Algorithmica最新文献

We consider a matching problem in a bipartite graph (G = (A cup B, E)) where vertices in A rank their neighbors in a strict order of preference while vertices in B are allowed to have weak rankings, i.e., ties are allowed in their rankings. Stable matchings always exist in G and are easy to find, however popular matchings need not exist in G and it is NP-complete to decide if one exists. This motivates the “approximately popular” matching problem. A well-known measure of approximate popularity is low unpopularity factor. We show that when each tie in G has length at most k, there always exists a stable matching whose unpopularity factor is at most k and such a matching can be computed in polynomial time. Thus when ties have bounded length, there always exists a near-popular stable matching. This can be considered to be a generalization of Gärdenfors’ result (1975) which showed that when rankings are strict, every stable matching is popular. We then extend our result to the hospitals/residents setting, i.e., vertices in B have capacities. There are several applications where the size of the matching is its most important attribute. When ties are one-sided and of length at most k, we show a polynomial time algorithm to find a maximum matching whose unpopularity factor within the set of maximum matchings is at most 2k.

Roman domination is one of the many variants of domination that keeps most of the complexity features of the classical domination problem. We prove that Roman domination behaves differently in two aspects: enumeration and extension. We develop non-trivial enumeration algorithms for minimal Roman dominating functions with polynomial delay and polynomial space. Recall that the existence of a similar enumeration result for minimal dominating sets is open for decades. Our result is based on a polynomial-time algorithm for Extension Roman Domination: Given a graph (G=(V,E)) and a function (f:Vrightarrow {0,1,2}), is there a minimal Roman dominating function (tilde{f}) with (fle tilde{f})? Here, (le ) lifts (0< 1< 2) pointwise; minimality is understood in this order. Our enumeration algorithm is also analyzed from an input-sensitive viewpoint, leading to a run-time estimate of (mathcal {O}(1.9332^n)) for graphs of order n; this is complemented by a lower bound example of (Omega (1.7441^n)).

We study the problem of solving consensus in synchronous directed dynamic networks, in which communication is controlled by an oblivious message adversary that picks the communication graph to be used in a round from a fixed set of graphs (textbf{D}) arbitrarily. In this fundamental model, determining consensus solvability and designing efficient consensus algorithms is surprisingly difficult. Enabled by a decision procedure that is derived from a well-established previous consensus solvability characterization for a given set (textbf{D}), we study, for the first time, the time complexity of solving consensus in this model: We provide both upper and lower bounds for this time complexity, and also relate it to the number of iterations required by the decision procedure. Among other results, we find that reaching consensus under an oblivious message adversary can take exponentially longer than both deciding consensus solvability and broadcasting the input value of some unknown process to all other processes.

We study the problem of approximating the eigenspectrum of a symmetric matrix (textbf{A} in mathbb {R}^{n times n}) with bounded entries (i.e., (Vert textbf{A}Vert _{infty } le 1)). We present a simple sublinear time algorithm that approximates all eigenvalues of (textbf{A}) up to additive error (pm epsilon n) using those of a randomly sampled ({tilde{O}}left( frac{log ^3 n}{epsilon ^3}right) times {{tilde{O}}}left( frac{log ^3 n}{epsilon ^3}right) ) principal submatrix. Our result can be viewed as a concentration bound on the complete eigenspectrum of a random submatrix, significantly extending known bounds on just the singular values (the magnitudes of the eigenvalues). We give improved error bounds of (pm epsilon sqrt{text {nnz}(textbf{A})}) and (pm epsilon Vert textbf{A}Vert _F) when the rows of (textbf{A}) can be sampled with probabilities proportional to their sparsities or their squared (ell _2) norms respectively. Here (text {nnz}(textbf{A})) is the number of non-zero entries in (textbf{A}) and (Vert textbf{A}Vert _F) is its Frobenius norm. Even for the strictly easier problems of approximating the singular values or testing the existence of large negative eigenvalues (Bakshi, Chepurko, and Jayaram, FOCS ’20), our results are the first that take advantage of non-uniform sampling to give improved error bounds. From a technical perspective, our results require several new eigenvalue concentration and perturbation bounds for matrices with bounded entries. Our non-uniform sampling bounds require a new algorithmic approach, which judiciously zeroes out entries of a randomly sampled submatrix to reduce variance, before computing the eigenvalues of that submatrix as estimates for those of (textbf{A}). We complement our theoretical results with numerical simulations, which demonstrate the effectiveness of our algorithms in practice.

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.