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33
NCApproximation Schemes for NP and PSPACEHard Problems for Geometric Graphs
, 1997
"... We present NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance tradeoff as the best known approximation schemes for planar gr ..."
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Cited by 116 (1 self)
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We present NC approximation schemes for a number of graph problems when restricted to geometric graphs including unit disk graphs and graphs drawn in a civilized manner. Our approximation schemes exhibit the same time versus performance tradeoff as the best known approximation schemes for planar graphs. We also define the concept of precision unit disk graphs and show that for such graphs the approximation schemes have a better time versus performance tradeoff than the approximation schemes for arbitrary unit disk graphs. Moreover, compared to unit disk graphs, we show that for precision unit disk graphs, many more graph problems have efficient approximation schemes. Our NC approximation schemes can also be extended to obtain efficient NC approximation schemes for several PSPACEhard problems on unit disk graphs specified using a restricted version of the hierarchical specification language of Bentley, Ottmann and Widmayer. The approximation schemes for hierarchically specified un...
Conjunctive Query Containment Revisited
, 1998
"... We consider the problems of conjunctive query containment and minimization, which are known to be NPcomplete, and show that these problems can be solved in polynomial time for the class of acyclic queries. We then generalize the notion of acyclicity and define a parameter called query width that ca ..."
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Cited by 115 (0 self)
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We consider the problems of conjunctive query containment and minimization, which are known to be NPcomplete, and show that these problems can be solved in polynomial time for the class of acyclic queries. We then generalize the notion of acyclicity and define a parameter called query width that captures the "degree of cyclicity" of a query: in particular, a query is acyclic if and only if its query width is 1. We give algorithms for containment and minimization that run in time polynomial in n k , where n is the input size and k is the query width. These algorithms naturally generalize those for acyclic queries, and are of practical significance because many queries have small query width compared to their sizes. We show that good bounds on the query width of Q can be obtained using the treewidth of the incidence graph of Q. We then consider the problem of finding an equivalent query to a given conjunctive query Q that has the least number of subgoals. We show that a polynomial tim...
Diameter and Treewidth in MinorClosed Graph Families
, 1999
"... It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a ..."
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Cited by 108 (2 self)
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It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a minorclosed family, if and only if some apex graph does not belong to the family. In particular, the O(D) bound above can be extended to boundedgenus graphs. As a consequence, we extend several approximation algorithms and exact subgraph isomorphism algorithms from planar graphs to other graph families.
The Approximability of Constraint Satisfaction Problems
, 2000
"... ... oftheoptimizationtask. Here weconsiderfourpossiblegoals: MaxCSP(MinCSP)isthe classofproblemswherethegoalistondanassignment maximizingthenumberofsatised factionproblemsdependingonthenatureofthe "underlying" constraintsaswellasonthegoal constraints(minimizingthenumberofunsatisedconstrain ..."
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Cited by 84 (1 self)
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... oftheoptimizationtask. Here weconsiderfourpossiblegoals: MaxCSP(MinCSP)isthe classofproblemswherethegoalistondanassignment maximizingthenumberofsatised factionproblemsdependingonthenatureofthe "underlying" constraintsaswellasonthegoal constraints(minimizingthenumberofunsatisedconstraints). MaxOnes(MinOnes)isthe classofoptimizationproblemswherethegoalistondan assignmentsatisfyingallconstraints withmaximum(minimum)numberofvariablesset to 1. Eachclassconsistsofinnitelymany thatdescribethepossibleconstraintsthatmaybeused. problemsandaproblemwithinaclass is specified by a finite collectionofniteBooleanfunctions pletelyclassiesalloptimizationproblems derived from Booleanconstraintsatisfaction.Our Creignou [11]. Inthisworkwedeterminetightboundsonthe "approximability"(i.e.,thera in MaxOnes,MinCSPandMinOnes.Combinedwiththeresultof Creignou,thiscomtiotowithinwhicheachproblemmay be approximatedinpolynomialtime)ofeveryproblem Tightboundsontheapproximabilityofeveryproblemin MaxCSPwereobtainedby resultscaptureadiversecollectionofoptimization problemssuchasMAX3SAT,MaxCut, (in)approximabilityoftheseoptimizationproblems andyieldacompactpresentationofmost MaxClique,MinCut,NearestCodewordetc. Ourresultsunifyrecentresultsonthe knownresults. Moreover, theseresultsprovideaformalbasistomanystatementsonthe behaviorofnaturaloptimizationproblems,thathaveso faronlybeenobservedempirically.
A New Rounding Procedure for the Assignment Problem with Applications to Dense Graph Arrangement Problems
, 2001
"... We present a randomized procedure for rounding fractional perfect matchings to (integral) matchings. If the original fractional matching satis es any linear inequality, then with high probability, the new matching satis es that linear inequality in an approximate sense. This extends the wellkn ..."
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Cited by 78 (3 self)
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We present a randomized procedure for rounding fractional perfect matchings to (integral) matchings. If the original fractional matching satis es any linear inequality, then with high probability, the new matching satis es that linear inequality in an approximate sense. This extends the wellknown LP rounding procedure of Raghavan and Thompson, which is usually used to round fractional solutions of linear programs.
Local treewidth, excluded minors, and approximation algorithms
 COMBINATORICA
, 2001
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Approximation schemes for NPhard geometric optimization problems: A survey
 Mathematical Programming
, 2003
"... NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (di ..."
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Cited by 44 (2 self)
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NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (distance between nodes (x1, y1) and (x2, y2) is ((x1−x2) 2 +(y1−y2) 2) 1/2) then the problem is called Euclidean TSP. More generally the distance could be defined using other norms, such as ℓp norms for any p> 1. All these are subcases of the more general notion of a geometric norm or Minkowski norm. We will refer to the version of the problem with a general geometric norm as geometric TSP. Some other NPhard geometric optimization problems are Minimum Steiner Tree (“Given n points, find the smallest network connecting them,”), kTSP(“Given n points and a number k, find the shortest salesman tour that visits k points”), kMST (“Given n points and a number k, find the shortest tree that contains k points”), vehicle routing, degree restricted minimum
Bidimensionality: New Connections between FPT Algorithms and PTASs
, 2005
"... We demonstrate a new connection between fixedparametertractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of socalled "bidimensional " problems to show that essentially ..."
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Cited by 43 (7 self)
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We demonstrate a new connection between fixedparametertractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of socalled &quot;bidimensional &quot; problems to show that essentially all such problems have both subexponential fixedparameter algorithms and PTASs. Bidimensional problems include e.g. feedbackvertex set, vertex cover, minimum maximal matching, face cover, a series of vertexremoval problems, dominating set,edge dominating set,
Constraint satisfaction: The approximability of minimization problems
 Proc. 12th Annual Conference on Structure in Complexity Theory, IEEE
, 1997
"... This paper continues the work initiated by Creignou [5] and Khanna, Sudan and Williamson [15] who classify maximization problems derived from Boolean constraint satisfaction. Here we study the approximability of minimization problems derived thence. A problem in this framework is characterized by a ..."
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Cited by 42 (5 self)
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This paper continues the work initiated by Creignou [5] and Khanna, Sudan and Williamson [15] who classify maximization problems derived from Boolean constraint satisfaction. Here we study the approximability of minimization problems derived thence. A problem in this framework is characterized by a collection�of “constraints” (i.e., functions����������) and an instance of a problem is constraints drawn from�applied to specified subsets of Boolean variables. We study the two minimization analogs of classes studied in [15]: in one variant, namely MIN CSP�, the objective is to find an assignment to minimize the number of unsatisfied constraints, while in the other, namely MIN ONES�, the goal is to find a satisfying assignment with minimum number of ones. These two classes together capture an entire spectrum of important minimization problems including Min Cut, vertex cover, hitting set with bounded size sets, integer programs with two variables per inequality, graph bipartization, clause deletion in CNF formulae, and nearest codeword. Our main result is that there exists a finite partition of the space of all constraint sets such that for any given�, the approximability of MIN CSP�and MIN ONES� is completely determined by the partition containing it. Moreover, we present a compact set of rules that determines which partition contains a given family�. Our classification identifies the central elements governing the approximability of problems in these classes, by unifying a large collection algorithmic and hardness of approximation results. When contrasted with the work of [15], our results also serve to formally highlight inherent differences between maximization and minimization problems.