A fast algorithm for maximum integral two-commodity flow in planar graphs

We study the following all-or-nothing multicommodity flow problem in planar graphs. Input: A graph G with n vertices and k pairs of vertices (s1, t1), (s2, t2),..., (sk, tk) in G. Find: A largest subset W of {1, ...., k such that for every i in W, we can send one unit of flow between si and ti. This problem is different from the well-known maximum edge-disjoint paths problem in that we do not require integral flows for the pairs. This problem is APX-hard even for trees, and a 2-approximation algorithm is known for trees. For general graphs, Chekuri et al. (STOC'04) give a poly-logarithmic factor approximation algorithm and show that a natural LP-relaxation has a poly-logarithmic integrality gap. This result is in contrast with the integrality gap Ω(√n) for the maximum edge-disjoint paths problem. Our main result considerably strengthens this result when an input graph is planar. Namely, for the all-or-nothing multicommodity flow problem in planar graphs, we give an O(1)-approximation algorithm and show that the integrality gap is O(1). In particular, in polynomial time, we can find an index set W with |W| = Ω(OPT) and eight si-ti paths for each i in W such that each edge is used at most eight times in these paths (with multiplicity), where OPT is the optimal value of the LP-relaxation of the all-or-nothing multicommodity flow problem. Our result can be compared to the recent result by S'eguin-Charbonneau and Shepherd (FOCS'11) who give an O(1)-approximation algorithm for the maximum edge-disjoint paths problem in planar graphs with congestion 2 (but not implied by this result).

Augmenting path algorithms, first introduced by Ford and Fulkerson (Ford, L. R., D. R. Fulkerson. 1962. Flows in Networks. Princeton University Press, Princeton, N.J.), are widely used in optimization. Examples include Schönsleben's polymatroid intersection algorithm (Schönsleben, P. 1980. Ganzzahlige polymatroid-intersection-algorithmen. Ph.D. thesis, Eidgenössischen Technischen Hochschule, Zürich.), the maximum polymatroidal network flow algorithm of Lawler and Martel (Lawler, E. L., C. U. Martel. 1982. Computing maximal polymatroidal network flows. Math. Oper. Res. 7 334–347.), Frank's algorithm for the Edmonds–Giles polyhedron (Frank, A. 1984. Finding feasible vectors of Edmonds-Giles polyhedra. J. Combin. Theory Ser. B 36 221-239.) and Cunningham's algorithm for testing membership in matroid polyhedra (Cunningham, H. W. 1984. Testing membership in matroid polyhedra. J. Combin. Theory Ser. B 36 161–188.). Here we give an order of magnitude improvement for the above algorithms by using an approach analogous to that of Dinits’ maximum flow algorithm (Dinits, E. A. 1970. Algorithm for solution of a problem of maximum flow in a network with power estimation. Soviet Math. Dokl. 11 1277–1280.).

With an increasing number of processing elements (PEs) integrated on a single chip, fault-tolerant techniques are critical to ensure the reliability of such complex systems. In current reconfigurable architectures, redundant PEs are utilized for fault tolerance. In the presence of faulty PEs, the physical topologies of various chips may be different, so the concept of virtual topology from network embedding problem has been used to alleviate the burden for the operating systems. With limited hardware resources, how to reconfigure a system into the most effective virtual topology such that the maximum repair rate can be reached presents a significant challenge. In this article, a new approach using a maximum flow (MF) algorithm is proposed for an efficient topology reconfiguration in reconfigurable architectures. In this approach, topology reconfiguration is converted into a network flow problem by constructing a directed graph; the solution is then found by using the MF algorithm. This approach optimizes the use of spare PEs with minimal impacts on area, throughput, and delay, and thus it significantly improves the repair rate of faulty PEs. In addition, it achieves a polynomial reconfiguration time. Experimental results show that compared to previous methods, the MF approach increases the probability to repair faulty PEs by up to 50% using the same redundant resources. Compared to a fault-free system, the throughput only decreases by less than 2.5% and latency increases by less than 4%. To consider various types of PEs in a practical application, a cost factor is introduced into the MF algorithm. An enhanced approach using a minimum-cost MF algorithm is further shown to be efficient in the fault-tolerant reconfiguration of heterogeneous reconfigurable architectures.

Greenʼs theorem and isolation in planar graphs

The $(\Delta,D)$ (degree/diameter) problem consists of finding the largest possible number of vertices $n$ among all the graphs with maximum degree $\Delta$ and diameter $D$. We consider the $(\Delta,D)$ problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the $(\Delta,2)$ problem, the number of vertices is $n=\Delta+2$; and for the $(\Delta,3)$ problem, $n= 3\Delta-1$ if $\Delta$ is odd and $n= 3\Delta-2$ if $\Delta$ is even. Then, we prove that, for the general case of the $(\Delta,D)$ problem, an upper bound on $n$ is approximately $3(2D+1)(\Delta-2)^{\lfloor D/2\rfloor}$, and another one is $C(\Delta-2)^{\lfloor D/2\rfloor}$ if $\Delta\geq D$ and $C$ is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on $n$ for maximal planar bipartite graphs, which is approximately $(\Delta-2)^{k}$ if $D=2k$, and $3(\Delta-3)^k$ if $D=2k+1$, for $\Delta$ and $D$ sufficiently large in both cases.

A finite metric space, (S,d) , contains a finite set of points and a distance function on pairs of points. A contraction is an embedding, h, of a finite metric space (S, d) into Rd where for any u, v E S, the Euclidean (&) distance between h(u) and h(v) is no more than d(u, v). The distortion of the embedding is the maximum over pairs of the ratio of d(u, w) and the Euclidean distance between h(u) and h(v). Bourgain showed that any graphical metric could be embedded with distortion O(logn). Linial, London and Rabinovich and Aumman and Rabani used such embeddings to prove an O(log k) approximate max-flow min-cut theorem for k commodity flow problems. A generalization of embeddings that preserve distances between pairs-of points are embeddings that preserve volumes of larger sets. In particular, A (k, c)-volume respecting embedding of n-points in any metric space is a contraction where every subset of k points has within an ck-’ factor of its maximal possible k l-dimensional volume. Feige invented these embeddings in devising a polylogarithmic approximation algorithm for the bandwidth problem using these embeddings. Feige’s methods have subsequently been used by Vempala for approximating versions of the VLSI layout problem. Feise showed that a (k, O(10,g~‘~ n,/m)) volume r&ecting embedding‘ eksted.” Be -recently found improved (k, 0( mdk log k + log n)) volume respecting embeddings. For metrics arising from planar graphs (planar metrics), we give (k,O(m)) volume respecting contractions. As a corollary, we give embeddings for Permission to makkr digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. XC’99 Miami Beach Florida Copyright ACM 1999 I-581 13-068-6/99/06...$5.00 planar metrics with distortion O(e). This gives rise to an O(e)-approximate max-flow min-cut theorem for multicommodity flow problems in planar graphs. We also give an improved bound for volume respecting embeddings for Euclidean metrics. In particular, we give an (k,O(dog klog D)) volume respecting embedding where D is the ratio of the largest distance to the smallest distance in the metric. Our results give improvements for Feige’s and Vempala’s approximation algorithms for planar and Euclidean metrics. For volume respecting embeddings, our embeddings do not degrade very fast when preserving the volumes of large subsets. This may be useful in the future for approximation algorithms or if volume .respecting embeddings prove to be of independent interest.

Deciding whether a grid is a topological subgraph of a planar graph is NP-complete

Faster Shortest-Path Algorithms for Planar Graphs

We consider dynamic subgraph connectivity problems for planar undirected graphs. In this model there is a fixed underlying planar graph, where each edge and vertex is either off (failed) or (recovered). We wish to answer connectivity queries with respect to the subgraph. The model has two natural variants, one in which there are d edge/vertex failures that precede all connectivity queries, and one in which failures/recoveries and queries are intermixed. We present a d-failure connectivity oracle for planar graphs that processes any d edge/vertex failures in sort(d,n) time so that connectivity queries can be answered in pred(d,n) time. (Here sort and pred are the time for integer sorting and integer predecessor search over a subset of [n] of size d.) Our algorithm has two discrete parts. The first is an algorithm tailored to triconnected planar graphs. It makes use of Barnette's theorem, which states that every triconnected planar graph contains a degree-3 spanning tree. The second part is a generic reduction from general (planar) graphs to triconnected (planar) graphs. Our algorithm is, moreover, provably optimal. An implication of Pǎtrascu and Thorup's lower bound on predecessor search is that no d-failure connectivity oracle (even on trees) can beat pred(d,n) query time. We extend our algorithms to the subgraph connectivity model where edge/vertex failures (but no recoveries) are intermixed with connectivity queries. In triconnected planar graphs each failure and query is handled in O(logn) time (amortized), whereas in general planar graphs both bounds become O(log2n).

The bulk power system is one of the largest man-made networks and its size makes control an extremely difficult task. This paper presents a method to control a power network using UPFCs set to levels determined by a maximum flow (max-flow) algorithm. The graph-theory-based max-flow is applied to the power system for UPFC placement and scheduling. A distributed version of max-flow is described to coordinate the actions of the UPFCs distributed in a power network. Two sample power systems were tested using max-flow for UPFC placement and settings. The resulting system characteristics are examined over all single-line contingencies and the appropriateness of the maximum flow algorithm for power flow control is discussed.

Parallelism and the maximal path problem

A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are at distance 2 from each other. The Square Root problem is that of deciding whether a given graph admits a square root. We consider this problem for planar graphs in the context of the “distance from triviality” framework. For an integer k, a planar+kv graph (or k-apex graph) is a graph that can be made planar by the removal of at most k vertices. We prove that a generalization of Square Root, in which some edges are prescribed to be either in or out of any solution, has a kernel of size O(k) for planar+kv graphs, when parameterized by k. Our result is based on a new edge reduction rule which, as we shall also show, has a wider applicability for the Square Root problem.

We study the natural problem of secure n-party computation (in the computationally unbounded attack model) of circuits over an arbitrary finite non-Abelian group (G,?), which we call G-circuits. Besides its intrinsic interest, this problem is also motivating by a completeness result of Barrington, stating that such protocols can be applied for general secure computation of arbitrary functions. For flexibility, we are interested in protocols which only require black-box access to the group G (i.e. the only computations performed by players in the protocol are a group operation, a group inverse, or sampling a uniformly random group element). Our investigations focus on the passive adversarial model, where up to t of the n participating parties are corrupted. Our results are as follows. We initiate a novel approach for the construction of black-box protocols for G-circuits based on k-of-k threshold secret-sharing schemes, which are efficiently implementable over any black-box (non-Abelian) group G. We reduce the problem of constructing such protocols to a combinatorial coloring problem in planar graphs. We then give three constructions for such colorings. Our first approach leads to a protocol with optimal resilience t<n/2, but it requires exponential communication complexity $O({\binom{2 t+1}{t}}^{2} \cdot N_{g})$ group elements and round complexity $O(\binom{2 t + 1}{t} \cdot N_{g})$ , for a G-circuit of size N g . Nonetheless, using this coloring recursively, we obtain another protocol to t-privately compute G-circuits with communication complexity $\mathcal{P}\mathit{oly}(n)\cdot N_{g}$ for any t?O(n 1?∈ ) where ∈ is any positive constant. For our third protocol, there is a probability ? (which can be made arbitrarily small) for the coloring to be flawed in term of security, in contrast to the first two techniques, where the colorings are always secure (we call this protocol probabilistic, and those earlier protocols deterministic). This third protocol achieves optimal resilience t<n/2. It has communication complexity O(n 5.056(n+log?? ?1)2?N g ) and the number of rounds is O(n 2.528?(n+log?? ?1)?N g ).

We study the natural problem of secure n-party computation (in the passive, computationally unbounded attack model) of the n-product function fG(x1, . . . , xn) = x1 ċ x2 . . . xn in an arbitrary finite group (G, ċ), where the input of party Pi is xi ∈ G for i = 1, . . . , n. For flexibility, we are interested in protocols for fG which require only black-box access to the group G (i.e. the only computations performed by players in the protocol are a group operation, a group inverse, or sampling a uniformly random group element). Our results are as follows. First, on the negative side, we show that if (G, ċ) is non-abelian and n ≥ 4, then no ⌈n/2⌉-private protocol for computing fG exists. Second, on the positive side, we initiate an approach for construction of black-box protocols for fG based on k-of-k threshold secret sharing schemes, which are efficiently implementable over any black-box group G. We reduce the problem of constructing such protocols to a combinatorial colouring problem in planar graphs. We then give two constructions for such graph colourings. Our first colouring construction gives a protocol with optimal collusion resistance t < n/2, but has exponential communication complexity O(n(2t+1/t)2) group elements (this construction easily extends to general adversary structures). Our second probabilistic colouring construction gives a protocol with (close to optimal) collusion resistance t < n/µ for a graph-related constant µ ≤ 2.948, and has efficient communication complexity O(nt2) group elements. Furthermore, we believe that our results can be improved by further study of the associated combinatorial problems.

We present a polynomial time approximation scheme (PTAS) for the Steiner tree problem with polygonal obstacles in the plane with running time O(n log2 n), where n denotes the number of terminals plus obstacle vertices. To this end, we show how a planar spanner of size O(n logn) can be constructed that contains a (1 + ε)-approximation of the optimal tree. Then one can find an approximately optimal Steiner tree in the spanner using the algorithm of Borradaile et al. (2007) for the Steiner tree problem in planar graphs. We prove this result for the Euclidean metric and also for all uniform orientation metrics, i.e. particularly the rectilinear and octilinear metrics.