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Joan Boyar
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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This paper is devoted to the online dominating set problem and its variants. We believe the paper represents the first systematic study of the effect of two limitations of online algorithms: making irrevocable decisions while not knowing the future, and being incremental, i.e., having to maintain solutions to all prefixes of the input. This is quantified through competitive analyses of online algorithms against two optimal algorithms, both knowing the entire input, but only one having to be incremental. We also consider the competitive ratio of the weaker of the two optimal algorithms against the other. We consider important graph classes, distinguishing between connected and not necessarily connected graphs. For the classic graph classes of trees, bipartite, planar, and general graphs, we obtain tight results in almost all cases. We also derive upper and lower bounds for the class of bounded-degree graphs. From these analyses, we get detailed information regarding the significance of the necessary requirement that online algorithms be incremental. In some cases, having to be incremental fully accounts for the online algorithms disadvantage.
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Dealing with the NP-complete Dominating Set problem on undirected graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set restricted to planar graphs has a so-called problem kernel of linear size, achieved by two simple and easy to implement reduction rules. Moreover, having implemented our reduction rules, first experiments indicate the impressive practical potential of these rules. Thus, this work seems to open up a new and prospective way how to cope with one of the most important problems in graph theory and combinatorial optimization.
We recently introduced the graph invariant twin-width, and showed that first-order model checking can be solved in time $f(d,k)n$ for $n$-vertex graphs given with a witness that the twin-width is at most $d$, called $d$-contraction sequence or $d$-sequence, and formulas of size $k$ [Bonnet et al., FOCS 20]. The inevitable price to pay for such a general result is that $f$ is a tower of exponentials of height roughly $k$. In this paper, we show that algorithms based on twin-width need not be impractical. We present $2^{O(k)}n$-time algorithms for $k$-Independent Set, $r$-Scattered Set, $k$-Clique, and $k$-Dominating Set when an $O(1)$-sequence is provided. We further show how to solve weighted $k$-Independent Set, Subgraph Isomorphism, and Induced Subgraph Isomorphism, in time $2^{O(k log k)}n$. These algorithms are based on a dynamic programming scheme following the sequence of contractions forward. We then show a second algorithmic use of the contraction sequence, by starting at its end and rewinding it. As an example, we establish that bounded twin-width classes are $chi$-bounded. This significantly extends the $chi$-boundedness of bounded rank-width classes, and does so with a very concise proof. The third algorithmic use of twin-width builds on the second one. Playing the contraction sequence backward, we show that bounded twin-width graphs can be edge-partitioned into a linear number of bicliques, such that both sides of the bicliques are on consecutive vertices, in a fixed vertex ordering. Given that biclique edge-partition, we show how to solve the unweighted Single-Source Shortest Paths and hence All-Pairs Shortest Paths in sublinear time $O(n log n)$ and time $O(n^2 log n)$, respectively. Finally we show that Min Dominating Set and related problems have constant integrality gaps on bounded twin-width classes, thereby getting constant approximations on these classes.
In this article, we study a generalized version of the maximum independent set and minimum dominating set problems, namely, the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem on unit disk graphs for a positive integer $d>0$. We first show that the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem belongs to NP-hard class. Next, we propose a simple polynomial-time constant-factor approximation algorithms and PTAS for both the problems.
Retraction note: After posting the manuscript on arXiv, we were informed by Erik Jan van Leeuwen that both results were known and they appeared in his thesis[vL09]. A PTAS for MDS is at Theorem 6.3.21 on page 79 and A PTAS for MCDS is at Theorem 6.3.31 on page 82. The techniques used are very similar. He noted that the idea for dealing with the connected version using a constant number of extra layers in the shifting technique not only appeared Zhang et al.[ZGWD09] but also in his 2005 paper [vL05]. Finally, van Leeuwen also informed us that the open problem that we posted has been resolved by Marx~[Mar06, Mar07] who showed that an efficient PTAS for MDS does not exist [Mar06] and under ETH, the running time of $n^{O(1/epsilon)}$ is best possible [Mar07]. We thank Erik Jan van Leeuwen for the information and we regret that we made this mistake. Abstract before retraction: We present two (exponentially) faster PTASs for dominating set problems in unit disk graphs. Given a geometric representation of a unit disk graph, our PTASs that find $(1+epsilon)$-approximate solutions to the Minimum Dominating Set (MDS) and the Minimum Connected Dominating Set (MCDS) of the input graph run in time $n^{O(1/epsilon)}$. This can be compared to the best known $n^{O(1/epsilon log {1/epsilon})}$-time PTAS by Nieberg and Hurink~[WAOA05] for MDS that only uses graph structures and an $n^{O(1/epsilon^2)}$-time PTAS for MCDS by Zhang, Gao, Wu, and Du~[J Glob Optim09]. Our key ingredients are improved dynamic programming algorithms that depend exponentially on more essential 1-dimensional widths of the problems.
We show that the k-Dominating Set problem is fixed parameter tractable (FPT) and has a polynomial kernel for any class of graphs that exclude K_{i,j} as a subgraph, for any fixed i, j >= 1. This strictly includes every class of graphs for which this problem has been previously shown to have FPT algorithms and/or polynomial kernels. In particular, our result implies that the problem restricted to bounded- degenerate graphs has a polynomial kernel, solving an open problem posed by Alon and Gutner.
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