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28
Approximating the Domatic Number
"... A set of vertices in a graph is a dominating set if every vertex outside the set has aneighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number ofvertices, ffi the minimum degree, and ..."
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Cited by 64 (7 self)
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A set of vertices in a graph is a dominating set if every vertex outside the set has aneighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number ofvertices, ffi the minimum degree, and \Delta the maximum degree.We show that every graph has a domatic partition with (1o(1))(ffi + 1) / ln n dominatingsets, and moreover, that such a domatic partition can be found in polynomial time. This implies a (1 + o(1)) ln n approximation algorithm for domatic number, since the domaticnumber is always at most ffi + 1. We also show this to be essentially best possible. Namely,extending the approximation hardness of set cover by combining multiprover protocols with zeroknowledge techniques, we show that for every ffl> 0, a (1 ffl) ln napproximation impliesthat N P ` DT IM E(nO(log log n)). This makes domatic number the first natural maximization problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better.We also show that every graph has a domatic partition with (1o(1))(ffi + 1) / ln \Delta dominating sets, where the " o(1) " term goes to zero as \Delta increases. This can be turned intoan efficient algorithm that produces a domatic partition of \Omega ( ffi / ln \Delta) sets.
Decomposition of Balanced Matrices
 J. COMBINATORIAL THEORY, SER. B
, 1999
"... A 0,1 matrix is balanced if it does not contain a square submatrix of odd order with two ones per row and per column. We show that a balanced 0,1 matrix is either totally unimodular or its bipartite representation has a cutset consisting of two adjacent nodes and some of their neighbors. This resul ..."
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Cited by 29 (5 self)
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A 0,1 matrix is balanced if it does not contain a square submatrix of odd order with two ones per row and per column. We show that a balanced 0,1 matrix is either totally unimodular or its bipartite representation has a cutset consisting of two adjacent nodes and some of their neighbors. This result yields a polytime recognition algorithm for balancedness. To prove the result, we first prove a decomposition theorem for balanced 0,1 matrices that are not strongly balanced.
On Powers of Chordal Graphs And Their Colorings
 Congr. Numer
, 2000
"... The kth power of a graph G is a graph on the same vertex set as G, where a pair of vertices is connected by an edge if they are of distance at most k in G. We study the structure of powers of chordal graphs and the complexity of coloring them. We start by giving new and constructive proofs of t ..."
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Cited by 26 (1 self)
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The kth power of a graph G is a graph on the same vertex set as G, where a pair of vertices is connected by an edge if they are of distance at most k in G. We study the structure of powers of chordal graphs and the complexity of coloring them. We start by giving new and constructive proofs of the known facts that any power of an interval graph is an interval graph, and that any odd power of a general chordal graph is again chordal. We then show that it is computationally hard to approximately color the even powers of n vertex chordal graphs within an n 1 2 \Gammaffl factor, for any ffl ? 0. We present two exact and closed formulas for the chromatic polynomial for the kth power of a tree on n vertices. Furthermore, we give an O(kn) algorithm for evaluating the polynomial. Keywords: Chordal graphs, chromatic number, chromatic polynomial, coloring, interval graphs, power of a graph, tree. 1 Introduction In this paper we study the structure of powers of chordal graphs a...
shellable and unmixed clutters with a perfect matching of König type
 J. Pure Appl. Alg
"... Abstract. Let C be a clutter with a perfect matching e1,..., eg of König type and let ∆C be the StanleyReisner complex of the edge ideal of C. If all cminors of C have a free vertex and C is unmixed, we show that ∆C is pure shellable. We are able to describe, in combinatorial and algebraic terms, ..."
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Cited by 16 (5 self)
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Abstract. Let C be a clutter with a perfect matching e1,..., eg of König type and let ∆C be the StanleyReisner complex of the edge ideal of C. If all cminors of C have a free vertex and C is unmixed, we show that ∆C is pure shellable. We are able to describe, in combinatorial and algebraic terms, when ∆C is pure. If C has no cycles of length 3 or 4, then it is shown that ∆C is pure if and only if ∆C is pure shellable (in this case ei has a free vertex for all i), and that ∆C is pure if and only if for any two edges f1, f2 of C and for any ei, one has that f1 ∩ei ⊂ f2 ∩ei or f2 ∩ei ⊂ f1 ∩ei. It is also shown that this ordering condition implies that ∆C is pure shellable, without any assumption on the cycles of C. Then we prove that complete admissible uniform clutters and their Alexander duals are unmixed. In addition, the edge ideals of complete admissible uniform clutters are facet ideals of shellable simplicial complexes, they are CohenMacaulay, and they have linear resolutions. Furthermore if C is admissible and complete, then C is unmixed. We characterize certain conditions that occur in a CohenMacaulay criterion for bipartite graphs of Herzog and Hibi, and extend some results of Faridi—on the structure of unmixed simplicial trees—to clutters with the König property without 3cycles or 4cycles. 1.
The Domatic Number Problem on Some Perfect Graph Families
 Information Processing Letters
, 1995
"... An extremely simple, linear time algorithm is given for constructing a domatic partition in totally balanced hypergraphs. This simplifies and generalizes previous algorithms for interval and strongly chordal graphs. On the other hand, the domatic number problem is shown to be NPcomplete for several ..."
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Cited by 11 (0 self)
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An extremely simple, linear time algorithm is given for constructing a domatic partition in totally balanced hypergraphs. This simplifies and generalizes previous algorithms for interval and strongly chordal graphs. On the other hand, the domatic number problem is shown to be NPcomplete for several families of perfect graphs, including chordal and bipartite graphs.
Resolution and the Integrality of Satisfiability Problems
 Mathematical Programming
, 1995
"... A satisfiability problem can be regarded as a nondisjoint union of set covering problems. We show that if the resolution method of theorem proving is applied to the satisfiability problem, its constraint set defines an integral polytope if and only if the constraint sets of the set covering problems ..."
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Cited by 8 (0 self)
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A satisfiability problem can be regarded as a nondisjoint union of set covering problems. We show that if the resolution method of theorem proving is applied to the satisfiability problem, its constraint set defines an integral polytope if and only if the constraint sets of the set covering problems do. In this sense, resolution reduces the integrality question for the satisfiability problem to that for the set covering problem. 1 Introduction The satisfiability problem of propositional logic asks whether a set of logical clauses can be true simultaneously. The clauses can be represented as linear inequalities that have a 01 solution if and only if the clauses are satisfiable. In many cases one is not only interested in the 01 solubility of this constraint set but in solving a 01 optimization problem subject to it. Such a problem is implicit, for instance, in the maximum satisfiability problem [13, 15], which assigns weights to the constraints and seeks the maximum weight feasible ...
Algorithmic Aspects of the ConsecutiveOnes Property
, 2009
"... We survey the consecutiveones property of binary matrices. Herein, a binary matrix has the consecutiveones property (C1P) if there is a permutation of its columns that places the 1s consecutively in every row. We provide an overview over connections to graph theory, characterizations, recognition ..."
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Cited by 8 (1 self)
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We survey the consecutiveones property of binary matrices. Herein, a binary matrix has the consecutiveones property (C1P) if there is a permutation of its columns that places the 1s consecutively in every row. We provide an overview over connections to graph theory, characterizations, recognition algorithms, and applications such as integer linear programming and solving Set Cover.
Optimized Crew Scheduling at Air New Zealand
"... The aircrewscheduling problem consists of two important subproblems: the toursofduty planning problem to generate minimumcost tours of duty (sequences of duty periods and rest periods) to cover all scheduled flights, and the rostering problem to assign tours of duty to individual crew members. B ..."
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Cited by 5 (0 self)
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The aircrewscheduling problem consists of two important subproblems: the toursofduty planning problem to generate minimumcost tours of duty (sequences of duty periods and rest periods) to cover all scheduled flights, and the rostering problem to assign tours of duty to individual crew members. Between 1986 and 1999, Air New Zealand staff and consultants in collaboration with the University of Auckland have developed eight applicationspecific optimizationbased computer systems to solve all aspects of the toursofduty planning and rostering processes for Air New Zealandâs national and international operations. These systems have saved NZ$15,655,000 per year while providing crew rosters that better respect crew membersâ preferences.
LAGRANGIAN RELAXATION AND PARTIAL COVER (EXTENDED ABSTRACT)
, 2008
"... Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover. We show that for Partial Cover in general no algorithm that uses Lagrangian relaxation and a Lagrangian Multiplier Preserving ( ..."
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Cited by 5 (2 self)
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Lagrangian relaxation has been used extensively in the design of approximation algorithms. This paper studies its strengths and limitations when applied to Partial Cover. We show that for Partial Cover in general no algorithm that uses Lagrangian relaxation and a Lagrangian Multiplier Preserving (LMP) αapproximation as a black box can yield an approximation factor better than 4 3 α. This matches the upper bound given by Könemann et al. (ESA 2006, pages 468–479). Faced with this limitation we study a specific, yet broad class of covering problems: Partial Totally Balanced Cover. By carefully analyzing the inner workings of the LMP algorithm we are able to give an almost tight characterization of the integrality gap of the standard linear relaxation of the problem. As a consequence we obtain improved approximations for the Partial version of Multicut and Path Hitting on Trees, Rectangle Stabbing, and Set Cover with ρBlocks.