Algorithmica最新文献

We initiate the study of the Diverse Pair of (Maximum/ Perfect) Matchings problems which given a graph G and an integer k, ask whether G has two (maximum/perfect) matchings whose symmetric difference is at least k. Diverse Pair of Matchings (asking for two not necessarily maximum or perfect matchings) is (textsf{NP})-complete on general graphs if k is part of the input, and we consider two restricted variants. First, we show that on bipartite graphs, the problem is polynomial-time solvable, and second we show that Diverse Pair of Maximum Matchings is (textsf{FPT}) parameterized by k. We round off the work by showing that Diverse Pair of Matchings has a kernel on ({mathcal {O}}(k^2)) vertices.

A finite group of order n can be represented by its Cayley table. In the word-RAM model the Cayley table of a group of order n can be stored using (O(n^2)) words and can be used to answer a multiplication query in constant time. It is interesting to ask if we can design a data structure to store a group of order n that uses (o(n^2)) space but can still answer a multiplication query in constant time. Das et al. (J Comput Syst Sci 114:137–146, 2020) showed that for any finite group G of order n and for any (delta in [1/log {n}, 1]), a data structure can be constructed for G that uses (O(n^{1+delta }/delta )) space and answers a multiplication query in time (O(1/delta )). Farzan and Munro (ISSAC, 2006) gave an information theoretic lower bound of (Omega (n)) on the number of words to store a group of order n. We design a constant query-time data structure that can store any finite group using O(n) words where n is the order of the group. Since our data structure achieves the information theoretic lower bound and answers queries in constant time, it is optimal in both space usage and query-time. A crucial step in the process is essentially to design linear space and constant query-time data structures for nonabelian simple groups. The data structures for nonabelian simple groups are designed using a lemma that we prove using the Classification Theorem for Finite Simple Groups.

Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighbor instances, that is, instances with one local modification? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems, most notably, graph theory’s core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted). We focus on two prototypical graph problems, colorability and vertex cover. For example, we show that it is (text {NP})-hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in (text {P}). We observe that vertex cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level. Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal-3-UnColorability is complete for (text {DP}) (differences of (text {NP}) sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For vertex cover, we show that recognizing (beta )-vertex-critical graphs is complete for (Theta _2^text {p}) (parallel access to (text {NP})), obtaining the first completeness result for a criticality problem for this class.

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.