Algorithmica最新文献

We study a fundamental question related to the feasibility of deterministic symmetry breaking in the infinite Euclidean plane for two robots that have minimal or no knowledge of the respective capabilities and “measuring instruments” of themselves and each other. Assume that two anonymous mobile robots are placed at different locations at unknown distance d from each other on the infinite Euclidean plane. Each robot knows neither the location of itself nor of the other robot. The robots cannot communicate wirelessly, but have a certain nonzero visibility radius r (with range r unknown to the robots). By rendezvous we mean that they are brought at distance at most r of each other by executing symmetric (identical) mobility algorithms. The robots are moving with unknown and constant but not necessarily identical speeds, their clocks and pedometers may be asymmetric, and their chirality inconsistent. We demonstrate that rendezvous for two robots is feasible under the studied model iff the robots have either: different speeds; or different clocks; or different orientations but equal chiralities. When the rendezvous is feasible, we provide a universal algorithm which always solves rendezvous despite the fact that the robots have no knowledge of which among their respective parameters may be different.

We consider the popular matching problem in a roommates instance G on n vertices, i.e., G is a graph where each vertex has a strict preference order over its neighbors. A matching M is popular if there is no matching N such that the vertices that prefer N to M outnumber those that prefer M to N. It is known that it is NP-hard to decide if G admits a popular matching or not. There is no better algorithm known for this problem than the brute force algorithm that enumerates all matchings and tests each for popularity—this could take n! time. Here we show an (O^*(k^n)) time algorithm for this problem, where (k < 7.32). We use the recent breakthrough result on the maximum number of stable matchings possible in a roommates instance to analyze our algorithm for the popular matching problem. We identify a natural (also, hard) subclass of popular matchings called truly popular matchings that are “popular fractional” and show an (O^*(2^n)) time algorithm for the truly popular matching problem in G. We also identify a subclass of max-size popular matchings called super-dominant matchings and show a linear time algorithm for the super-dominant roommates problem.

We study the impact of merging routines in merge-based sorting algorithms. More precisely, we focus on the galloping routine that TimSort uses to merge monotonic sub-arrays, hereafter called runs, and on the impact on the number of element comparisons performed if one uses this routine instead of a naïve merging routine. This routine was introduced in order to make TimSort more efficient on arrays with few distinct values. Alas, we prove that, although it makes TimSort sort array with two values in linear time, it does not prevent TimSort from requiring up to (Theta (n log (n))) element comparisons to sort arrays of length n with three distinct values. However, we also prove that slightly modifying TimSort ’s galloping routine results in requiring only (mathcal {O}(n + n log (sigma ))) element comparisons in the worst case, when sorting arrays of length n with (sigma ) distinct values. We do so by focusing on the notion of dual runs, which was introduced in the 1990s, and on the associated dual run-length entropy. This notion is both related to the number of distinct values and to the number of runs in an array, which came with its own run-length entropy that was used to explain TimSort ’s otherwise “supernatural” efficiency. We also introduce new notions of fast- and middle-growth for natural merge sorts (i.e., algorithms based on merging runs), which are found in several sorting algorithms similar to TimSort. We prove that algorithms with the fast- or middle-growth property, provided that they use our variant of TimSort ’s galloping routine for merging runs, are as efficient as possible at sorting arrays with low run-induced or dual-run-induced complexities.

Let (mathscr {B}) be a set of n unit balls in ({mathbb {R}}^3). We present a linear-size data structure for storing (mathscr {B}) that can determine in (O^*(sqrt{n})) time whether a query line intersects any ball of (mathscr {B}) and report all k such balls in additional O(k) time. The data structure can be constructed in (O(nlog n)) time. (The (O^*(cdot )) notation hides subpolynomial factors, e.g., of the form (O(n^{{varepsilon }})), for arbitrarily small ({varepsilon }> 0), and their coefficients which depend on ({varepsilon }).) We also consider the dual problem: Let (mathscr {L}) be a set of n lines in ({mathbb {R}}^3). We preprocess (mathscr {L}), in (O^*(n^2)) time, into a data structure of size (O^*(n^2)) that can determine in (O(log {n})) time whether a query unit ball intersects any line of (mathscr {L}), or report all k such lines in additional O(k) time.

Estimation of distribution algorithms (EDAs) are general-purpose optimizers that maintain a probability distribution over a given search space. This probability distribution is updated through sampling from the distribution and a reinforcement learning process which rewards solution components that have shown to be part of good quality samples. The compact genetic algorithm (cGA) is a non-elitist EDA able to deal with difficult multimodal fitness landscapes that are hard to solve by elitist algorithms. We investigate the cGA on the Cliff function for which it was shown recently that non-elitist evolutionary algorithms and artificial immune systems optimize it in expected polynomial time. We point out that the cGA faces major difficulties when solving the Cliff function and investigate its dynamics both experimentally and theoretically. Our experimental results indicate that the cGA requires exponential time for all values of the update strength 1/K. We show theoretically that, under sensible assumptions, there is a negative drift when sampling around the location of the cliff. Experiments further suggest that there is a phase transition for K where the expected optimization time drops from (n^{Theta (n)}) to (2^{Theta (n)}).

The Maker–Breaker domination game is a positional game played on a graph by two players called Dominator and Staller. The players alternately select a vertex of the graph that has not yet been chosen. Dominator wins if at some point the vertices she has chosen form a dominating set of the graph. Staller wins if Dominator cannot form a dominating set. Deciding if Dominator has a winning strategy has been shown to be a PSPACE-complete problem even when restricted to chordal or bipartite graphs. In this paper, we consider strategies for Dominator based on partitions of the graph into basic subgraphs where Dominator wins as the second player. Using partitions into cycles and edges (also called perfect [1,2]-factors), we show that Dominator always wins in regular graphs and that deciding whether Dominator has a winning strategy as a second player can be computed in polynomial time for outerplanar and block graphs. We then study partitions into subgraphs with two universal vertices, which is equivalent to considering the existence of pairing dominating sets with adjacent pairs. We show that in interval graphs, Dominator wins if and only if such a partition exists. In particular, this implies that deciding whether Dominator has a winning strategy playing second is in NP for interval graphs. We finally provide an algorithm in (n^{k+3}) for interval graphs with at most k nested intervals.

We study the b-matching problem, which generalizes classical online matching introduced by Karp, Vazirani and Vazirani (STOC 1990). Consider a bipartite graph (G=(Sdot{cup }R,E)). Every vertex (sin S) is a server with a capacity (b_s), indicating the number of possible matching partners. The vertices (rin R) are requests that arrive online and must be matched immediately to an eligible server. The goal is to maximize the cardinality of the constructed matching. In contrast to earlier work, we study the general setting where servers may have arbitrary, individual capacities. We prove that the most natural and simple online algorithms achieve optimal competitive ratios. As for deterministic algorithms, we give a greedy algorithm RelativeBalance and analyze it by extending the primal-dual framework of Devanur, Jain and Kleinberg (SODA 2013). In the area of randomized algorithms we study the celebrated Ranking algorithm by Karp, Vazirani and Vazirani. We prove that the original Ranking strategy, simply picking a random permutation of the servers, achieves an optimal competitiveness of (1-1/e), independently of the server capacities. Hence it is not necessary to resort to a reduction, replacing every server s by (b_s) vertices of unit capacity and to then run Ranking on this graph with (sum _{sin S} b_s) vertices on the left-hand side. Additionally, we extend this result to the vertex-weighted b-matching problem. Technically, we formulate a new configuration LP for the b-matching problem and conduct a primal-dual analysis.

In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna (ACM Trans Comput Theory 9:1–36, 2018), that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, Capacitated Vertex Cover and Bipartite Bandwidth.

In the spanning tree congestion problem, given a connected graph G, the objective is to compute a spanning tree T in G that minimizes its maximum edge congestion, where the congestion of an edge e of T is the number of edges in G for which the unique path in T between their endpoints traverses e. The problem is known to be (mathbb{N}mathbb{P})-hard, but its approximability is still poorly understood, and it is not even known whether the optimum solution can be efficiently approximated with ratio o(n). In the decision version of this problem, denoted ({varvec{K}-textsf {STC}}), we need to determine if G has a spanning tree with congestion at most K. It is known that ({varvec{K}-textsf {STC}}) is (mathbb{N}mathbb{P})-complete for (Kge 8), and this implies a lower bound of 1.125 on the approximation ratio of minimizing congestion. On the other hand, ({varvec{3}-textsf {STC}}) can be solved in polynomial time, with the complexity status of this problem for (Kin { left{ 4,5,6,7 right} }) remaining an open problem. We substantially improve the earlier hardness results by proving that ({varvec{K}-textsf {STC}}) is (mathbb{N}mathbb{P})-complete for (Kge 5). This leaves only the case (K=4) open, and improves the lower bound on the approximation ratio to 1.2. Motivated by evidence that minimizing congestion is hard even for graphs of small constant radius, we also consider ({varvec{K}-textsf {STC}}) restricted to graphs of radius 2, and we prove that this variant is (mathbb{N}mathbb{P})-complete for all (Kge 6).

Let M be a perfect matching on a set of points in the plane where every edge is a line segment between two points. We say that M is globally maximum if it is a maximum-length matching on all points. We say that M is k-local maximum if for any subset (M'={a_1b_1,dots ,a_kb_k}) of k edges of M it holds that (M') is a maximum-length matching on points ({a_1,b_1,dots ,a_k,b_k}). We show that local maximum matchings are good approximations of global ones. Let (mu _k) be the infimum ratio of the length of any k-local maximum matching to the length of any global maximum matching, over all finite point sets in the Euclidean plane. It is known that (mu _kgeqslant frac{k-1}{k}) for any (kgeqslant 2). We show the following improved bounds for (kin {2,3}): (sqrt{3/7}leqslant mu _2< 0.93 ) and (sqrt{3}/2leqslant mu _3< 0.98). We also show that every pairwise crossing matching is unique and it is globally maximum. Towards our proof of the lower bound for (mu _2) we show the following result which is of independent interest: If we increase the radii of pairwise intersecting disks by factor (2/sqrt{3}), then the resulting disks have a common intersection.