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2009

Volume 38, Issue 6, pp. 2113-2547

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Separating ${AC}^0$ from Depth-2 Majority Circuits

Alexander A. Sherstov

SIAM J. Comput. 38, pp. 2113-2129 (17 pages) | Cited 2 times

Online Publication Date: February 11, 2009

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We construct a function in ${AC}^0$ that cannot be computed by a depth-2 majority circuit of size less than $\exp(\Theta(n^{1/5}))$. This solves an open problem due to Krause and Pudlák [Theoret. Comput. Sci., 174 (1997), pp. 137–156] and matches Allender's classic result [A note on the power of threshold circuits, in Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Research Triangle Park, NC, 1989, pp. 580–584] that ${AC}^0$ can be efficiently simulated by depth-3 majority circuits. To obtain our result, we develop a novel technique for proving lower bounds on communication complexity. This technique, the Degree/Discrepancy Theorem, is of independent interest. It translates lower bounds on the threshold degree of any Boolean function into upper bounds on the discrepancy of a related function. Upper bounds on the discrepancy, in turn, immediately imply lower bounds on communication and circuit size. In particular, we exhibit the first known function in ${AC}^0$ with exponentially small discrepancy, $\exp(-\Omega(n^{1/5}))$, thereby establishing the separations $\Sigma_2^{cc}\not\subseteq{PP}^{cc}$ and $\Pi_2^{cc}\not\subseteq{PP}^{cc}$ in communication complexity.

Interpolation of Depth-3 Arithmetic Circuits with Two Multiplication Gates

Amir Shpilka

SIAM J. Comput. 38, pp. 2130-2161 (32 pages)

Online Publication Date: February 11, 2009

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In this paper we consider the problem of constructing a small arithmetic circuit for a polynomial for which we have oracle access. Our focus is on $n$-variate polynomials, over a finite field $\mathbb{F}$, that have depth-3 arithmetic circuits (with an addition gate at the top) with two multiplication gates of degree at most $d$. We obtain the following results: 1. Multilinear case. When the circuit is multilinear (multiplication gates compute multilinear polynomials) we give an algorithm that outputs, with probability $1-o(1)$, all the depth-3 circuits with two multiplication gates computing the polynomial. The running time of the algorithm is $\operatorname{poly}(n,|\mathbb{F}|)$. 2. General case. When the circuit is not multilinear we give a quasi-polynomial (in $n,d,|\mathbb{F}|$) time algorithm that outputs, with probability $1-o(1)$, a succinct representation of the polynomial. In particular, if the depth-3 circuit for the polynomial is not of small depth-3 rank (namely, after removing the g.c.d. (greatest common divisor) of the two multiplication gates, the remaining linear functions span a not too small linear space), then we output the depth-3 circuit itself. In the case that the rank is small we output a depth-3 circuit with a quasi-polynomial number of multiplication gates. $\diamond$ Prior to our work there have been several interpolation algorithms for restricted models. However, all the techniques used there completely fail when dealing with depth-3 circuits with even just two multiplication gates. Our proof technique is new and relies on the factorization algorithm for multivariate black-box polynomials, on lower bounds on the length of linear locally decodable codes with two queries, and on a theorem regarding the structure of identically zero depth-3 circuits with four multiplication gates.

Breaking a Time-and-Space Barrier in Constructing Full-Text Indices

Wing-Kai Hon, Kunihiko Sadakane, and Wing-Kin Sung

SIAM J. Comput. 38, pp. 2162-2178 (17 pages)

Online Publication Date: February 11, 2009

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Suffix trees and suffix arrays are the most prominent full-text indices, and their construction algorithms are well studied. In the literature, the fastest algorithm runs in $O(n)$ time, while it requires $O(n\log n)$-bit working space, where $n$ denotes the length of the text. On the other hand, the most space-efficient algorithm requires $O(n)$-bit working space while it runs in $O(n\log n)$ time. It was open whether these indices can be constructed in both $o(n\log n)$ time and $o(n\log n)$-bit working space. This paper breaks the above time-and-space barrier under the unit-cost word RAM. We give an algorithm for constructing the suffix array, which takes $O(n)$ time and $O(n)$-bit working space, for texts with constant-size alphabets. Note that both the time and the space bounds are optimal. For constructing the suffix tree, our algorithm requires $O(n\log^{\epsilon}n)$ time and $O(n)$-bit working space for any $0<\epsilon<1$. Apart from that, our algorithm can also be adopted to build other existing full-text indices, such as compressed suffix tree, compressed suffix arrays, and FM-index. We also study the general case where the size of the alphabet $\Sigma$ is not constant. Our algorithm can construct a suffix array and a suffix tree using optimal $O(n\log|\Sigma|)$-bit working space while running in $O(n\log\log|\Sigma|)$ time and $O(n(\log^{\epsilon}n+\log|\Sigma|))$ time, respectively. These are the first algorithms that achieve $o(n\log n)$ time with optimal working space. Moreover, for the special case where $\log|\Sigma|=O((\log\log n)^{1-\epsilon})$, we can speed up our suffix array construction algorithm to the optimal $O(n)$.

Linear-Time Haplotype Inference on Pedigrees without Recombinations and Mating Loops

Mee Yee Chan, Wun-Tat Chan, Francis Y. L. Chin, Stanley P. Y. Fung, and Ming-Yang Kao

SIAM J. Comput. 38, pp. 2179-2197 (19 pages)

Online Publication Date: March 04, 2009

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In this paper, an optimal linear-time algorithm is presented to solve the haplotype inference problem for pedigree data when there are no recombinations and the pedigree has no mating loops. The approach is based on the use of graphs to capture SNP, Mendelian, and parity constraints of the given pedigree. This representation allows us to capture the constraints as the edges in a graph, rather than as a system of linear equations as in previous approaches. Graph traversals are then used to resolve the parity of these edges, resulting in an optimal running time.

Efficient Algorithms for Reconstructing Zero-Recombinant Haplotypes on a Pedigree Based on Fast Elimination of Redundant Linear Equations

Jing Xiao, Lan Liu, Lirong Xia, and Tao Jiang

SIAM J. Comput. 38, pp. 2198-2219 (22 pages)

Online Publication Date: March 04, 2009

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Computational inference of haplotypes from genotypes has attracted a great deal of attention in the computational biology community recently, partially driven by the international HapMap project. In this paper, we study the question of how to efficiently infer haplotypes from genotypes of individuals related by a pedigree, assuming that the hereditary process was free of mutations (i.e., the Mendelian law of inheritance) and recombinants. The problem has recently been formulated as a system of linear equations over the finite field of $F(2)$ and solved in $O(m^3n^3)$ time by using standard Gaussian elimination, where $m$ is the number of loci (or markers) in a genotype and $n$ the number of individuals in the pedigree. We give a much faster algorithm with running time $O(mn^2+n^3\log^2n\log\log n)$. The key ingredients of our construction are (i) a new system of linear equations based on some spanning tree of the pedigree graph and (ii) an efficient method for eliminating redundant equations in a system of $O(mn)$ linear equations over $O(n)$ variables. Although such a fast elimination method is not known for general systems of linear equations, we take advantage of the underlying pedigree graph structure and recent progress on low-stretch spanning trees.

Polylogarithmic Independence Can Fool DNF Formulas

Louay M. J. Bazzi

SIAM J. Comput. 38, pp. 2220-2272 (53 pages)

Online Publication Date: March 04, 2009

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We show that any $k$-wise independent probability distribution on $\{0,1\}^n$ $O(m^{2.2}$ $2^{-\sqrt{k}/10})$-fools any boolean function computable by an $m$-clause disjunctive normal form (DNF) (or conjunctive normal form (CNF)) formula on $n$ variables. Thus, for each constant $e>0$, there is a constant $c>0$ such that any boolean function computable by an $m$-clause DNF (or CNF) formula is $m^{-e}$-fooled by any $c\log^2m$-wise probability distribution. This resolves up to an $O(\log m)$ factor the depth-2 circuit case of a conjecture due to Linial and Nisan [Combinatorica, 10 (1990), pp. 349–365]. The result is equivalent to a new characterization of DNF (or CNF) formulas by low degree polynomials. It implies a similar statement for probability distributions with the small bias property. Using known explicit constructions of small probability spaces having the limited independence property or the small bias property, we directly obtain a large class of explicit pseudorandom generators of $O(\log^2m\log n)$-seed length for $m$-clause DNF (or CNF) formulas on $n$ variables, improving previously known seed lengths.

On the Value of Coordination in Network Design

Susanne Albers

SIAM J. Comput. 38, pp. 2273-2302 (30 pages)

Online Publication Date: March 04, 2009

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We study network design games where $n$ self-interested agents have to form a network by purchasing links from a given set of edges. We consider Shapley cost sharing mechanisms that split the cost of an edge in a fair manner among the agents using the edge. It is well known that the price of anarchy of these games is as high as $n$. Another line of research has focused on evaluating the price of stability, i.e., the cost of the best Nash equilibrium relative to the social optimum. In this paper we investigate to which extent coordination among agents can improve the quality of solutions. We resort to the concept of strong Nash equilibria, which were introduced by Aumann and are resilient to deviations by coalitions of agents. We analyze the price of anarchy of strong Nash equilibria and develop lower and upper bounds for unweighted and weighted games in both directed and undirected graphs. These bounds are tight or nearly tight for many scenarios. It shows that, by using coordination, the price of anarchy drops from linear to logarithmic bounds. We complement these results by also proving the first superconstant lower bound on the price of stability of standard equilibria (without coordination) in undirected graphs. More specifically, we show a lower bound of $\Omega(\log W/\log\log W)$ for weighted games, where $W$ is the total weight of all the agents. This almost matches the known upper bound of $O(\log W)$. Our results imply that, for most settings, the worst-case performance ratios of strong coordinated equilibria are essentially always as good as the performance ratios of the best equilibria achievable without coordination. These settings include unweighted games in directed graphs as well as weighted games in both directed and undirected graphs.

Metric Embeddings with Relaxed Guarantees

T.-H. Hubert Chan, Kedar Dhamdhere, Anupam Gupta, Jon Kleinberg, and Aleksandrs Slivkins

SIAM J. Comput. 38, pp. 2303-2329 (27 pages)

Online Publication Date: March 04, 2009

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We consider the problem of embedding finite metrics with slack: We seek to produce embeddings with small dimension and distortion while allowing a (small) constant fraction of all distances to be arbitrarily distorted. This definition is motivated by recent research in the networking community, which achieved striking empirical success at embedding Internet latencies with low distortion into low-dimensional Euclidean space, provided that some small slack is allowed. Answering an open question of Kleinberg, Slivkins, and Wexler [in Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, 2004], we show that provable guarantees of this type can in fact be achieved in general: Any finite metric space can be embedded, with constant slack and constant distortion, into constant-dimensional Euclidean space. We then show that there exist stronger embeddings into $\ell_1$ which exhibit gracefully degrading distortion: There is a single embedding into $\ell_1$ that achieves distortion at most $O(\log\frac{1}{\epsilon})$ on all but at most-1.5pt an $\epsilon$ fraction of distances simultaneously for all $\epsilon>0$. We extend this with distortion1pt $O(\log\frac{1}{\epsilon})^{1/p}$ to maps into general $\ell_p$, $p\geq1$, for several classes of metrics, including those with bounded doubling dimension and those arising from the shortest-path metric of a graph with an excluded minor. Finally, we show that many of our constructions are tight and give a general technique to obtain lower bounds for $\epsilon$-slack embeddings from lower bounds for low-distortion embeddings.

The Connectivity of Boolean Satisfiability: Computational and Structural Dichotomies

Parikshit Gopalan, Phokion G. Kolaitis, Elitza Maneva, and Christos H. Papadimitriou

SIAM J. Comput. 38, pp. 2330-2355 (26 pages)

Online Publication Date: March 04, 2009

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Boolean satisfiability problems are an important benchmark for questions about complexity, algorithms, heuristics, and threshold phenomena. Recent work on heuristics and the satisfiability threshold has centered around the structure and connectivity of the solution space. Motivated by this work, we study structural and connectivity-related properties of the space of solutions of Boolean satisfiability problems and establish various dichotomies in Schaefer's framework. On the structural side, we obtain dichotomies for the kinds of subgraphs of the hypercube that can be induced by the solutions of Boolean formulas, as well as for the diameter of the connected components of the solution space. On the computational side, we establish dichotomy theorems for the complexity of the connectivity and $st$-connectivity questions for the graph of solutions of Boolean formulas. Our results assert that the intractable side of the computational dichotomies is PSPACE-complete, while the tractable side—which includes but is not limited to all problems with polynomial-time algorithms for satisfiability—is in P for the $st$-connectivity question, and in coNP for the connectivity question. The diameter of components can be exponential for the PSPACE-complete cases, whereas in all other cases it is linear; thus, diameter and complexity of the connectivity problems are remarkably aligned. The crux of our results is an expressibility theorem showing that in the tractable cases, the subgraphs induced by the solution space possess certain good structural properties, whereas in the intractable cases, the subgraphs can be arbitrary.

The Undecidability of the Infinite Ribbon Problem: Implications for Computing by Self-Assembly

Leonard Adleman, Jarkko Kari, Lila Kari, Dustin Reishus, and Petr Sosik

SIAM J. Comput. 38, pp. 2356-2381 (26 pages)

Online Publication Date: March 20, 2009

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Self-assembly, the process by which objects autonomously come together to form complex structures, is omnipresent in the physical world. Recent experiments in self-assembly demonstrate its potential for the parallel creation of a large number of nanostructures, including possibly computers. A systematic study of self-assembly as a mathematical process has been initiated by L. Adleman and E. Winfree. The individual components are modeled as square tiles on the infinite two-dimensional plane. Each side of a tile is covered by a specific “glue,” and two adjacent tiles will stick iff they have matching glues on their abutting edges. Tiles that stick to each other may form various two-dimensional “structures” such as squares and rectangles, or may cover the entire plane. In this paper we focus on a special type of structure, called a ribbon: a non-self-crossing rectilinear sequence of tiles on the plane, in which successive tiles are adjacent along an edge and abutting edges of consecutive tiles have matching glues. We prove that it is undecidable whether an arbitrary finite set of tiles with glues (infinite supply of each tile type available) can be used to assemble an infinite ribbon. While the problem can be proved undecidable using existing techniques if the ribbon is required to start with a given “seed” tile, our result settles the “unseeded” case, an open problem formerly known as the “unlimited infinite snake problem.” The proof is based on a construction, due to R. Robinson, of a special set of tiles that allow only aperiodic tilings of the plane. This construction is used to create a special set of directed tiles (tiles with arrows painted on the top) with the “strong plane-filling property”—a variation of the “plane-filling property” previously defined by J. Kari. A construction of “sandwich” tiles is then used in conjunction with this special tile set, to reduce the well-known undecidable tiling problem to the problem of the existence of an infinite directed zipper (a special kind of ribbon). A “motif” construction is then introduced that allows one tile system to simulate another by using geometry to represent glues. Using motifs, the infinite directed zipper problem is reduced to the infinite ribbon problem, proving the latter undecidable. An immediate consequence of our result is the undecidability of the existence of arbitrarily large structures self-assembled using tiles from a given tile set.

Dynamic Programming Optimization over Random Data: The Scaling Exponent for Near-Optimal Solutions

David J. Aldous, Charles Bordenave, and Marc Lelarge

SIAM J. Comput. 38, pp. 2382-2410 (29 pages)

Online Publication Date: March 20, 2009

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A very simple example of an algorithmic problem solvable by dynamic programming is to maximize, over $A\subseteq\{1,2,\ldots,n\}$, the objective function $|A|-\sum_i\xi_i{\rm1\hspace{-0.90ex}1}(i\in A,i+1\in A)$ for given $\xi_i>0$. This problem, with random $(\xi_i)$, provides a test example for studying the relationship between optimal and near-optimal solutions of combinatorial optimization problems. We show that, amongst solutions differing from the optimal solution in a small proportion $\delta$ of places, we can find near-optimal solutions whose objective function value differs from the optimum by a factor of order $\delta^2$ but not of smaller order. We conjecture this relationship holds widely in the context of dynamic programming over random data, and Monte Carlo simulations for the Kauffman–Levin NK model are consistent with the conjecture. This work is a technical contribution to a broad program initiated in [D. J. Aldous and A. G. Percus, Proc. Natl. Acad. Sci. USA, 100 (2003), pp. 11211–11215] of relating such scaling exponents to the algorithmic difficulty of optimization problems.

Random Hyperplane Search Trees

Luc Devroye, James King, and Colin McDiarmid

SIAM J. Comput. 38, pp. 2411-2425 (15 pages)

Online Publication Date: March 27, 2009

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A hyperplane search tree is a binary tree used to store a set $S$ of $n$ $d$-dimensional data points. In a random hyperplane search tree for $S$, the root represents a hyperplane defined by $d$ data points drawn uniformly at random from $S$. The remaining data points are split by the hyperplane, and the definition is used recursively on each subset. We assume that the data are points in general position in $\mathbb{R}^d$. We show that, uniformly over all such data sets $S$, the expected height of the hyperplane tree is not worse than that of the $k$-d tree or the ordinary one-dimensional random binary search tree, and that, for any fixed $d\ge3$, the expected height improves over that of the standard random binary search tree by an asymptotic factor strictly greater than one.

A Constant Factor Approximation for the Single Sink Edge Installation Problem

Sudipto Guha, Adam Meyerson, and Kamesh Munagala

SIAM J. Comput. 38, pp. 2426-2442 (17 pages) | Cited 1 time

Online Publication Date: March 27, 2009

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We present the first constant approximation to the single sink buy-at-bulk network design problem, where we have to design a network by buying pipes of different costs and capacities per unit length to route demands at a set of sources to a single sink. The distances in the underlying network form a metric. This result improves the previous bound of $O(\log|R|)$, where $R$ is the set of sources. We also present a better constant approximation to the related Access Network Design problem. Our algorithms are randomized and combinatorial. As a subroutine in our algorithm, we use an interesting variant of facility location with lower bounds on the amount of demand an open facility needs to serve. We call this variant load balanced facility location and present a constant factor approximation for it, while relaxing the lower bounds by a constant factor.

A 2EXPTIME Complete Varietal Membership Problem

Marcin Kozik

SIAM J. Comput. 38, pp. 2443-2467 (25 pages)

Online Publication Date: April 01, 2009

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We construct a finite algebra generating a variety with 2EXPTIME complete membership problem. This proves that the universal membership problem for varieties and the varietal equivalence problem are 2EXPTIME complete as well, answering the question of Bergman and Slutzki from 2000.

Stateless Distributed Gradient Descent for Positive Linear Programs

Baruch Awerbuch and Rohit Khandekar

SIAM J. Comput. 38, pp. 2468-2486 (19 pages)

Online Publication Date: April 01, 2009

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We develop a framework of distributed and stateless solutions for packing and covering linear programs (LPs), which are solved by multiple agents operating in a cooperative but uncoordinated manner. Our model has a separate “agent” controlling each variable, and an agent is allowed to read off the current values only of those constraints in which it has nonzero coefficients. This is a natural model for many distributed applications like flow control, maximum bipartite matching, and dominating sets. The most appealing features of our algorithms are their simplicity and polylogarithmic convergence. For the packing LP $\max\{c\cdot x\mid Ax\leq b,$ $x\geq0\}$, the algorithm associates a dual variable $y_i=\exp[\frac{1}{\epsilon}(\frac{A_ix}{b_i}-1)]$ for each constraint $i$, and each agent $j$ iteratively increases (resp., decreases) $x_j$ multiplicatively if $A_j^\top y$ is too small (resp., large) as compared to $c_j$. Our algorithm, starting from a feasible solution, always maintains feasibility and computes a $(1+\epsilon)$ approximation in $\mathrm{poly}(\frac{\ln(mn\cdot A_{\max})}{\epsilon})$ rounds. Here $m$ and $n$ are number of rows and columns of $A$, and $A_{\max}$, also known as the “width” of the LP, is the ratio of the maximum and the minimum nonzero values taken by the expression $A_{ij}/(b_ic_j)$ as the pair $i,j$ varies over the matrix. A similar algorithm works for the covering LP $\min\{b\cdot y\mid A^\top y\geq c,$ $y\geq0\}$ as well. While exponential dual variables have been used in several packing/covering linear programming (LP) algorithms before [S. Plotkin, D. Shmoys, and E. Tardos, Math. Oper. Res., 20 (1995), pp. 257–301; Y. Bartal, J. W. Byers, and D. Raz, Proceedings of the IEEE Symposium on Foundations of Computer Science, 1997; N. Garg and J. Könemann, SIAM J. Comput., 37 (2007), pp. 630–652; L. K. Fleischer, SIAM J. Discrete Math., 13 (2000), pp. 505–520; N. E. Young, Proceedings of the IEEE Symposium on Foundations of Computer Science, 2001; C. Koufogiannakis and N. E. Young, Proceedings of the IEEE Symposium on Foundations of Computer Science, 2007], this is the first algorithm which is both stateless and has polylogarithmic convergence. Our algorithms can be thought of as applying distributed gradient descent/ascent on a carefully chosen potential. Our analysis differs from those of previous multiplicative update based algorithms and argues that while the current solution is far away from optimality, the potential function decreases/increases by a significant factor.

Improved Lower Bounds for Embeddings into $L_1$

Robert Krauthgamer and Yuval Rabani

SIAM J. Comput. 38, pp. 2487-2498 (12 pages)

Online Publication Date: April 10, 2009

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We improve upon recent lower bounds on the minimum distortion of embedding certain finite metric spaces into $L_1$. In particular, we show that for every $n\ge1$, there is an $n$-point metric space of negative type that requires a distortion of $\Omega(\log\log n)$ for such an embedding, implying the same lower bound on the integrality gap of a well-known semidefinite programming relaxation for sparsest cut. This result builds upon and improves the recent lower bound of $(\log\log n)^{1/6-o(1)}$ due to Khot and Vishnoi [The unique games conjecture, integrality gap for cut problems and the embeddability of negative type metrics into $l_1$, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 2005, pp. 53–62]. We also show that embedding the edit distance metric on $\{0,1\}^n$ into $L_1$ requires a distortion of $\Omega(\log n)$. This result improves a very recent $(\log n)^{1/2-o(1)}$ lower bound by Khot and Naor [Nonembeddability theorems via Fourier analysis, in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 2005, pp. 101–112].

Testing Hereditary Properties of Nonexpanding Bounded-Degree Graphs

Artur Czumaj, Asaf Shapira, and Christian Sohler

SIAM J. Comput. 38, pp. 2499-2510 (12 pages) | Cited 2 times

Online Publication Date: April 10, 2009

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We study graph properties that are testable for bounded-degree graphs in time independent of the input size. Our goal is to distinguish between graphs having a predetermined graph property and graphs that are far from every graph having that property. It is well known that in the bounded-degree graph model (where two graphs are considered “far” if they differ in $\varepsilon n$ edges for a positive constant $\varepsilon$), many graph properties cannot be tested even with a constant or even with a polylogarithmic number of queries. Therefore in this paper we focus our attention on testing graph properties for special classes of graphs. Specifically, we show that every hereditary graph property is testable with a constant number of queries provided that every sufficiently large induced subgraph of the input graph has poor expansion. This result implies that, for example, any hereditary property (e.g., $k$-colorability, $H$-freeness, etc.) is testable in the bounded-degree graph model for planar graphs, graphs with bounded genus, interval graphs, etc. No such results have been known before, and prior to our work, very few graph properties have been known to be testable with a constant number of queries for general graph classes in the bounded-degree graph model.

Size-Space Tradeoffs for Resolution

Eli Ben-Sasson

SIAM J. Comput. 38, pp. 2511-2525 (15 pages) | Cited 1 time

Online Publication Date: May 06, 2009

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We investigate tradeoffs of various basic complexity measures such as size, space, and width. We show examples of formulas that have optimal proofs with respect to any one of these parameters, but optimizing one parameter must cost an increase in the other. These results have implications to the efficiency (or rather, inefficiency) of some commonly used SAT solving heuristics. Our proof relies on a novel connection of the variable space of a proof to the black-white pebbling measure of an underlying graph.

Randomized Divide-and-Conquer: Improved Path, Matching, and Packing Algorithms

Jianer Chen, Joachim Kneis, Songjian Lu, Daniel Mölle, Stefan Richter, Peter Rossmanith, Sing-Hoi Sze, and Fenghui Zhang

SIAM J. Comput. 38, pp. 2526-2547 (22 pages)

Online Publication Date: May 13, 2009

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We propose a randomized divide-and-conquer technique that leads to improved randomized and deterministic algorithms for NP-hard path, matching, and packing problems. For the parameterized max-path problem, our randomized algorithm runs in time $O(4^{k}k^{2.7}m)$ and polynomial space (where $m$ is the number of edges in the input graph), improving the previous best randomized algorithm for the problem that runs in time $O(5.44^{k}km)$ and exponential space. Our randomized algorithms for the parameterized max $r$-d matching and max $r$-set packing problems run in time $4^{(r-1)k}n^{O(1)}$ and polynomial space, improving the previous best algorithms for the problems that run in time $10.88^{rk}n^{O(1)}$ and exponential space. Moreover, our randomized algorithms can be derandomized to result in significantly improved deterministic algorithms for the problems, and they can be extended to solve other matching and packing problems.
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