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29
The Importance of Being Biased
, 2002
"... The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NPhard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1 ..."
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The Minimum Vertex Cover problem is the problem of, given a graph, finding a smallest set of vertices that touches all edges. We show that it is NPhard to approximate this problem 1.36067, improving on the previously known hardness result for a 6 factor. 1
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 85 (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.
Rounding via Trees: Deterministic Approximation Algorithms for Group Steiner Trees and kmedian
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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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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.
Multicuts in Unweighted Graphs and Digraphs with Bounded Degree and Bounded TreeWidth
, 1998
"... this paper. Also, we show that Directed Edge Multicut is NPhard in digraphs with treewidth one and maximum in and out degree three. Other hardness results indicate why we cannot eliminate any of the three restrictionsunweighted, bounded degree and bounded treewidthon the input graph and sti ..."
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Cited by 23 (0 self)
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this paper. Also, we show that Directed Edge Multicut is NPhard in digraphs with treewidth one and maximum in and out degree three. Other hardness results indicate why we cannot eliminate any of the three restrictionsunweighted, bounded degree and bounded treewidthon the input graph and still obtain a PTAS. It is known [1] that for a Max SNPhard problem, unless P=NP, no PTAS exists. We have already seen that Unweighted Edge Multicut is Max SNPhard in stars [9], so letting the input graph have unbounded degree makes the problem harder. We show that Weighted Edge Multicut is Max SNPhard in binary trees, therefore letting the input graph be weighted makes the problem harder. Finally, we show that Unweighted Edge Multicut is Max SNPhard if the input graphs are walls. Walls, to be formally defined in Section 6, have degree at most three and unbounded treewidth. We conclude that letting the input graph have unbounded treewidth makes the problem significantly harder
Approximating Minimum Subset Feedback Sets in Undirected Graphs with Applications to Multicuts
, 1996
"... Let G = (V; E) be a weighted undirected graph where all weights are at least one. We consider the following generalization of feedback set problems. Let S ae V be a subset of the vertices. A cycle is called interesting if it intersects the set S. A subset feedback edge (vertex) set is a subset of th ..."
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Cited by 19 (0 self)
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Let G = (V; E) be a weighted undirected graph where all weights are at least one. We consider the following generalization of feedback set problems. Let S ae V be a subset of the vertices. A cycle is called interesting if it intersects the set S. A subset feedback edge (vertex) set is a subset of the edges (vertices) that intersects all interesting cycles. In minimum subset feedback problems the goal is to find such sets of minimumweight. The case in which S consists of a single vertex is equivalent to the multiway cut problem, in which the goal is to separate a given set of terminals. Hence, the subset feedback problem is NPcomplete, and also generalizes the multiway cut problem. We provide a polynomialtime algorithm for approximating the subset feedback edge set problem that achieves an approximation factor of two. For the subset feedback vertex set problem we achieve an approximation factor of minf2\Delta; O(log jSj); O(log ø )g, where \Delta is the maximum degree in G and ø ...
Oblivious routing on nodecapacitated and directed graphs
 IN PROCEEDINGS OF THE 16TH ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA), 2005
, 2005
"... Oblivious routing algorithms for general undirected networks were introduced by Räcke [17], and this work has led to many subsequent improvements and applications. Comparatively little is known about oblivious routing in general directed networks, or even in undirected networks with node capacities. ..."
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Cited by 16 (8 self)
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Oblivious routing algorithms for general undirected networks were introduced by Räcke [17], and this work has led to many subsequent improvements and applications. Comparatively little is known about oblivious routing in general directed networks, or even in undirected networks with node capacities. We present the first nontrivial upper bounds for both these cases, providing algorithms for kcommodity oblivious routing problems with competitive ratio O (√ k log(n)) for undirected nodecapacitated graphs and O (√ k n 1/4 log(n)) for directed graphs. In the special case that all commodities have a common source or sink, our upper bound becomes O ( √ n log(n)) in both cases, matching the lower bound up to a factor of log(n). The lower bound (which first appeared in [6]) is obtained on a graph with very high degree. We show that in fact the degree of a graph is a crucial parameter for nodecapacitated oblivious routing in undirected graphs, by providing an O(∆ polylog(n))competitive oblivious routing scheme for graphs of degree ∆. For the directed case, however, we show that the lower bound of Ω (√ n) still holds in lowdegree graphs. Finally, we settle an open question about routing problems in which all commodities share a common source or sink. We show that even in this simplified scenario there are networks in which no oblivious routing algorithm can achieve a competitive ratio better than Ω(log n).
Pay Today for a Rainy Day: Improved Approximation Algorithms for DemandRobust MinCut and Shortest Path Problems
 STACS
, 2006
"... Abstract. Demandrobust versions of common optimization problems were recently introduced by Dhamdhere et al. [4] motivated by the worstcase considerations of twostage stochastic optimization models. We study the demand robust mincut and shortest path problems, and exploit the nature of the robus ..."
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Abstract. Demandrobust versions of common optimization problems were recently introduced by Dhamdhere et al. [4] motivated by the worstcase considerations of twostage stochastic optimization models. We study the demand robust mincut and shortest path problems, and exploit the nature of the robust objective to give improved approximation factors. Specifically, we give a (1 + √ 2) approximation for robust mincut and a 7.1 approximation for robust shortest path. Previously, the best approximation factors were O(log n) for robust mincut and 16 for robust shortest paths, both due to Dhamdhere et al. [4]. Our main technique can be summarized as follows: We investigate each of the second stage scenarios individually, checking if it can be independently serviced in the second stage within an acceptable cost (namely, a guess of the optimal second stage costs). For the costly scenarios that cannot be serviced in this way (“rainy days”), we show that they can be fully taken care of in a nearoptimal first stage solution (i.e., by ”paying today”). We also consider “hittingset ” extensions of the robust mincut and shortest path problems and show that our techniques can be combined with algorithms for Steiner multicut and group Steiner tree problems to give similar approximation guarantees for the hittingset versions of robust mincut and shortest path problems respectively. 1