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19
Subexponential algorithms for Unique Games and related problems
 IN 51 ST IEEE FOCS
, 2010
"... We give subexponential time approximation algorithms for the unique games and the small set expansion problems. Specifically, for some absolute constant c, we give: 1. An exp(kn ε)time algorithm that, given as input a kalphabet unique game on n variables that has an assignment satisfying 1 − ε c f ..."
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Cited by 82 (7 self)
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We give subexponential time approximation algorithms for the unique games and the small set expansion problems. Specifically, for some absolute constant c, we give: 1. An exp(kn ε)time algorithm that, given as input a kalphabet unique game on n variables that has an assignment satisfying 1 − ε c fraction of its constraints, outputs an assignment satisfying 1 − ε fraction of the constraints. 2. An exp(n ε /δ)time algorithm that, given as input an nvertex regular graph that has a set S of δn vertices with edge expansion at most ε c, outputs a set S ′ of at most δn vertices with edge expansion at most ε. We also obtain a subexponential algorithm with improved approximation for the MultiCut problem, as well as subexponential algorithms with improved approximations to MaxCut, SparsestCut and Vertex Cover on some interesting subclasses of instances. Khot’s Unique Games Conjecture (UGC) states that it is NPhard to achieve approximation guarantees such as ours for unique games. While our results stop short of refusing the UGC, they do suggest that Unique Games is significantly easier than NPhard problems such as 3SAT,3LIN, Label Cover and more, that are believed not to have a subexponential algorithm achieving a nontrivial approximation ratio. The main component in our algorithms is a new result on graph decomposition that may have other applications. Namely we show that for every δ> 0 and a regular nvertex graph G, by changing at most δ fraction of G’s edges, one can break G into disjoint parts so that the induced graph on each part has at most n ε eigenvalues larger than 1 − η (where ε, η depend polynomially on δ). Our results are based on combining this decomposition with previous algorithms for unique games on graphs with few large eigenvalues (Kolla and Tulsiani 2007, Kolla 2010).
Parameterized graph separation problems
 In Proc. 1st IWPEC, volume 3162 of LNCS
, 2004
"... We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k vertices such that (a) each of the given ℓ terminals is separated from the others, (b) each of the given ℓ pairs of terminal ..."
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Cited by 51 (4 self)
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We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k vertices such that (a) each of the given ℓ terminals is separated from the others, (b) each of the given ℓ pairs of terminals is separated, (c) exactly ℓ vertices are cut away from the graph, (d) exactly ℓ connected vertices are cut away from the graph, (e) the graph is separated into at least ℓ components. We show that if both k and ℓ are
Alpha Coverage: Bounding the Interconnection Gap for Vehicular Internet Access
, 2009
"... Abstract—Vehicular Internet access via open WLAN access points (APs) 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 ..."
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Cited by 19 (4 self)
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Abstract—Vehicular Internet access via open WLAN access points (APs) 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 (full coverage) is not very practical due to the prohibitive deployment and operational cost. 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 a deployment of APs that will provide αcoverage. We compare αcoverage with opportunistic access of open WLAN APs (modeled as a random deployment) via simulations over a realworld road network and show that using the same number of APs as random deployment, αcoverage bounds the interconnection gap to a much smaller distance than that in a random deployment. I.
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).
FixedParameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset
"... Given a directed graph G, a set of k terminals and an integer p, the DIRECTED VERTEX MULTIWAY CUT problem asks if there is a set S of at most p (nonterminal) vertices whose removal disconnects each terminal from all other terminals. DIRECTED EDGE MULTIWAY CUT is the analogous problem where S is a se ..."
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Cited by 14 (6 self)
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Given a directed graph G, a set of k terminals and an integer p, the DIRECTED VERTEX MULTIWAY CUT problem asks if there is a set S of at most p (nonterminal) vertices whose removal disconnects each terminal from all other terminals. DIRECTED EDGE MULTIWAY CUT is the analogous problem where S is a set of at most p edges. These two problems indeed are known to be equivalent. A natural generalization of the multiway cut is the multicut problem, in which we want to disconnect only a set of k given pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in undirected graphs multiway cut is fixedparameter tractable (FPT) parameterized by p. Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and directed multicut is W[1]hard parameterized by p. We complete the picture here by our main result which is that both DIRECTED VERTEX MULTIWAY CUT and DIRECTED EDGE MULTIWAY CUT can be solved in time 22O(p) nO(1) , i.e., FPT parameterized by size p of the cutset of the solution. This answers an open question raised by Marx (Theor. Comp. Sci. 2006) and Marx and Razgon (STOC 2011). It follows from our result that DIRECTED MULTICUT is FPT for the case of k = 2 terminal pairs, which answers another open problem raised in Marx and Razgon (STOC 2011). 1
Computing Minimum Multiway Cuts in Hypergraphs from Hypertree Packings
, 2009
"... Hypergraph multiway cut problem is a problem of finding a minimum capacity set of hyperedges whose removal divides a given hypergraph into a specified number of connected components. We present an algorithm for this problem which runs in strongly polynomialtime if both the specified number of conne ..."
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Cited by 6 (0 self)
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Hypergraph multiway cut problem is a problem of finding a minimum capacity set of hyperedges whose removal divides a given hypergraph into a specified number of connected components. We present an algorithm for this problem which runs in strongly polynomialtime if both the specified number of connected components and the maximum size of hyperedges in the hypergraph are constants. Our algorithm extends the algorithm due to Thorup (2008) for computing minimum multiway cuts of graphs from greedy packings of spanning trees.
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.
Divideandconquer algorithms for partitioning hypergraphs and submodular systems
, 2009
"... The submodular system kpartition problem is a problem of partitioning a given finite set V into k nonempty subsets V1, V2,..., Vk so that P k i=1 f(Vi) is minimized where f is a nonnegative submodular function on V, and k is a fixed integer. This problem contains the hypergraph kcut problem. In ..."
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Cited by 4 (1 self)
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The submodular system kpartition problem is a problem of partitioning a given finite set V into k nonempty subsets V1, V2,..., Vk so that P k i=1 f(Vi) is minimized where f is a nonnegative submodular function on V, and k is a fixed integer. This problem contains the hypergraph kcut problem. In this paper, we design the first exact algorithm for k = 3 and approximation algorithms for k ≥ 4. We also analyze the approximation factor for the hypergraph kcut problem. 1
Cardinality constrained and multicriteria (multi)cut problems
 J. of Discrete Algorithms
, 2009
"... In this paper, we consider multicriteria and cardinality constrained multicut problems. Let G be a graph where each edge is weighted by R positive costs corresponding to R criteria and consider k sourcesink pairs of vertices of G and R integers B1,..., BR. The problem RCriMultiCut consists in find ..."
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Cited by 2 (0 self)
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In this paper, we consider multicriteria and cardinality constrained multicut problems. Let G be a graph where each edge is weighted by R positive costs corresponding to R criteria and consider k sourcesink pairs of vertices of G and R integers B1,..., BR. The problem RCriMultiCut consists in finding a set of edges whose removal leaves no path between the i th source and the i th sink for each i, and whose cost, with respect to the j th criterion, is at most Bj, for 1 ≤ j ≤ R. We prove this problem to be N Pcomplete in paths and cycles even if R = 2. When R = 2 and the edge costs of the second criterion are all 1, the problem can be seen as a monocriterion multicut problem subject to a cardinality constraint. In this case, we show that the problem is strongly N Pcomplete if k = 1 and that, for arbitrary k, it remains strongly N Pcomplete in directed stars but can be solved by (polynomial) dynamic programming algorithms in paths and cycles. For k = 1, we also prove that RCriMultiCut is strongly N Pcomplete in planar bipartite graphs and remains N Pcomplete in K2,d even for R = 2.
Enumerating Minimal Subset Feedback Vertex Sets
"... Abstract. The Subset Feedback Vertex Set problem takes as input a weighted graph G and a vertex subset S of G, andthetaskis to find a set of vertices of total minimum weight to be removed from G such that in the remaining graph no cycle contains a vertex of S. This problem is a generalization of two ..."
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Abstract. The Subset Feedback Vertex Set problem takes as input a weighted graph G and a vertex subset S of G, andthetaskis to find a set of vertices of total minimum weight to be removed from G such that in the remaining graph no cycle contains a vertex of S. This problem is a generalization of two classical NPcomplete problems: Feedback Vertex Set and Multiway Cut. Weshowthatitcanbe solved in time O(1.8638 n) for input graphs on n vertices. To the best of our knowledge, no exact algorithm breaking the trivial 2 n n O(1)time barrier has been known for Subset Feedback Vertex Set, eveninthe case of unweighted graphs. The mentioned running time is a consequence of the more general main result of this paper: we show that all minimal subset feedback vertex sets of a graph can be enumerated in O(1.8638 n) time. 1