Results 1  10
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13
Homology flows, cohomology cuts
 ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2009
"... We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time fo ..."
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We describe the first algorithms to compute maximum flows in surfaceembedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)flow in O(g 7 n log 2 n log 2 C) time for integer capacities that sum to C, or in (g log n) O(g) n time for real capacities. Except for the special case of planar graphs, for which an O(n log n)time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surfaceembedded graphs follow from algorithms for general sparse graphs. Our key insight is to optimize the relative homology class of the flow, rather than directly optimizing the flow itself. A dual formulation of our algorithm computes the minimumcost cycle or circulation in a given (real or integer) homology class.
Flows in onecrossingminorfree graphs
 In ISAAC (1
, 2010
"... Abstract. We study the maximum flow problem in directed Hminorfree graphs where H can be drawn in the plane with one crossing. If a structural decomposition of the graph as a cliquesum of planar graphs and graphs of constant complexity is given, we show that a maximum flow can be computed in O(n ..."
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Abstract. We study the maximum flow problem in directed Hminorfree graphs where H can be drawn in the plane with one crossing. If a structural decomposition of the graph as a cliquesum of planar graphs and graphs of constant complexity is given, we show that a maximum flow can be computed in O(n logn) time. In particular, maximum flows in directed K3,3minorfree graphs and directed K5minorfree graphs can be computed in O(n logn) time without additional assumptions. 1
Preserving Terminal Distances using Minors ⋆
"... Abstract. We introduce the following notion of compressing an undirected graph G with (nonnegative) edgelengths and terminal vertices R ⊆ V (G). A distancepreserving minor is a minor G ′ (of G) with possibly different edgelengths, such that R ⊆ V (G ′ ) and the shortestpath distance between ever ..."
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Abstract. We introduce the following notion of compressing an undirected graph G with (nonnegative) edgelengths and terminal vertices R ⊆ V (G). A distancepreserving minor is a minor G ′ (of G) with possibly different edgelengths, such that R ⊆ V (G ′ ) and the shortestpath distance between every pair of terminals is exactly the same in G and in G ′. We ask: what is the smallest f ∗ (k) such that every graph G with k = R  terminals admits a distancepreserving minor G ′ with at most f ∗ (k) vertices? Simple analysis shows that f ∗ (k) ≤ O(k 4). Our main result proves that f ∗ (k) ≥ Ω(k 2), significantly improving over the trivial f ∗ (k) ≥ k. Our lower bound holds even for planar graphs G, in contrast to graphs G of constant treewidth, for which we prove that O(k) vertices suffice. 1
Capacitated network design on undirected graphs
 In APPROXRANDOM
, 2013
"... In this paper, we study the approximability of the capacitated network design problem (CapNDP) on undirected graphs: GivenG = (V,E) with nonnegative costs c and capacities u on its edges, sourcesink pairs (si, ti) with demand ri, the goal is to find the minimum cost subgraph where the minimum (si ..."
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In this paper, we study the approximability of the capacitated network design problem (CapNDP) on undirected graphs: GivenG = (V,E) with nonnegative costs c and capacities u on its edges, sourcesink pairs (si, ti) with demand ri, the goal is to find the minimum cost subgraph where the minimum (si, ti) cut with ucapacities is at least ri. When u ≡ 1, we get usual SNDP for which Jain gave a 2approximation algorithm [9]. Prior to our work, the approximability of undirected CapNDP was not well understood even in the single sourcesink pair case. In this paper, we show that the singlesource pair CapNDP is labelcover hard in undirected graphs. An important special case of single sourcesink pair undirected CapNDP is the following source location problem. Given an undirected graph, a collection of sources S and a sink t, find the minimum cardinality subset S ′ ⊆ S such that flow(S′, t), the maximum flow from S ′ to t, equals flow(S, t). In general, the problem is known to be setcover hard. We give a O(ρ)approximation when flow(s, t) ≈ρ flow(s′, t) for s, s ′ ∈ S, that is, all sources have maxflow values to the sink within a multiplicative ρ factor of each other. The main technical ingredient of our algorithmic result is the following theorem which may have other application. Given a capacitated, undirected graph G with a dedicated sink t, call a subset X ⊆ V irreducible if the maximum flow f(X) from X to t is strictly greater than that from any strict subset X ′ ⊂ X, to t. We prove that for any irreducible set, X, the flow f(X) ≥ 12 i∈X fi, where fi is the maxflow from i to t. That is, undirected flows are quasiadditive on irreducible sets. 1
Mimicking networks and succinct representations of terminal cuts
 CoRR
"... Given a large edgeweighted network G with k terminal vertices, we wish to compress it and store, using little memory, the value of the minimum cut (or equivalently, maximum flow) between every bipartition of terminals. One appealing methodology to implement a compression of G is to construct a mimi ..."
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Given a large edgeweighted network G with k terminal vertices, we wish to compress it and store, using little memory, the value of the minimum cut (or equivalently, maximum flow) between every bipartition of terminals. One appealing methodology to implement a compression of G is to construct a mimicking network: a small network G ′ with the same k terminals, in which the minimum cut value between every bipartition of terminals is the same as in G. This notion was introduced by Hagerup, Katajainen, Nishimura, and Ragde [JCSS ’98], who proved that such G ′ of size at most 22k always exists. Obviously, by having access to the smaller network G ′, certain computations involving cuts can be carried out much more efficiently. We provide several new bounds, which together narrow the previously known gap from doublyexponential to only singlyexponential, both for planar and for general graphs. Our first and main result is that every kterminal planar network admits a mimicking network G ′ of size O(k 2 2 2k), which is moreover a minor of G. On the other hand, some planar networks G require E(G ′)  ≥ Ω(k 2). For general networks, we show that certain bipartite graphs only admit mimicking networks of size V (G ′)  ≥ 2 Ω(k) , and moreover, every data structure that stores the minimum cut value between all bipartitions of the terminals must use 2Ω(k) machine words. 1
Towards (1 + ε)Approximate Flow Sparsifiers
"... A useful approach to “compress ” a large network G is to represent it with a flowsparsifier, i.e., a small network H that supports the same flows as G, up to a factor q ≥ 1 called the quality of sparsifier. Specifically, we assume the network G contains a set of k terminals T, shared with the netwo ..."
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A useful approach to “compress ” a large network G is to represent it with a flowsparsifier, i.e., a small network H that supports the same flows as G, up to a factor q ≥ 1 called the quality of sparsifier. Specifically, we assume the network G contains a set of k terminals T, shared with the network H, i.e., T ⊆ V (G)∩V (H), and we want H to preserve all multicommodity flows that can be routed between the terminals T. The challenge is to construct H that is small. These questions have received a lot of attention in recent years, leading to some known tradeoffs between the sparsifier’s quality q and its size V (H). Nevertheless, it remains an outstanding question whether every G admits a flowsparsifier H with quality q = 1 + ε, or even q = O(1), and size V (H)  ≤ f(k, ε) (in particular, independent of V (G)  and the edge capacities). Making a first step in this direction, we present new constructions for several scenarios: • Our main result is that for quasibipartite networks G, one can construct a (1 + ε)flowsparsifier of size poly(k/ε). In contrast, exact (q = 1) sparsifiers for this family of networks are known to require size 2 Ω(k). • For networks G of bounded treewidth w, we construct a flowsparsifier with quality q = O(log w / log log w) and size O(w · poly(k)). • For general networks G, we construct a sketch sk(G), that stores all the feasible multicommodity flows up to factor q = 1 + ε, and its size (storage requirement) is f(k, ε). ∗ A full version of this paper is submitted to arXiv [AGK13].
Degree3 Treewidth Sparsifiers∗
, 2014
"... We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, V (H)  is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on nodedisj ..."
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We study treewidth sparsifiers. Informally, given a graph G of treewidth k, a treewidth sparsifier H is a minor of G, whose treewidth is close to k, V (H)  is small, and the maximum vertex degree in H is bounded. Treewidth sparsifiers of degree 3 are of particular interest, as routing on nodedisjoint paths, and computing minors seems easier in subcubic graphs than in general graphs. In this paper we describe an algorithm that, given a graph G of treewidth k, computes a topological minor H of G such that (i) the treewidth of H is Ω(k/polylog(k)); (ii) V (H)  = O(k4); and (iii) the maximum vertex degree in H is 3. The running time of the algorithm is polynomial in V (G)  and k. Our result is in contrast to the known fact that unless NP ⊆ coNP/poly, treewidth does not admit polynomialsize kernels. One of our key technical tools, which is of independent interest, is a construction of a small minor that preserves nodedisjoint routability between two pairs of vertex subsets. This is closely related to the open question of computing small goodquality vertexcut sparsifiers that are also minors of the original graph. 1
Inferring Tree Topologies Using Flow Tests
 Proc. 14th ACMSIAM Symposium on Discrete Algorithms
, 2003
"... Introduction We consider the problem of discovering the structure of an unknown hierarchical network by means of measuring the maximum flow between the root and selected subsets of leaf nodes. More precisely, we are given a root node and a set of n leaf nodes, each identified by a unique label. The ..."
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Introduction We consider the problem of discovering the structure of an unknown hierarchical network by means of measuring the maximum flow between the root and selected subsets of leaf nodes. More precisely, we are given a root node and a set of n leaf nodes, each identified by a unique label. The leaf nodes are the leaves of a capacirated hierarchical network. We do not have any additional information about the structure of the network, including the degrees of any internal nodes, the edge capacities, or which leaf nodes are in the same subtree. Our goal is to infer the structure of the network (up to certain equivalences discussed below) by using only a simple test operation, in which we "switch on" a selected subset of the leaf nodes, causing these nodes to transmit data to the root at maximum speed, and then measure the total rate of data arriving at the root. We will refer to this operation as a flow test. In this note, we derive upper and lower bounds for the number of flow te
Technical report version
"... Abstract We present a new technique for determining how much informationabout a program's secret inputs is revealed by its public outputs. In contrast to previous techniques based on reachability from secretinputs (tainting), it achieves a more precise quantitative result by computing a maximum ..."
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Abstract We present a new technique for determining how much informationabout a program's secret inputs is revealed by its public outputs. In contrast to previous techniques based on reachability from secretinputs (tainting), it achieves a more precise quantitative result by computing a maximum flow of information between the inputs andoutputs. The technique uses static controlflow regions to soundly account for implicit flows via branches and pointer operations, butoperates dynamically by observing one or more program executions and giving numeric flow bounds specific to them (e.g., &quot;17bits&quot;). The maximum flow in a network also gives a minimum cut (a set of edges that separate the secret input from the output), whichcan be used to efficiently check that the same policy is satisfied on future executions. We performed case studies on 5 real C, C++,and Objective C programs, 3 of which had more than 250K lines of code. The tool checked multiple security policies, including onethat was violated by a previously unknown bug. 1. Introduction The goal of informationflow security is to enforce limits on thedissemination of information. For instance, a confidentiality property requires that a program that is entrusted with secrets shouldnot &quot;leak &quot; those secrets into public outputs. Absolute prohibitions on information flow are rarely satisfied by real programs: if a sensitive input does not affect a program's output at all, it is better to simply omit it, and unrelated computations of different securitylevels should be performed by separate processes. Rather, the key challenge for informationflow security is to distinguish acceptablefrom unacceptable flows.
Counting and Sampling Minimum Cuts in Genus g Graphs
, 2012
"... Let G be a directed graph with n vertices embedded on an orientable surface of genus g with two designated vertices s and t. We show that counting the number of minimum (s, t)cuts in G is fixed parameter tractable in g. Specially, we give a 2 O(g) n 2 + min { n 2 log n, g O(g) n 3/2} time algorithm ..."
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Let G be a directed graph with n vertices embedded on an orientable surface of genus g with two designated vertices s and t. We show that counting the number of minimum (s, t)cuts in G is fixed parameter tractable in g. Specially, we give a 2 O(g) n 2 + min { n 2 log n, g O(g) n 3/2} time algorithm for this problem. Our algorithm requires counting sets of cycles in a particular integer homology class. That we can count these cycles is an interesting result in itself as there are few prior results that are fixed parameter tractable and deal directly with integer homology. We also describe an algorithm which, after running our algorithm to count the number of cuts once, can sample repeatedly for a minimum cut in O(g 2 n) time per sample.