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Fixedparameter tractability of multicut parameterized by the size of the cutset
, 2011
"... Given an undirected graph G, a collection {(s1, t1),...,(sk, tk)} of pairs of vertices, and an integer p, the EDGE MULTICUT problem ask if there is a set S of at most p edges such that the removal of S disconnects every si from the corresponding ti. VERTEX MULTICUT is the analogous problem where S i ..."
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Cited by 32 (6 self)
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Given an undirected graph G, a collection {(s1, t1),...,(sk, tk)} of pairs of vertices, and an integer p, the EDGE MULTICUT problem ask if there is a set S of at most p edges such that the removal of S disconnects every si from the corresponding ti. VERTEX MULTICUT is the analogous problem where S is a set of at most p vertices. Our main result is that both problems can be solved in time 2O(p3) · nO(1), i.e., fixedparameter tractable parameterized by the size p of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form f (p) · nO(1) exists for the directed version of the problem, as we show it to be W[1]hard parameterized by the size of the cutset.
Tractable Hypergraph Properties for Constraint Satisfaction and Conjunctive Queries
, 2010
"... An important question in the study of constraint satisfaction problems (CSP) is understanding how the graph or hypergraph describing the incidence structure of the constraints influences the complexity of the problem. For binary CSP instances (i.e., where each constraint involves only two variables ..."
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Cited by 31 (4 self)
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An important question in the study of constraint satisfaction problems (CSP) is understanding how the graph or hypergraph describing the incidence structure of the constraints influences the complexity of the problem. For binary CSP instances (i.e., where each constraint involves only two variables), the situation is well understood: the complexity of the problem essentially depends on the treewidth of the graph of the constraints [19, 24]. However, this is not the correct answer if constraints with unbounded number of variables are allowed, and in particular, for CSP instances arising from query evaluation problems in database theory. Formally, if H is a class of hypergraphs, then let CSP(H) be CSP restricted to instances whose hypergraph is in H. Our goal is to characterize those classes of hypergraphs for which CSP(H) is polynomialtime solvable or fixedparameter tractable, parameterized by the number of variables. In the applications related to database query evaluation, we usually assume that the number of variables is much smaller than the size of the instance, thus parameterization by the number of variables is a meaningful question. The most general known property of H that makes CSP(H) polynomialtime solvable is bounded fractional hypertree width. Here we introduce a new hypergraph measure called submodular width, and show that bounded submodular width of H (which is a strictly more general property than bounded fractional hypertree width) implies that CSP(H) is fixedparameter tractable. In a matching hardness result, we show that if H has unbounded submodular width, then CSP(H) is not fixedparameter tractable (and hence not polynomialtime solvable), unless the Exponential Time Hypothesis (ETH) fails. The algorithmic result uses tree decompositions in a novel way: instead of using a single decomposition depending on the hypergraph, the instance is split into a set of
Polynomial Flowcut Gaps and Hardness of Directed Cut Problems
 In Proc. of STOC, 2007
"... We study the multicut and the sparsest cut problems in directed graphs. In the multicut problem, we are a given an nvertex graph G along with k sourcesink pairs, and the goal is to find the minimum cardinality subset of edges whose removal separates all sourcesink pairs. The sparsest cut problem ..."
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Cited by 24 (0 self)
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We study the multicut and the sparsest cut problems in directed graphs. In the multicut problem, we are a given an nvertex graph G along with k sourcesink pairs, and the goal is to find the minimum cardinality subset of edges whose removal separates all sourcesink pairs. The sparsest cut problem has the same input, but the goal is to find a subset of edges to delete so as to minimize the ratio of the number of deleted edges to the number of sourcesink pairs that are separated by this deletion. The natural linear programming relaxation for multicut corresponds, by LPduality, to the wellstudied maximum (fractional) multicommodity flow problem, while the standard LPrelaxation for sparsest cut corresponds to maximum concurrent flow. Therefore, the integrality gap of the linear programming relaxation for multicut/sparsest cut is also the flowcut gap: the largest gap, achievable for any graph, between the maximum flow value and the minimum cost solution for the corresponding cut problem. Our first result is that the flowcut gap between maximum multicommodity flow and minimum multicut is ˜ Ω(n 1/7) in directed graphs. We show a similar result for the gap between maximum concurrent flow and sparsest cut in directed graphs. These results improve upon a
Improved approximation for directed cut problems
, 2007
"... We present improved approximation algorithms for directed multicut and directed sparsest cut. The current best known approximation ratio for these problems is O(n 1/2). We obtain an Õ(n11/23)approximation. Our algorithm works with the natural LP relaxation used in prior work. We use a randomized ro ..."
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Cited by 21 (0 self)
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We present improved approximation algorithms for directed multicut and directed sparsest cut. The current best known approximation ratio for these problems is O(n 1/2). We obtain an Õ(n11/23)approximation. Our algorithm works with the natural LP relaxation used in prior work. We use a randomized rounding algorithm with a more sophisticated charging scheme and analysis to obtain our improvement. This also implies a Õ(n11/23) upper bound on the ratio between the maximum multicommodity flow and minimum multicut in directed graphs.
Approximation and Hardness Results for Label Cut and Related Problems
"... We investigate a natural combinatorial optimization problem called the Label Cut problem. Given an input graph G with a source s and a sink t, the edges of G are classified into different categories, represented by a set of labels. The labels may also have weights. We want to pick a subset of labels ..."
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Cited by 8 (0 self)
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We investigate a natural combinatorial optimization problem called the Label Cut problem. Given an input graph G with a source s and a sink t, the edges of G are classified into different categories, represented by a set of labels. The labels may also have weights. We want to pick a subset of labels of minimum cardinality (or minimum total weight), such that the removal of all edges with these labels disconnects s and t. We give the first nontrivial approximation and hardness results for the Label Cut problem. Firstly, we present an O ( √ m)approximation algorithm for the Label Cut problem, where m is the number of edges in the input graph. Secondly, we show that it is NPhard to approximate Label Cut within 2 log1−1 / log logc n n for any constant c < 1/2, where n is the input length of the problem. Thirdly, our techniques can be applied to other previously considered optimization problems. In particular we show that the Minimum Label Path problem has the same approximation hardness as that of Label Cut, simultaneously improving and unifying two known hardness results for this problem which were previously the best (but incomparable due to different complexity assumptions). 1
Sparse WiFi Deployment for Vehicular Internet Access with Bounded Interconnection Gap
"... Vehicular Internet access via open WLAN access points (AP) has been demonstrated to be a feasible solution to provide opportunistic data service to moving vehicles. Using an in situ deployment, however, such a solution does not provide worstcase performance guarantees due to unpredictable intermitt ..."
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Cited by 8 (1 self)
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Vehicular Internet access via open WLAN access points (AP) has been demonstrated to be a feasible solution to provide opportunistic data service to moving vehicles. Using an in situ deployment, however, such a solution does not provide worstcase performance guarantees due to unpredictable intermittent connectivity. On the other hand, a solution that tries to cover every point in an entire road network with APs (a full coverage) is not very practical due to prohibitive deployment and operational costs. In this paper, we introduce a new notion of intermittent coverage for mobile users, called αcoverage, which provides worstcase guarantees on the interconnection gap while using significantly fewer APs than needed for full coverage. We propose efficient algorithms to verify whether a given deployment provides αcoverage and approximation algorithms for determining an economic deployment of APs that will provide αcoverage. Our algorithms can also be used to supplement open WLAN APs in a region with appropriate number of additional APs that will provide worstcase guarantees on interconnection gap. We compare αcoverage with opportunistic access of open WLAN APs (modeled as a random deployment) via simulations over realworld road networks and show that using the same number of APs as in case of random deployment, αcoverage limits the interconnection gap to a much smaller distance.
On the MaxFlow MinCut Ratio for Directed Multicommodity Flows
 Theor. Comput. Sci
, 2003
"... We give a pure combinatorial problem whose solution determines maxflow mincut ratio for directed multicommodity flows. In addition, this combinatorial problem has applications in improving the approximation factor of Greedy algorithm for maximum edge disjoint path problem. More precisely, our u ..."
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Cited by 7 (1 self)
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We give a pure combinatorial problem whose solution determines maxflow mincut ratio for directed multicommodity flows. In addition, this combinatorial problem has applications in improving the approximation factor of Greedy algorithm for maximum edge disjoint path problem. More precisely, our upper bound improves the approximation factor for this problem to O(n ). Finally, we demonstrate how even for very simple graphs the aforementioned ratio might be very large.
An O( √ n)Approximation Algorithm For Directed Sparsest Cut
"... We give an O (√ n)approximation algorithm for the Sparsest Cut Problem on directed graphs. A naïve reduction from Sparsest Cut to Minimum Multicut would only give an approximation ratio of O (√ n log D), where D is the sum of the demands. We obtain the improvement using a novel LProunding method f ..."
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Cited by 7 (1 self)
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We give an O (√ n)approximation algorithm for the Sparsest Cut Problem on directed graphs. A naïve reduction from Sparsest Cut to Minimum Multicut would only give an approximation ratio of O (√ n log D), where D is the sum of the demands. We obtain the improvement using a novel LProunding method for fractional Sparsest Cut, the dual of
Completely inapproximable monotone and antimonotone parameterized problems
"... We prove that weighted monotone/antimonotone circuit satisfiability has no fixedparameter tractable approximation algorithm with any approximation ratio function ρ, unless FPT 6 = W [1]. In particular, not having such an fptapproximation algorithm implies that these problems have no polynomialti ..."
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Cited by 6 (0 self)
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We prove that weighted monotone/antimonotone circuit satisfiability has no fixedparameter tractable approximation algorithm with any approximation ratio function ρ, unless FPT 6 = W [1]. In particular, not having such an fptapproximation algorithm implies that these problems have no polynomialtime approximation algorithms with ratio ρ(OPT) for any nontrivial function ρ.
FixedParameter and Approximation Algorithms: A New Look
"... A FixedParameter Tractable (FPT) ρapproximation algorithm for a minimization (resp. maximization) parameterized problem P is an FPTalgorithm that, given an instance (x, k) ∈ P computes a solution of cost at most k · ρ(k) (resp. k/ρ(k)) if a solution of cost at most (resp. at least) k exists; ot ..."
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Cited by 4 (1 self)
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A FixedParameter Tractable (FPT) ρapproximation algorithm for a minimization (resp. maximization) parameterized problem P is an FPTalgorithm that, given an instance (x, k) ∈ P computes a solution of cost at most k · ρ(k) (resp. k/ρ(k)) if a solution of cost at most (resp. at least) k exists; otherwise the output can be arbitrary. For wellknown intractable problems such as the W[1]hard Clique and W[2]hard Set Cover problems, the natural question is whether we can get any FPTapproximation. It is widely believed that both Clique and SetCover admit no FPT ρapproximation algorithm, for any increasing function ρ. However, to the best of our knowledge, there has been no progress towards proving this conjecture. Assuming standard conjectures such as the Exponential Time Hypothesis (ETH) [18] and the Projection Games Conjecture (PGC) [27], we make the first progress towards proving this conjecture by showing that – Under the ETH and PGC, there exist constants F1, F2> 0 such that the Set Cover problem does not admit a FPT approximation algorithm with ratio k F1 k in 2 F2 · poly(N, M) time, where N is the size of the universe and M is the number of sets. – Unless NP ⊆ SUBEXP, for every 1> δ> 0 there exists a constant F (δ)> 0 such that Clique has no FPT cost approximation with ratio k 1−δ in 2 kF · poly(n) time, where n is the number of vertices in the graph. In the second part of the paper we consider various W[1]hard problems