Algorithmica最新文献

In the Determinant Maximization problem, given an (n times n) positive semi-definite matrix ({textbf {A}} ) in (mathbb {Q}^{n times n}) and an integer k, we are required to find a (k times k) principal submatrix of ({textbf {A}} ) having the maximum determinant. This problem is known to be NP-hard and further proven to be W[1]-hard with respect to k by Koutis (Inf Process Lett 100:8–13, 2006); i.e., a (f(k)n^{{{,mathrm{mathcal {O}},}}(1)})-time algorithm is unlikely to exist for any computable function f. However, there is still room to explore its parameterized complexity in the restricted case, in the hope of overcoming the general-case parameterized intractability. In this study, we rule out the fixed-parameter tractability of Determinant Maximization even if an input matrix is extremely sparse or low rank, or an approximate solution is acceptable. We first prove that Determinant Maximization is NP-hard and W[1]-hard even if an input matrix is an arrowhead matrix; i.e., the underlying graph formed by nonzero entries is a star, implying that the structural sparsity is not helpful. By contrast, Determinant Maximization is known to be solvable in polynomial time on tridiagonal matrices (Al-Thani and Lee, in: LAGOS, 2021). Thereafter, we demonstrate the W[1]-hardness with respect to the rank r of an input matrix. Our result is stronger than Koutis’ result in the sense that any (k times k) principal submatrix is singular whenever (k > r). We finally give evidence that it is W[1]-hard to approximate Determinant Maximization parameterized by k within a factor of (2^{-csqrt{k}}) for some universal constant (c > 0). Our hardness result is conditional on the Parameterized Inapproximability Hypothesis posed by Lokshtanov et al. (in: SODA, 2020), which asserts that a gap version of Binary Constraint Satisfaction Problem is W[1]-hard. To complement this result, we develop an (varepsilon )-additive approximation algorithm that runs in (varepsilon ^{-r^2} cdot r^{{{,mathrm{mathcal {O}},}}(r^3)} cdot n^{{{,mathrm{mathcal {O}},}}(1)}) time for the rank r of an input matrix, provided that the diagonal entries are bounded.

We consider the problem of finding a “treasure” at an unknown point of an n-dimensional infinite grid, (nge 3), by initially collocated finite automaton (FA) agents. Recently, the problem has been well characterized for 2 dimensions for deterministic as well as randomized FA agents, both in synchronous and semi-synchronous models (Brandt et al. in Proceedings of 32nd International Symposium on Distributed Computing (DISC) LIPCS 121:13:1–13:17, 2018; Emek et al. in Theor Comput Sci 608:255–267, 2015). It has been conjectured that (n+1) randomized FA agents are necessary to solve this problem in the n-dimensional grid (Cohen et al. in Proceedings of the 28th SODA, SODA ’17, pp 207–224, 2017). In this paper we disprove the conjecture in a strong sense: we show that three randomized synchronous FA agents suffice to explore an n-dimensional grid for any n. Our algorithm is optimal in terms of the number of the agents. Our key insight is that a constant number of FA agents can, by their positions and movements, implement a stack, which can store the path being explored. We also show how to implement our algorithm using: four randomized semi-synchronous FA agents; four deterministic synchronous FA agents; or five deterministic semi-synchronous FA agents. We give a different, no-stack algorithm that uses 4 deterministic semi-synchronous FA agents for the 3-dimensional grid. This is provably optimal in the number of agents and the exploration cost, and surprisingly, matches the result for 2 dimensions. For (nge 4), the time complexity of the stack-based algorithms mentioned above is exponential in distance D of the treasure from the starting point of the agents. We show that in the deterministic case, one additional finite automaton agent brings the time down to a polynomial. We also show that any algorithm using 3 synchronous deterministic FA agents in 3 dimensions must travel beyond (Omega (D^{3/2})) from the origin. Finally, we show that all the above algorithms can be generalized to unoriented grids. More specifically, six deterministic semi-synchronous FA agents are sufficient to locate the treasure in an unoriented n-dimensional grid.

The Feedback Vertex Set problem is undoubtedly one of the most well-studied problems in Parameterized Complexity. In this problem, given an undirected graph G and a non-negative integer k, the objective is to test whether there exists a subset (Ssubseteq V(G)) of size at most k such that (G-S) is a forest. After a long line of improvement, recently, Li and Nederlof [TALG, 2022] designed a randomized algorithm for the problem running in time ({mathcal {O}}^{star }(2.7^k)^{*}). In the Parameterized Complexity literature, several problems around Feedback Vertex Set have been studied. Some of these include Independent Feedback Vertex Set (where the set S should be an independent set in G), Almost Forest Deletion and Pseudoforest Deletion. In Pseudoforest Deletion, each connected component in (G-S) has at most one cycle in it. However, in Almost Forest Deletion, the input is a graph G and non-negative integers (k,ell in {{mathbb {N}}}), and the objective is to test whether there exists a vertex subset S of size at most k, such that (G-S) is (ell ) edges away from a forest. In this paper, using the methodology of Li and Nederlof [TALG, 2022], we obtain the current fastest algorithms for all these problems. In particular we obtain the following randomized algorithms.

1. 1.

   Independent Feedback Vertex Set can be solved in time ({mathcal {O}}^{star }(2.7^k)).

2. 2.

   Pseudo Forest Deletion can be solved in time ({mathcal {O}}^{star }(2.85^k)).

3. 3.

   Almost Forest Deletion can be solved in time ({mathcal {O}}^{star }(min {2.85^k cdot 8.54^ell ,2.7^k cdot 36.61^ell ,3^k cdot 1.78^ell })).

We study truthful mechanisms for welfare maximization in online bipartite matching. In our (multi-parameter) setting, every buyer is associated with a (possibly private) desired set of items, and has a private value for being assigned an item in her desired set. Unlike most online matching settings, where agents arrive online, in our setting the items arrive one by one in an adversarial order while the buyers are present for the entire duration of the process. This poses a significant challenge to the design of truthful mechanisms, due to the ability of buyers to strategize over future rounds. We provide an almost full picture of the competitive ratios in different scenarios, including myopic vs. non-myopic agents, tardy vs. prompt payments, and private vs. public desired sets. Among other results, we identify the frontier up to which the celebrated (e/(e-1)) competitive ratio for the vertex-weighted online matching of Karp, Vazirani and Vazirani extends to truthful agents and online items.

We consider the predecessor problem on the ultra-wide word RAM model of computation, which extends the word RAM model with ultrawords consisting of (w^2) bits (TAMC, 2015). The model supports arithmetic and boolean operations on ultrawords, in addition to scattered memory operations that access or modify w (potentially non-contiguous) memory addresses simultaneously. The ultra-wide word RAM model captures (and idealizes) modern vector processor architectures. Our main result is a simple, linear space data structure that supports predecessor in constant time and updates in amortized, expected constant time. This improves the space of the previous constant time solution that uses space in the order of the size of the universe. Our result holds even in a weaker model where ultrawords consist of (w^{1+epsilon }) bits for any (epsilon > 0 ). It is based on a new implementation of the classic x-fast trie data structure of Willard (Inform Process Lett 17(2):81–84, https://doi.org/10.1016/0020-0190(83)90075-3, 1983) combined with a new dictionary data structure that supports fast parallel lookups.

In this paper we study the classical problem of throughput maximization. In this problem we have a collection J of n jobs, each having a release time (r_j), deadline (d_j), and processing time (p_j). They have to be scheduled non-preemptively on m identical parallel machines. The goal is to find a schedule which maximizes the number of jobs scheduled entirely in their ([r_j,d_j]) window. This problem has been studied extensively (even for the case of (m=1)). Several special cases of the problem remain open. Bar-Noy et al. (Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, May 1–4, 1999, Atlanta, Georgia, USA, pp. 622–631. ACM, 1999, https://doi.org/10.1145/301250.301420) presented an algorithm with ratio (1-1/(1+1/m)^m) for m machines, which approaches (1-1/e) as m increases. For (m=1), Chuzhoy et al. (42nd Annual Symposium on Foundations of Computer Science (FOCS) 2001, 14–17 October 2001, Las Vegas, Nevada, USA, pp. 348–356. IEEE Computer Society, 2001) presented an algorithm with approximation with ratio (1-frac{1}{e}-varepsilon ) (for any (varepsilon >0)). Recently Im et al. (SIAM J Discrete Math 34(3):1649–1669, 2020) presented an algorithm with ratio (1-1/e+varepsilon _0) for some absolute constant (varepsilon _0>0) for any fixed m. They also presented an algorithm with ratio (1-O(sqrt{log m/m})-varepsilon ) for general m which approaches 1 as m grows. The approximability of the problem for (m=O(1)) remains a major open question. Even for the case of (m=1) and (c=O(1)) distinct processing times the problem is open (Sgall in: Algorithms - ESA 2012 - 20th Annual European Symposium, Ljubljana, Slovenia, September 10–12, 2012. Proceedings, pp 2–11, 2012). In this paper we study the case of (m=O(1)) and show that if there are c distinct processing times, i.e. (p_j)’s come from a set of size c, then there is a randomized ((1-{varepsilon }))-approximation that runs in time (O(n^{mc^7{varepsilon }^{-6}}log T)), where T is the largest deadline. Therefore, for constant m and constant c this yields a PTAS. Our algorithm is based on proving structural properties for a near optimum solution that allows one to use a dynamic programming with pruning.

In this paper, we study the Harmless Set problem from a parameterized complexity perspective. Given a graph (G = (V,E)), a threshold function(~t~:~ V rightarrow {mathbb {N}}) and an integer k, we study Harmless Set, where the goal is to find a subset of vertices (S subseteq V) of size at least k such that every vertex (vin V) has fewer than t(v) neighbours in S. On the positive side, we obtain fixed-parameter algorithms for the problem when parameterized by the neighbourhood diversity, the twin-cover number and the vertex integrity of the input graph. We complement two of these results from the negative side. On dense graphs, we show that the problem is W[1]-hard parameterized by cluster vertex deletion number—a natural generalization of the twin-cover number. We show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, and treedepth—a natural generalization of the vertex integrity. We thereby resolve one open question stated by Bazgan and Chopin (Discrete Optim 14(C):170–182, 2014) concerning the complexity of Harmless Set parameterized by the treewidth of the input graph. We also show that Harmless Set for a special case where each vertex has the threshold set to half of its degree (the so-called Majority Harmless Set problem) is W[1]-hard when parameterized by the treewidth of the input graph. Given a graph G and an irredundant c-expression of G, we prove that Harmless Set can be solved in XP-time when parameterized by clique-width.

For a set ({mathcal {Q}}) of points in the plane and a real number (delta ge 0), let ({mathbb {G}}_delta ({mathcal {Q}})) be the graph defined on ({mathcal {Q}}) by connecting each pair of points at distance at most (delta ).We consider the connectivity of ({mathbb {G}}_delta ({mathcal {Q}})) in the best scenario when the location of a few of the points is uncertain, but we know for each uncertain point a line segment that contains it. More precisely, we consider the following optimization problem: given a set ({mathcal {P}}) of (n-k) points in the plane and a set ({mathcal {S}}) of k line segments in the plane, find the minimum (delta ge 0) with the property that we can select one point (p_sin s) for each segment (sin {mathcal {S}}) and the corresponding graph ({mathbb {G}}_delta ( {mathcal {P}}cup { p_smid sin {mathcal {S}}})) is connected. It is known that the problem is NP-hard. We provide an algorithm to exactly compute an optimal solution in ({{,mathrm{{mathcal {O}}},}}(f(k) n log n)) time, for a computable function (f(cdot )). This implies that the problem is FPT when parameterized by k. The best previous algorithm uses ({{,mathrm{{mathcal {O}}},}}((k!)^k k^{k+1}cdot n^{2k})) time and computes the solution up to fixed precision.

The leafage of a chordal graph G is the minimum integer (ell ) such that G can be realized as an intersection graph of subtrees of a tree with (ell ) leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time (2^{mathcal {O}(ell ^2)} cdot n^{mathcal {O}(1)}). We present a conceptually much simpler algorithm that runs in time (2^{mathcal {O}(ell )} cdot n^{mathcal {O}(1)}). We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple (n^{mathcal {O}(ell )})-time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in (n^{{{mathcal {O}}}(1)})-time.