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21
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
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 6 (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.
Multicommodity Flows and Cuts in Polymatroidal Networks
, 2011
"... We consider multicommodity flow and cut problems in polymatroidal networks where there are submodular capacity constraints on the edges incident to a node. Polymatroidal networks were introduced by Lawler and Martel [24] and Hassin [18] in the singlecommodity setting and are closely related to the ..."
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Cited by 6 (3 self)
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We consider multicommodity flow and cut problems in polymatroidal networks where there are submodular capacity constraints on the edges incident to a node. Polymatroidal networks were introduced by Lawler and Martel [24] and Hassin [18] in the singlecommodity setting and are closely related to the submodular flow model of Edmonds and Giles [10]; the wellknown maxflowmincut theorem generalizes to this more general setting. Polymatroidal networks for the multicommodity case have not, as far as the authors are aware, been previously explored. Our work is primarily motivated by applications to information flow in wireless networks. We also consider the notion of undirected polymatroidal networks and observe that they provide a natural way to generalize flows and cuts in edge and node capacitated undirected networks. We establish polylogarithmic flowcut gap results in several scenarios that have been previously considered in the standard network flow models where capacities are on the edges or nodes [25, 26, 14, 23, 13]. Our results from a preliminary version have already found applications in wireless network information flow [20, 21] and we anticipate more in the future. On the technical side our key tools are the formulation and analysis of the dual of the flow relaxations via continuous extensions of submodular functions, in particular the Lovász extension. For directed graphs we rely on a simple yet useful reduction from
Designing FPT algorithms for cut problems using randomized contractions
 In FOCS [1
"... We introduce a new technique for designing fixedparameter algorithms for cut problems, namely randomized contractions. We apply our framework to obtain the first FPT algorithm for the Unique Label Cover problem and new FPT algorithms with exponential speed up for the Steiner Cut and Node Multiway C ..."
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Cited by 5 (3 self)
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We introduce a new technique for designing fixedparameter algorithms for cut problems, namely randomized contractions. We apply our framework to obtain the first FPT algorithm for the Unique Label Cover problem and new FPT algorithms with exponential speed up for the Steiner Cut and Node Multiway CutUncut problems. More precisely, we show the following: • We prove that the parameterized version of the Unique Label Cover problem, which is the base of the Unique Games Conjecture, can be solved in 2O(k 2 log Σ)n4 log n deterministic time (even in the stronger, vertexdeletion variant) where k is the number of unsatisfied edges and Σ  is the size of the alphabet. As a consequence, we show that one can in polynomial time solve instances of Unique Games where the number of edges allowed not to be satisfied is upper bounded by O( log n) to optimality, which improves over the trivial O(1) upper bound.
An informationtheoretic metatheorem on edgecut bounds
 IN PROCEEDINGS IEEE INTERNATIONAL SYMPOSIUM ON INFORMATION THEORY (ISIT
, 2012
"... We consider the problem of multiple unicast in wireline networks. Edgecut based bounds which are simple bounds on the rates achievable by routing flow are not in general, fundamental, i.e. they are not outer bounds on the capacity region. It has been observed that when the problem has some kind of ..."
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We consider the problem of multiple unicast in wireline networks. Edgecut based bounds which are simple bounds on the rates achievable by routing flow are not in general, fundamental, i.e. they are not outer bounds on the capacity region. It has been observed that when the problem has some kind of symmetry involved, then flows and edgecut based bounds are ‘close’, i.e. within a constant or polylogarithmic factor of each other. In this paper, we make the observation that in these very cases, such edgecut based bounds are actually ‘close ’ to fundamental yielding an approximate characterization of the capacity region for these problems. We demonstrate this in the case of kunicast in undirected networks, kpair unicast in directed networks with symmetric demands i.e. for every source communicating to a destination at a certain rate, the destination communicates an independent message back to the source at the same rate, and sumrate of kgroupcast in directed networks, i.e. a group of nodes, each of which has an independent message for every other node in the group. We place our work in context of existing results to suggest a metatheorem: if there is inherent symmetry either in the network connectivity or in the traffic pattern, then edgecut bounds are nearfundamental and flows approximately achieve capacity.
The Checkpoint Problem
"... In this paper we consider the checkpoint problem. The input consists of an undirected graph G, a set of sourcedestination pairs {(s1, t1),..., (sk, tk)}, and a collection P of paths connecting the (si, ti) pairs. A feasible solution is a multicut E ′ ; namely, a set of edges whose removal disconnec ..."
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In this paper we consider the checkpoint problem. The input consists of an undirected graph G, a set of sourcedestination pairs {(s1, t1),..., (sk, tk)}, and a collection P of paths connecting the (si, ti) pairs. A feasible solution is a multicut E ′ ; namely, a set of edges whose removal disconnects every sourcedestination pair. For each p ∈ P we define cpE ′(p) = p ∩ E ′ . In the sum checkpoint (SCP) problem the goal is to minimize ∑ p∈P cpE ′(p), while in the maximum checkpoint (MCP) problem the goal is to minimize maxp∈P cpE ′(p). These problem have several natural applications, e.g., in urban transportation and network security. In a sense, they combine the multicut problem and the minimum membership set cover problem. For the sum objective we show that weighted SCP is equivalent, with respect to approximability, to undirected multicut. Thus there exists an O(log n) approximation for SCP in general graphs. Our current approximability results for the max objective have a wide gap: we provide an approximation factor of O ( √ n log n/opt) for MCP and a hardness of 2 under the assumption P ̸ = NP. The hardness holds for trees, in which case we can obtain an asymptotic approximation factor of 2. Finally we show strong hardness for the wellknown problem of finding a path with minimum forbidden pairs, which in a sense can be considered the dual to the checkpoint problem. Despite various works on this problem, hardness of approximation was not known prior to this work. We show that the problem cannot be approximated within c n for some constant c> 0, unless P = NP. This is the strongest type of hardness possible. It carries over to directed acyclic graphs and is a huge improvement over the plain NPhardness of Gabow (SIAM J. Comp 2007, pages 1648–1671).