Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials

Jansen, Bart M. P.; Pieterse, Astrid

doi:10.1007/s00453-019-00578-5

Cited by 11 publications

(4 citation statements)

References 19 publications

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“…Another parameter of interest is size of a minimum vertex cover. Recently, Jansen and Pieterse [101] gave a kernel parameterized on the number q ≥ 3 of colors and the size k of a minimum vertex cover, having size O(k q−1 log k) bits, which is optimal up to a factor of k O(1) [99]. Their result also applies for a tighter parameter, when k is the size of the twin cover.…”

Section: Open Problem 17 How Effective Is Dynamic Programming Over a ...mentioning

confidence: 99%

Recent Advances in Practical Data Reduction

Abu-Khzam¹,

Lamm²,

Mnich³

et al. 2020

Preprint

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Over the last two decades, significant advances have been made in the design and analysis of fixedparameter algorithms for a wide variety of graph-theoretic problems. This has resulted in an algorithmic toolbox that is by now well-established. However, these theoretical algorithmic ideas have received very little attention from the practical perspective. We survey recent trends in data reduction engineering results for selected problems. Moreover, we describe concrete techniques that may be useful for future implementations in the area and give open problems and research questions.2 Recent Advances for NP-Hard Problems Maximum Independent Set and Minimum Vertex CoverGiven an undirected graph G = (V, E), the goal of the maximum independent set (MIS) problem is to compute a set of vertices I ⊆ V such that (1) no two vertices in I are adjacent to one another, (2) the set I has maximum cardinality among all such sets. The complement of an independent set I is called a vertex cover V \ I. The MIS problem and the complementary problem of finding a minimum vertex cover (MVC) are well-studied NP-hard optimization problems [75] that attract both researchers and practitioners alike. Furthermore, there is no polynomial time approximation algorithm for MIS that can provide an O(n 1−ε ) guarantee for any constant ε > 0, unless P=NP [174]. Finally, MIS is W [1]-hard [58] when parameterized by solution size k. This makes it unlikely that the problem is fixed-parameter tractable in k [58]. On the other hand, MVC is fixed-parameter tractable in solution size k [58]. Exact ApproachesIn recent years, the bridge between theoretically efficient algorithms and their practical applicability has been significantly reduced. In particular, the branch-and-reduce paradigm, i.e., branching algorithms that use a wide variety of reduction rules, have been (1) shown to achieve theoretical running times that are among the best for both MIS and MVC [71,170], and (2) are able to solve large real-world networks in practice [5]. However, most often the approaches used in practice only use a small subset of the reduction rules that have been proposed to achieve good theoretical running times.Abu-Khzam et al.[4] introduced and analyzed the crown reduction rule (and the usage of data reduction rules in this context in practice). Even though the crown rule is not as powerful as the linear programming (LP)-based rule [135] when considering the worst-case size of the resulting kernel, they experimentally verified that it often performs as well as the LP-based rule and is significantly faster in many cases. Furthermore, they show that the LP-based rule is most useful for fairly sparse graphs and should be avoided for dense graphs, as it yields little to no reduction in size.Later, Akiba and Iwata [5] were the first to show the practicality of the branch-and-reduce paradigm for MVC (and MIS) compared to other state-of-the-art approaches like branch-and-bound and branch-and-cut. Their algorithm uses a wide spectrum of reduction rules that form the foundation...

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Section: Open Problem 17 How Effective Is Dynamic Programming Over a ...mentioning

confidence: 99%

Recent Advances in Practical Data Reduction

Abu-Khzam¹,

Lamm²,

Mnich³

et al. 2020

Preprint

View full text Add to dashboard Cite

show abstract

“…It has recently been shown (Jansen & Pieterse, 2018) that graph coloring is fixed parameter tractable with parameter k = q + s where q is the size of the coloring the graph admits and s is the cardinality of the minimum vertex cover. The idea of that algorithm is based on the observation that, similar to peeling, for any vertex v of G, a q−coloring of the graph induced by V \ {v} (that we denote G | V \{v} ) can be extended to a q−coloring of G, if at most q − 1 colors are used to color the vertices N (v).…”

Section: Independent Set Extractionmentioning

confidence: 99%

Constraint and Satisfiability Reasoning for Graph Coloring

Hébrard

Katsirelos

2020

jair

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Graph coloring is an important problem in combinatorial optimization and a major component of numerous allocation and scheduling problems. In this paper we introduce a hybrid CP/SAT approach to graph coloring based on the addition-contraction recurrence of Zykov. Decisions correspond to either adding an edge between two non-adjacent vertices or contracting these two vertices, hence enforcing inequality or equality, respectively. This scheme yields a symmetry-free tree and makes learnt clauses stronger by not committing to a particular color. We introduce a new lower bound for this problem based on Mycielskian graphs; a method to produce a clausal explanation of this bound for use in a CDCL algorithm; a branching heuristic emulating Br´elaz’ heuristic on the Zykov tree; and dedicated pruning techniques relying on marginal costs with respect to the bound and on reasoning about transitivity when unit propagating learnt clauses. The combination of these techniques in both a branch-and-bound and in a bottom-up search outperforms other SAT-based approaches and Dsatur on standard benchmarks both for finding upper bounds and for proving lower bounds.

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“…Apart from L(p, 1)-Labeling, twin cover number is a relatively new graph parameter, which is introduced in [20] as a stronger parameter than vertex cover number. In the same paper, many problems are shown to be FPT when parameterized by twin cover number, and it is getting to be a standard parameter (e.g., [1,4,12,21,29,31]). Recently, for Imbalance, which is one of graph layout problems, a parameterized algorithm is presented [34].…”

Section: Related Workmentioning

confidence: 99%

Computing $L(p,1)$-Labeling with Combined Parameters

Hanaka¹,

Kawai²,

Ono³

2020

Preprint

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Given a graph, an L(p, 1)-labeling of the graph is an assignment f from the vertex set to the set of nonnegative integers such that for any pair of vertices (u, v), |f (u) − f (v)| ≥ p if u and v are adjacent, and f (u) = f (v) if u and v are at distance 2. The L(p, 1)-labeling problem is to minimize the span of f (i.e.,maxu∈V (f (u)) − minu∈V (f (u)) + 1). It is known to be NP-hard even for graphs of maximum degree 3 or graphs with tree-width 2, whereas it is fixedparameter tractable with respect to vertex cover number. Since vertex cover number is a kind of the strongest parameter, there is a large gap between tractability and intractability from the viewpoint of parameterization. To fill up the gap, in this paper, we propose new fixed-parameter algorithms for L(p, 1)-Labeling by the twin cover number plus the maximum clique size and by the tree-width plus the maximum degree. These algorithms reduce the gap in terms of several combinations of parameters.

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Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials

Cited by 11 publications

References 19 publications

Recent Advances in Practical Data Reduction

Recent Advances in Practical Data Reduction

Constraint and Satisfiability Reasoning for Graph Coloring

Computing $L(p,1)$-Labeling with Combined Parameters

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