Journal of Computer and System Sciences最新文献

The original possible winner problem consists of an unweighted election with partial preferences and a distinguished candidate c and asks whether the preferences can be extended to total ones such that c wins the given election. We introduce a novel variant of this problem: possible winner with uncertain weights. In this variant, for a given weighted election, not some of the preferences but some of the preferences' weights are uncertain. We introduce a general framework to study this problem for nonnegative integer and rational weights as well as for four different variations of the problem itself: with and without given upper bounds on the total weight and with and without given ranges to choose weights from. We study the complexity of these problems for important voting systems such as scoring protocols, (simplified) Bucklin and fallback voting, plurality with runoff and veto with runoff, ${\text{Copeland}}^{\alpha }$, ranked pairs, and Borda.

Tile Automata is a recently defined model of self-assembly that borrows many concepts from cellular automata to create active self-assembling systems where changes may be occurring within an assembly without requiring attachment. This model has been shown to be powerful even with limited assembly size, but many fundamental questions have yet to be explored. Here, we study the state complexity of assembling $n\times n$ squares in seeded Tile Automata systems where growth starts from a seed and tiles attach one at a time, similar to the abstract Tile Assembly Model. We provide optimal bounds for three classes of seeded Tile Automata systems (all without detachment), which vary in the amount of complexity allowed in the transition rules. We show that, in general, seeded Tile Automata systems require $\Theta ({\log }^{\frac{1}{4}}n)$ states. For single-transition systems, where only one state may change in a transition rule, we show a bound of $\Theta ({\log }^{\frac{1}{3}}n)$, and for deterministic systems, where each pair of states may only have one associated transition rule, a bound of $\Theta \left({\left(\frac{\log n}{\log \log n}\right)}^{\frac{1}{2}}\right)$. Along the way, we provide optimal bounds for the subroutines of building binary strings and building $O(\log n)\times n$ rectangles.

Graph-deletion problems involve deleting a small number of vertices so that the resulting graph belong to a given hereditary graph class. We initiate a study of a natural variation of the problem of deletion to scattered graph classes. We want to delete at most k vertices so that each connected component of the resulting graph belongs to one of the constant number of graph classes. As our main result, we show that this problem is non-uniformly fixed-parameter tractable (FPT) when the deletion problem corresponding to each of the constant number of graph classes is known to be FPT and the properties that a graph belongs to these classes are expressible in Counting Monodic Second Order (CMSO) logic. While this is shown using some black box theorems in parameterized complexity, we give a faster FPT algorithm when each of the graph classes has a finite forbidden set.

We revisit the k-Secluded Tree problem. Given a vertex-weighted undirected graph G, its objective is to find a maximum-weight induced subtree T whose open neighborhood has size at most k. We present a fixed-parameter tractable algorithm that solves the problem in time $2^{O(k\log k)}\cdot n^{O\left(1\right)}$, improving on a double-exponential running time from earlier work by Golovach, Heggernes, Lima, and Montealegre. Starting from a single vertex, our algorithm grows a k-secluded tree by branching on vertices in the open neighborhood of the current tree T. To bound the branching depth, we prove a structural result that can be used to identify a vertex that belongs to the neighborhood of any k-secluded supertree $T^{\prime }\supseteq T$ once the open neighborhood of T becomes sufficiently large. We extend the algorithm to enumerate compact descriptions of all maximum-weight k-secluded trees, which allows us to count them as well.

Semi-online models where decisions may be revoked in a limited way have been studied extensively in the last years. A well-studied measure of the amount of decisions that can be revoked is the (constant) migration factor. When an object arrives, the decisions for objects of total size at most the migration factor times its size may be revoked. This means that a small object only leads to small changes. We extensively study the bin covering problem with migration in different scenarios. We develop algorithms both for the static case where only insertions are allowed, and for the dynamic case, where items may also depart. We also develop lower bounds for these scenarios both for amortized migration and for worst-case migration showing that our algorithms have nearly optimal migration factor and asymptotic competitive ratio. We therefore resolve the competitiveness of the bin covering problem with migration.

The generic limit set of a dynamical system is the smallest set that attracts most of the space in a topological sense: it is the smallest closed set with a comeager basin of attraction. Introduced by Milnor, it has been studied in the context of one-dimensional cellular automata by Djenaoui and Guillon, Delacourt, and Törmä. In this article we present complexity bounds on realizations of generic limit sets of cellular automata with prescribed properties. We show that generic limit sets have a ${\Pi }_2^0$ language if they are inclusion-minimal, a ${\Sigma }_1^0$ language if the cellular automaton has equicontinuous points, and that these bounds are tight. We also prove that many chain mixing ${\Pi }_2^0$ subshifts and all chain mixing ${\Delta }_2^0$ subshifts are realizable as generic limit sets. As a corollary, we characterize the minimal subshifts that occur as generic limit sets.

We study fundamental clustering problems for incomplete data. Specifically, given a set of incomplete d-dimensional vectors (representing rows of a matrix), the goal is to complete the missing vector entries in a way that admits a partitioning of the vectors into at most k clusters with radius or diameter at most r. We give characterizations of the parameterized complexity of these problems with respect to the parameters k, r, and the minimum number of rows and columns needed to cover all the missing entries. We show that the considered problems are fixed-parameter tractable when parameterized by the three parameters combined, and that dropping any of the three parameters results in parameterized intractability. A byproduct of our results is that, for the complete data setting, all problems under consideration are fixed-parameter tractable parameterized by $k+r$.

A $0,1$ matrix is said to be regular if all of its rows and columns have the same number of ones. We prove that for infinitely many integers k, there exists a square regular $0,1$ matrix with binary rank k, such that the Boolean rank of its complement is $k^{\tilde{\Omega }(\log k)}$. This settles, in a strong form, a question of Pullman (1988) [27] and a conjecture of Hefner et al. (1990) [18]. The result can be viewed as a regular analogue of a recent result of Balodis et al. (2021) [2], motivated by the clique vs. independent set problem in communication complexity and by the (disproved) Alon-Saks-Seymour conjecture in graph theory. As an application of the produced regular matrices, we obtain regular counterexamples to the Alon-Saks-Seymour conjecture and prove that for infinitely many integers k, there exists a regular graph with biclique partition number k and chromatic number $k^{\tilde{\Omega }(\log k)}$.

We study the general setting of weighted search in which a number of weighted targets are hidden in a star-like environment, and a mobile searcher must locate a subset of targets with aggregate weight at least a given value W. The cost of the strategy is the distance traversed by the searcher, and its performance is measured by the worst-case ratio of the cost incurred by the searcher over the cost of an on optimal, offline strategy. This is the first study of a setting that generalizes several problems in search theory such as searching for a single target and searching for unit-weighted targets. We present and analyze a near-optimal strategy using an approach based on parameterized analysis. This problem formulates settings of resource allocation among parallel tasks under uncertainty; specifically, we demonstrate further applications in the design of interruptible systems based on adaptive scheduling of contract algorithms.

We consider a variation of balls-into-bins which randomly allocates m balls into n bins. Following Godfrey's model (SODA, 2008), we assume that each ball t, $1⩽t⩽m$, comes with a hypergraph $H^{\left(t\right)}=\{B_1,B_2,\dots ,B_{s_t}\}$, and each edge $B\in H^{\left(t\right)}$ contains at least a logarithmic number of bins. Given $d⩾2$, our d-choice algorithm chooses an edge $B\in H^{\left(t\right)}$, uniformly at random, and then chooses a set D of d random bins from the selected edge B. The ball is allocated to a least-loaded bin from D. We prove that if the hypergraphs $H^{\left(1\right)},\dots ,H^{\left(m\right)}$ satisfy a balancedness condition and have low pair visibility, then after allocating $m=\Theta \left(n\right)$ balls, the maximum load of any bin is at most ${\log }_d\log n+O\left(1\right)$, with high probability. Moreover, we establish a lower bound for the maximum load attained by the balanced allocation for a sequence of hypergraphs in terms of pair visibility.