A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for t...
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A vertex set D in a finite undirected graph G is an efficient dominating set (e.d.s for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d.s. in G, is known to be NP-complete even for very restricted graph classes such as for 2P(3)-free chordal graphs while it is solvable in polynomialtime for P-6-free chordal graphs (and even for P-6-free graphs). A standard reduction from the NP-complete Exact Cover problem shows that ED is NP-complete for a very special subclass of chordal graphs generalizing split graphs. The reduction implies that ED is NP-complete e.g. for double-gem-free chordal graphs while it is solvable in linear time for gem-free chordal graphs (by various reasons such as bounded clique-width, distance-hereditary graphs, chordal square etc.), and ED is NP-complete for butterfly-free chordal graphs while it is solvable in linear time for 2P(2)-free graphs. We show that (weighted) ED can be solved in polynomialtime for H-free chordal graphs when H is net, extended gem, or S-1,S-2,S-3. (c) 2019 Elsevier B.V. All rights reserved.
We propose bipartite analogues of comparability and cocomparability graphs. Surprisingly, the two classes coincide. We call these bipartite graphs cocomparability bigraphs. We characterize cocomparability bigraphs in ...
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We propose bipartite analogues of comparability and cocomparability graphs. Surprisingly, the two classes coincide. We call these bipartite graphs cocomparability bigraphs. We characterize cocomparability bigraphs in terms of vertex orderings, forbidden substructures, and orientations of their complements. In particular, we prove that cocomparability bigraphs are precisely those bipartite graphs that do not have edge-asteroids;this is analogous to Gallai's structural characterization of cocomparability graphs by the absence of (vertex-) asteroids. Our characterizations imply a robust polynomial-time recognition algorithm for the class of cocomparability bigraphs. Finally, we also discuss a natural relation of cocomparability bigraphs to interval containment bigraphs, resembling a well-known relation of cocomparability graphs to interval graphs.
While every instance of the Hospitals/Residents problem admits a stable matching, the problem with lower quotas (HR-LQ) has instances with no stable matching. For such an instance, we expect the existence of an envy-f...
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While every instance of the Hospitals/Residents problem admits a stable matching, the problem with lower quotas (HR-LQ) has instances with no stable matching. For such an instance, we expect the existence of an envy-free matching, which is a relaxation of a stable matching preserving a kind of fairness property. In this paper, we investigate the existence of an envy-free matching in several settings, in which hospitals have lower quotas and not all doctor-hospital pairs are acceptable. We first provide an algorithm that decides whether a given HR-LQ instance has an envy-free matching or not. Then, we consider envy-freeness in the Classified Stable Matching model due to Huang (in: Procedings of 21st annual ACM-SIAM symposium on discrete algorithms (SODA2010), SIAM, Philadelphia, pp 1235-1253, 2010), i.e., each hospital has lower and upper quotas on subsets of doctors. We show that, for this model, deciding the existence of an envy-free matching is NP-hard in general, but solvable in polynomialtime if quotas are paramodular.
We investigate a single-machine scheduling problem,where a deteriorating rate-modifying activity can be performed on the machine to reduce the processing times of *** objective is to minimize the number of tardy *** t...
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We investigate a single-machine scheduling problem,where a deteriorating rate-modifying activity can be performed on the machine to reduce the processing times of *** objective is to minimize the number of tardy *** the assumption that the duration of the rate-modifying activity is a nonnegative and nondecreasing function on its starting time,we propose an optimal polynomial time algorithm running in O(n^(3))time.
A set D of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to a vertex in D and the subgraph induced by D contains a perfect matching (not necessarily as an induced subgraph). A ...
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A set D of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to a vertex in D and the subgraph induced by D contains a perfect matching (not necessarily as an induced subgraph). A paired-dominating set of G is minimal if no proper subset of it is a paired-dominating set of G. The upper paired-domination number of G, denoted by Gamma(pr)(G), is the maximum cardinality of a minimal paired-dominating set of G. In UPPER PDS, it is required to compute a minimal paired dominating set with cardinality Gamma(pr)(G) for a given graph G. In this paper, we show that UPPER-PDS cannot be approximated within a factor of n(1-epsilon) for any epsilon > 0, unless P=NP and UPPER-PDS is APX-complete for bipartite graphs of maximum degree 4. On the positive side, we show that UPPER PDS can be approximated within O(Delta)-factor for graphs with maximum degree Delta. We also show that UPPER-PDS is solvable in polynomialtime for threshold graphs, chain graphs, and proper interval graphs. (C) 2019 Elsevier B.V. All rights reserved.
In this paper, we address the weighted linear matroid intersection problem from computation of the degree of the determinant of a symbolic matrix. We show that a generic algorithm computing the degree of noncommutativ...
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In this paper, we address the weighted linear matroid intersection problem from computation of the degree of the determinant of a symbolic matrix. We show that a generic algorithm computing the degree of noncommutative determinants, proposed by the second author, becomes an O(mn(3) log n) timealgorithm for the weighted linear matroid intersection problem, where two matroids are given by column vectors of n x m matrices A, B. We reveal that our algorithm is viewed as a "nonstandard" implementation of Frank's weight splitting algorithm for linear matroids. This gives a linear algebraic reasoning to Frank's algorithm. Although our algorithm is slower than existing algorithms in the worst case estimate, it has a notable feature. Contrary to existing algorithms, our algorithm works on different matroids represented by another "sparse" matrices A(0), B-0, which skips unnecessary Gaussian eliminations for constructing residual graphs.
Given a graph G = (V, E), a function f : V -> {0, 1, 2, 3} is called a double Roman dominating function on G if (i) for every v. V with f (v) = 0, there are at least two neighbors of v that are assigned 2 under f o...
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Given a graph G = (V, E), a function f : V -> {0, 1, 2, 3} is called a double Roman dominating function on G if (i) for every v. V with f (v) = 0, there are at least two neighbors of v that are assigned 2 under f or at least a neighbor of v that is assigned 3 under f, and (ii) for every vertex v with f (v) = 1, there is at least one neighbor w of v with f (w) >= 2. The weight of a double Roman dominating function f is f (V) = Sigma(u is an element of V) f (u). The double Roman domination number of G, denoted by gamma(dR)(G) is the minimum weight of a double Roman dominating function on G. For a graph G = (V, E), MIN-DOUBLE-RDF is to find a double Roman dominating function f with f (V) = gamma(dR)(G). The decision version of MIN-DOUBLE-RDF is shown to be NP-complete for chordal graphs and bipartite graphs. In this paper, we first strengthen the known NP-completeness of the decision version of MIN-DOUBLE-RDF by showing that the decision version of MIN-DOUBLE-RDF remains NP-complete for undirected path graphs, chordal bipartite graphs, and circle graphs. We then present linear timealgorithms for computing the double Roman domination number in proper interval graphs and block graphs. We then discuss on the approximability of MIN-DOUBLE-RDF and present a 2-approximation algorithm in 3-regular bipartite graphs.
Two vertices in a graph are said to 2-step dominate each other if they are at distance 2 apart. A set S of vertices in a graph G = (V, E) is a hop dominating set of G if every vertex outside S is 2-step dominated by s...
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Two vertices in a graph are said to 2-step dominate each other if they are at distance 2 apart. A set S of vertices in a graph G = (V, E) is a hop dominating set of G if every vertex outside S is 2-step dominated by some vertex of S. Given a graph G, MIN-HDS is the problem of finding a hop dominating set of G of minimum cardinality. The decision version of MIN-HDS is known to be NP-complete for planar bipartite graphs and planar chordal graphs, and hence for bipartite graphs and chordal graphs. In this paper, we present a linear timealgorithm for computing a minimum hop dominating set in bipartite permutation graphs, which is a subclass of bipartite graphs. We also show that MIN-HDS cannot be approximated within a factor of (1 - epsilon) In vertical bar V vertical bar, unless P=NP and can be approximated within a factor of 1 + In(Delta(Delta - 1) + 1), where Delta denotes the maximum degree in the graph G. Finally, we show that MIN-HDS is APX-complete for bipartite graphs of maximum degree 3. (C) 2019 Elsevier B.V. All rights reserved.
A set D subset of V of a graph G = (V, E) is called a neighborhood total dominating set of G if D is a dominating set and the subgraph of G induced by the open neighborhood of D has no isolated vertex. Given a graph G...
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A set D subset of V of a graph G = (V, E) is called a neighborhood total dominating set of G if D is a dominating set and the subgraph of G induced by the open neighborhood of D has no isolated vertex. Given a graph G, MIN-NTDS is the problem of finding a neighborhood total dominating set of G of minimum cardinality. The decision version of MIN-NTDS is known to be NP-complete for bipartite graphs and chordal graphs. In this paper, we extend this NP-completeness result to undirected path graphs, chordal bipartite graphs, and planar graphs. We also present a linear timealgorithm for computing a minimum neighborhood total dominating set in proper interval graphs. We show that for a given graph G = (V, E), MIN-NTDS cannot be approximated within a factor of (1 - epsilon) log vertical bar V vertical bar, unless NP subset of Dtime(vertical bar V vertical bar(O(loglog vertical bar V vertical bar))) and can be approximated within a factor of O(log Delta), where Delta is the maximum degree of the graph G. Finally, we show that MIN-NTDS is APX-complete for graphs of degree at most 3. (C) 2020 Elsevier B.V. All rights reserved.
We provide a polynomial time algorithm for deciding the equation solvability problem over finite groups that are semidirect products of a p-group and an Abelian group. As a consequence, we obtain a polynomialtime alg...
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We provide a polynomial time algorithm for deciding the equation solvability problem over finite groups that are semidirect products of a p-group and an Abelian group. As a consequence, we obtain a polynomial time algorithm for deciding the equivalence problem over semidirect products of a finite nilpotent group and a finite Abelian group. The key ingredient of the proof is to represent group expressions using a special polycyclic, presentation of these finite solvable groups.
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