants of it, Multistage Vertex Cover is computationally hard even in fairly restricted settings. On the positive side, however, we also spot several fixed-parameter tractability results based on some of the most natural parameterizations.

Addressing a quest by Gupta et al. [ICALP'14], we provide a first, comprehensive study of finding a short s-t path in the multistage graph model, referred to as the Multistage s-t Path problem. Herein, given a sequence of graphs over the same vertex set but changing edge sets, the task is to find short s-t paths in each graph ("snapshot") such that in the found path sequence the consecutive s-t paths are "similar". We measure similarity by the size of the symmetric difference of either the vertex set (vertex-similarity) or the edge set (edge-similarity) of any two consecutive paths. We prove that these two variants of Multistage s-t Path are already NP-hard for an input sequence of only two graphs and maximum vertex degree four. Motivated by this fact and natural applications of this scenario e.g. in traffic route planning, we perform a parameterized complexity analysis. Among other results, for both variants, vertex- and edge-similarity, we prove parameterized hardness (W[1]-hardness) regarding the parameter path length (solution size) for both variants, vertex- and edge-similarity. As a further conceptual study, we then modify the multistage model by asking for dissimilar consecutive paths. As one of the main technical results (employing so-called representative sets known from non-temporal settings), we prove that dissimilarity allows for fixed-parameter tractability for the parameter solution size, contrasting our W[1]-hardness proof of the corresponding similarity case. We also provide partially positive results concerning efficient and effective data reduction (kernelization).

Some classical graph problems such as finding minimal spanning tree, shortest path or maximal flow can be done efficiently. We describe slight variations of such problems which are shown to be NP-complete. Our proofs use straightforward reduction from 3 -SAT.

Kingdomino is a board game designed by Bruno Cathala and edited by Blue Orange since 2016. The goal is to place 2×1 dominoes on a grid layout, and get a better score than other players. Each 1×1 domino cell has a color that must match at least one adjacent cell, and an integer number of crowns (possibly none) used to compute the score. We prove that even with full knowledge of the future of the game, in order to maximize their score at Kingdomino, players are faced with an NP-complete optimization problem.

In this paper, we seek a natural problem and a natural distribution of instances such that any O( n c−ϵ ) -time algorithm fails to solve most instances drawn from the distribution, while the problem admits an n c+o(1) -time algorithm that correctly solves all instances. Specifically, we consider the K a,b counting problem in a random bipartite graph, where K a,b is a complete bipartite graph for constants a and b . We proved that the K a,b counting problem admits an n a+o(1) -time algorithm if a≥8 , while any n a−ϵ -time algorithm fails to solve it even on random bipartite graph for any constant ϵ>0 under the Strong Exponential Time Hypotheis. Then, we amplify the hardness of this problem using the direct product theorem and Yao's XOR lemma by presenting a general framework of hardness amplification in the setting of fine-grained complexity.