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2006

Volume 36, Issue 2, pp. 281-561


Dominating Sets in Planar Graphs: Branch-Width and Exponential Speed-Up

Fedor V. Fomin and Dimitrios M. Thilikos

SIAM J. Comput. 36, pp. 281-309 (29 pages) | Cited 8 times

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We introduce a new approach to design parameterized algorithms on planar graphs which builds on the seminal results of Robertson and Seymour on graph minors. Graph minors provide a list of powerful theoretical results and tools. However, the widespread opinion in the graph algorithms community about this theory is that it is of mainly theoretical importance. In this paper we show how deep min-max and duality theorems from graph minors can be used to obtain exponential speed-up to many known practical algorithms for different domination problems. Our use of branch-width instead of the usual tree-width allows us to obtain much faster algorithms. By using this approach, we show that the k-dominating set problem on planar graphs can be solved in time O(215.13 \sqrt k + n3).

Between O(nm) and O(nalpha)

Dieter Kratsch and Jeremy Spinrad

SIAM J. Comput. 36, pp. 310-325 (16 pages)

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This paper uses periodic matrix multiplication to improve the time complexities for a number of graph problems. The time for finding an asteroidal triple is reduced from O(nm) to O(n2.82), and the time for finding a star cutset, a two-pair, and a dominating pair is reduced from O(nm) to O(n2.79). It is also shown that each of these problems is at least as hard as one of three basic graph problems for which the best known algorithms run in times O(nm) and O(nalpha). We note that the fast matrix multiplication algorithms do not seem to be practical because of the enormous constants needed to achieve the asymptotic time bounds. These results are important theoretically for breaking the n3 barrier rather than giving efficient algorithms for a user.

Certifying Algorithms for Recognizing Interval Graphs and Permutation Graphs

Dieter Kratsch, Ross M. McConnell, Kurt Mehlhorn, and Jeremy P. Spinrad

SIAM J. Comput. 36, pp. 326-353 (28 pages) | Cited 7 times

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A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give linear-time certifying algorithms for recognition of interval graphs and permutation graphs, and for a few other related problems. Previous algorithms fail to provide supporting evidence when they claim that the input graph is not a member of the class. We show that our certificates of nonmembership can be authenticated in O(|V|) time.

Lower Bounds for On-line Graph Problems with Application to On-line Circuit and Optical Routing

Yair Bartal, Amos Fiat, and Stefano Leonardi

SIAM J. Comput. 36, pp. 354-393 (40 pages)

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We present lower bounds on the competitive ratio of randomized algorithms for a wide class of on-line graph optimization problems, and we apply such results to on-line virtual circuit and optical routing problems. Lund and Yannakakis [The approximation of maximum subgraph problems, in Proceedings of the 20th International Colloquium on Automata, Languages and Programming, 1993, pp. 40-51] give inapproximability results for the problem of finding the largest vertex induced subgraph satisfying any nontrivial, hereditary property pi--e.g., independent set, planar, acyclic, bipartite. We consider the on-line version of this family of problems, where some graph G is fixed and some subgraph H of G is presented on-line, vertex by vertex. The on-line algorithm must choose a subset of the vertices of H, choosing or rejecting a vertex when it is presented, whose vertex induced subgraph satisfies property pi. Furthermore, we study the on-line version of graph coloring whose off-line version has also been shown to be inapproximable [C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, in Proceedings of the 25th ACM Symposium on Theory of Computing, 1993], on-line max edge-disjoint paths, and on-line path coloring problems. Irrespective of the time complexity, we show an Omega(nepsilon) lower bound on the competitive ratio of randomized on-line algorithms for any of these problems. As a consequence, we obtain an Omega(nepsilon) lower bound on the competitive ratio of randomized on-line algorithms for virtual circuit routing on general networks, in contrast to the known results for some specific networks. Similar lower bounds are obtained for on-line optical routing as well.

Efficient Bundle Sorting

Yossi Matias, Eran Segal, and Jeffrey Scott Vitter

SIAM J. Comput. 36, pp. 394-410 (17 pages)

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Many data sets to be sorted consist of a limited number of distinct keys. Sorting such data sets can be thought of as bundling together identical keys and having the bundles placed in order; we therefore denote this as bundle sorting. We describe an efficient algorithm for bundle sorting in external memory, which requires at most c(N/B) logM/Bk disk accesses, where N is the number of keys, M is the size of internal memory, k is the number of distinct keys, B is the transfer block size, and 2 < c < 4. For moderately sized k, this bound circumvents the Theta((N/B) logM/B (N/B)) I/O lower bound known for general sorting. We show that our algorithm is optimal by proving a matching lower bound for bundle sorting. The improved running time of bundle sorting over general sorting can be significant in practice, as demonstrated by experimentation. An important feature of the new algorithm is that it is executed "in-place," requiring no additional disk space.

Approximation Algorithms for Metric Facility Location Problems

Mohammad Mahdian, Yinyu Ye, and Jiawei Zhang

SIAM J. Comput. 36, pp. 411-432 (22 pages) | Cited 21 times

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In this paper we present a 1.52-approximation algorithm for the metric uncapacitated facility location problem, and a 2-approximation algorithm for the metric capacitated facility location problem with soft capacities. Both these algorithms improve the best previously known approximation factor for the corresponding problem, and our soft-capacitated facility location algorithm achieves the integrality gap of the standard linear programming relaxation of the problem. Furthermore, we will show, using a result of Thorup, that our algorithms can be implemented in quasi-linear time.

An Unconditional Lower Bound on the Time-Approximation Trade-off for the Distributed Minimum Spanning Tree Problem

Michael Elkin

SIAM J. Comput. 36, pp. 433-456 (24 pages) | Cited 2 times

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The design of distributed approximation protocols is a relatively new and rapidly developing area of research. However, so far, little progress has been made in the study of the hardness of distributed approximation. In this paper we initiate the systematic study of this subject and show strong unconditional lower bounds on the time-approximation trade-off of the distributed minimum spanning tree problem, and show some of its variants.

Toward a Topological Characterization of Asynchronous Complexity

Gunnar Hoest and Nir Shavit

SIAM J. Comput. 36, pp. 457-497 (41 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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This paper introduces the use of topological models and methods, formerly used to analyze computability, as tools for the quantification and classification of asynchronous complexity. We present the first asynchronous complexity theorem, applied to decision tasks in the iterated immediate snapshot (IIS) model of Borowsky and Gafni. We do so by introducing a novel form of topological tool called the nonuniform chromatic subdivision. Building on the framework of Herlihy and Shavit’s topological computability model, our theorem states that the time complexity of any asynchronous algorithm is directly proportional to the level of nonuniform chromatic subdivisions necessary to allow a simplicial map from a task’s input complex to its output complex. To show the power of our theorem, we use it to derive a new tight bound on the time to achieve $n$ process approximate agreement in the IIS model: $\bigl\lceil \log_d \frac{\max\_input - \min\_input}{\epsilon} \bigr\rceil$, where $d = 3$ for two processes and $d = 2$ for three or more. This closes an intriguing gap between the known upper and lower bounds implied by the work of Aspnes and Herlihy. More than the new bounds themselves, the importance of our asynchronous complexity theorem is that the algorithms and lower bounds it allows us to derive are intuitive and simple, with topological proofs that require no mention of concurrency at all.

Covering Problems with Hard Capacities

Julia Chuzhoy and Joseph (Seffi) Naor

SIAM J. Comput. 36, pp. 498-515 (18 pages) | Cited 3 times

Online Publication Date: July 31, 2006

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We consider the classical vertex cover and set cover problems with hard capacity constraints. This means that a set (vertex) can cover only a limited number of its elements (adjacent edges), and the number of available copies of each set (vertex) is bounded. This is a natural generalization of the classical problems which also captures resource limitations in practical scenarios. We obtain the following results. For the unweighted vertex cover problem with hard capacities we give a $3$‐approximation algorithm that is based on randomized rounding with alterations. We prove that the weighted version is at least as hard as the set cover problem, yielding an interesting separation between the approximability of weighted and unweighted versions of a “natural” graph problem. A logarithmic approximation factor for both the set cover and the weighted vertex cover problem with hard capacities follows from the work of Wolsey [Combinatorica, 2 (1982), pp. 385–393] on submodular set cover. We provide here a simple and intuitive proof for this bound.

Properties of NP‐Complete Sets

Christian Glaßer, A. Pavan, Alan L. Selman, and Samik Sengupta

SIAM J. Comput. 36, pp. 516-542 (27 pages)

Online Publication Date: July 31, 2006

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We study several properties of sets that are complete for NP. We prove that if $L$ is an NP‐complete set and S \not\supseteq L is a p‐selective sparse set, then $L - S$ is $\leq^{p}_{m}$‐hard for NP. We demonstrate the existence of a sparse set $S \in \mathrm{DTIME}(2^{2^{n}})$ such that for every $L \in \mbox{NP} - \mbox{P}$, L - S is not $\leq^p_m$‐hard for NP. Moreover, we prove for every $L \in \mbox{NP} - \mbox{P}$ that there exists a sparse $S \in $ EXP such that L - S is not $\leq^p_m$‐hard for NP. Hence, removing sparse information in P from a complete set leaves the set complete, while removing sparse information in EXP from a complete set may destroy its completeness. Previously, these properties were known only for exponential time complexity classes. We use hypotheses about pseudorandom generators and secure one‐way permutations to derive consequences for longstanding open questions about whether NP‐complete sets are immune. For example, assuming that pseudorandom generators and secure one‐way permutations exist, it follows easily that NP‐complete sets are not p‐immune. Assuming only that secure one‐way permutations exist, we prove that no NP‐complete set is DTIME$(2^{n^{\epsilon}})$‐immune. Also, using these hypotheses we show that no NP‐complete set is quasi‐polynomial‐close to P. We introduce a strong but reasonable hypothesis and infer from it that disjoint Turing‐complete sets for NP are not closed under union. Our hypothesis asserts the existence of a UP‐machine $M$ that accepts $0^*$ such that for some $0 < \epsilon < 1$, no $2^{n^{\epsilon}}$ time‐bounded machine can correctly compute infinitely many accepting computations of $M$. We show that if $\UP \cap \co\UP$ contains DTIME$(2^{n^{\epsilon}})$‐bi‐immune sets, then this hypothesis is true.

The Directed Steiner Network Problem is Tractable for a Constant Number of Terminals

Jon Feldman and Matthias Ruhl

SIAM J. Comput. 36, pp. 543-561 (19 pages) | Cited 2 times

Online Publication Date: August 07, 2006

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We consider the Directed Steiner Network problem, also called the Point‐to‐Point Connection problem. Given a directed graph $G$ and $p$ pairs $\{ (s_1,t_1), \dotsc, (s_p,t_p) \}$ of nodes in the graph, one has to find the smallest subgraph $H$ of $G$ that contains paths from $s_i$ to $t_i$ for all $i$. The problem is NP‐hard for general $p$, since the Directed Steiner Tree problem is a special case. Until now, the complexity was unknown for constant $p \geq 3$. We prove that the problem is polynomially solvable if $p$ is any constant number, even if nodes and edges in $G$ are weighted and the goal is to minimize the total weight of the subgraph $H$. In addition, we give an efficient algorithm for the Strongly Connected Steiner Subgraph problem for any constant $p$, where given a directed graph and $p$ nodes in the graph, one has to compute the smallest strongly connected subgraph containing the $p$ nodes.
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