Improved Approximation Algorithms for Label Cover Problems

In this paper we consider both the maximization variant Max Rep and the minimization variant Min Rep of the famous Label Cover problem. So far the best approximation ratios known for these two problems were \(O(\sqrt{n})\) and indeed some authors suggested the possibility that this ratio is the best approximation factor for these two problems. We show, in fact, that there are a O(n ^1/3)-approximation algorithm for Max Rep and a O(n ^1/3log?^2/3 n)-approximation algorithm for Min Rep. In addition, we also exhibit a randomized reduction from Densest k-Subgraph to Max Rep, showing that any approximation factor for Max Rep implies the same factor (up to a constant) for Densest k-Subgraph.     

Vertex Cover Kernelization Revisited

An important result in the study of polynomial-time preprocessing shows that there is an algorithm which given an instance (G,k) of Vertex Cover outputs an equivalent instance (G′,k′) in polynomial time with the guarantee that G′ has at most 2k′ vertices (and thus $\mathcal{O}((k')^{2})$ edges) with k′≤k. Using the terminology of parameterized complexity we say that k-Vertex Cover has a kernel with 2k vertices. There is complexity-theoretic evidence that both 2k vertices and Θ(k ²) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)$ of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number $\mathop{\mathrm{\mbox {\textsc{vc}}}}(G)$ since $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)\leq\mathop{\mathrm{\mbox{\textsc{vc}}}}(G)$ and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)$ : an instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into an equivalent instance (G′,X′,k′) such that |V(G′)|≤2k and $|V(G')| \in\mathcal{O}(|X'|^{3})$ . A similar result holds when the feedback vertex set X is not given along with the input. In sharp contrast we show that the Weighted Vertex Cover problem does not have a polynomial kernel when parameterized by the cardinality of a given vertex cover of the graph unless NP ? coNP/poly and the polynomial hierarchy collapses to the third level.     

Faster Parameterized Algorithms for <Emphasis Type="SmallCaps">Minimum Fill-in</Emphasis>

We present two parameterized algorithms for the Minimum Fill-in problem, also known as Chordal Completion: given an arbitrary graph G and integer k, can we add at most k edges to G to obtain a chordal graph? Our first algorithm has running time \(\mathcal {O}(k^{2}nm+3.0793^{k})\), and requires polynomial space. This improves the base of the exponential part of the best known parameterized algorithm time for this problem so far. We are able to improve this running time even further, at the cost of more space. Our second algorithm has running time \(\mathcal {O}(k^{2}nm+2.35965^{k})\) and requires \(\mathcal {O}^{\ast}(1.7549^{k})\) space. To achieve these results, we present a new lemma describing the edges that can safely be added to achieve a chordal completion with the minimum number of edges, regardless of k.     

Parameterized Measure & Conquer for Problems with No Small Kernels

Measure & Conquer (M&C) is a prominent technique for analyzing exact algorithms for computationally hard problems, in particular, graph problems. It tries to balance worse and better situations within the algorithm analysis. This has led, e.g., to algorithms for Minimum Vertex Cover with a running time of $\mathcal{O}(c^{n})$ for some constant c??1.2, where n is the number of vertices in the graph. Several obstacles prevent the application of this technique in parameterized algorithmics, making it rarely applied in this area. However, these difficulties can be handled in some situations. We will exemplify this with two problems related to Vertex Cover, namely Connected Vertex Cover and Edge Dominating Set. For both problems, several parameterized algorithms have been published, all based on the idea of first enumerating minimal vertex covers. Using M&C in this context will allow us to improve on the hitherto published running times. In contrast to some of the earlier suggested algorithms, ours will use polynomial space.     

Ambiguity and Communication

The ambiguity of a nondeterministic finite automaton (NFA) N for input size n is the maximal number of accepting computations of N for inputs of size n. For every natural number k we construct a family \((L_{r}^{k}\;|\;r\in \mathbb{N})\) of languages which can be recognized by NFA’s with size k?poly(r) and ambiguity O(n ^k ), but \(L_{r}^{k}\) has only NFA’s with size exponential in r, if ambiguity o(n ^k ) is required. In particular, a hierarchy for polynomial ambiguity is obtained, solving a long standing open problem (Ravikumar and Ibarra, SIAM J. Comput. 19:1263–1282, 1989, Leung, SIAM J. Comput. 27:1073–1082, 1998).     

Distributed computing connected components with linear communication cost

The paper studies three fundamental problems in graph analytics, computing connected components (CCs), biconnected components (BCCs), and 2-edge-connected components (ECCs) of a graph. With the recent advent of big data, developing efficient distributed algorithms for computing CCs, BCCs and ECCs of a big graph has received increasing interests. As with the existing research efforts, we focus on the Pregel programming model, while the techniques may be extended to other programming models including MapReduce and Spark. The state-of-the-art techniques for computing CCs and BCCs in Pregel incur \(O(m\times \#\text {supersteps})\) total costs for both data communication and computation, where m is the number of edges in a graph and #supersteps is the number of supersteps. Since the network communication speed is usually much slower than the computation speed, communication costs are the dominant costs of the total running time in the existing techniques. In this paper, we propose a new paradigm based on graph decomposition to compute CCs and BCCs with O(m) total communication cost. The total computation costs of our techniques are also smaller than that of the existing techniques in practice, though theoretically almost the same. Moreover, we also study distributed computing ECCs. We are the first to study this problem and an approach with O(m) total communication cost is proposed. Comprehensive empirical studies demonstrate that our approaches can outperform the existing techniques by one order of magnitude regarding the total running time.     

Approximating the Sparsest k-Subgraph in Chordal Graphs

Given a simple undirected graph G = (V, E) and an integer k < |V|, the Sparsest k-Subgraph problem asks for a set of k vertices which induces the minimum number of edges. As a generalization of the classical independent set problem, Sparsest k-Subgraph is ????-hard and even not approximable unless ?????? in general graphs. Thus, we investigate Sparsest k-Subgraph in graph classes where independent set is polynomial-time solvable, such as subclasses of perfect graphs. Our two main results are the ????-hardness of Sparsest k-Subgraph on chordal graphs, and a greedy 2-approximation algorithm. Finally, we also show how to derive a P T A S for Sparsest k-Subgraph on proper interval graphs.     

Parameterized Domination in Circle Graphs

A circle graph is the intersection graph of a set of chords in a circle. Keil [Discrete Appl. Math., 42(1):51–63, 1993] proved that Dominating Set, Connected Dominating Set, and Total Dominating Set are NP-complete in circle graphs. To the best of our knowledge, nothing was known about the parameterized complexity of these problems in circle graphs. In this paper we prove the following results, which contribute in this direction:

• Dominating Set, Independent Dominating Set, Connected Dominating Set, Total Dominating Set, and Acyclic Dominating Set are W[1]-hard in circle graphs, parameterized by the size of the solution.
• Whereas both Connected Dominating Set and Acyclic Dominating Set are W[1]-hard in circle graphs, it turns out that Connected Acyclic Dominating Set is polynomial-time solvable in circle graphs.
• If T is a given tree, deciding whether a circle graph G has a dominating set inducing a graph isomorphic to T is NP-complete when T is in the input, and FPT when parameterized by t=|V(T)|. We prove that the FPT algorithm runs in subexponential time, namely $2^{\mathcal{O}(t \cdot\frac{\log\log t}{\log t})} \cdot n^{\mathcal{O}(1)}$ , where n=|V(G)|.

     

Incompressible Functions,Relative-Error Extractors,and the Power of Nondeterministic Reductions

A circuit C compresses a function \({f : \{0,1\}^n\rightarrow \{0,1\}^m}\) if given an input \({x\in \{0,1\}^n}\), the circuit C can shrink x to a shorter ?-bit string x′ such that later, a computationally unbounded solver D will be able to compute f(x) based on x′. In this paper we study the existence of functions which are incompressible by circuits of some fixed polynomial size \({s=n^c}\). Motivated by cryptographic applications, we focus on average-case \({(\ell,\epsilon)}\) incompressibility, which guarantees that on a random input \({x\in \{0,1\}^n}\), for every size s circuit \({C:\{0,1\}^n\rightarrow \{0,1\}^{\ell}}\) and any unbounded solver D, the success probability \({\Pr_x[D(C(x))=f(x)]}\) is upper-bounded by \({2^{-m}+\epsilon}\). While this notion of incompressibility appeared in several works (e.g., Dubrov and Ishai, STOC 06), so far no explicit constructions of efficiently computable incompressible functions were known. In this work, we present the following results:

1. (1)
   Assuming that E is hard for exponential size nondeterministic circuits, we construct a polynomial time computable boolean function \({f:\{0,1\}^n\rightarrow \{0,1\}}\) which is incompressible by size n ^c circuits with communication \({\ell=(1-o(1)) \cdot n}\) and error \({\epsilon=n^{-c}}\). Our technique generalizes to the case of PRGs against nonboolean circuits, improving and simplifying the previous construction of Shaltiel and Artemenko (STOC 14).
    
2. (2)
   We show that it is possible to achieve negligible error parameter \({\epsilon=n^{-\omega(1)}}\) for nonboolean functions. Specifically, assuming that E is hard for exponential size \({\Sigma_3}\)-circuits, we construct a nonboolean function \({f:\{0,1\}^n\rightarrow \{0,1\}^m}\) which is incompressible by size n ^c circuits with \({\ell=\Omega(n)}\) and extremely small \({\epsilon=n^{-c} \cdot 2^{-m}}\). Our construction combines the techniques of Trevisan and Vadhan (FOCS 00) with a new notion of relative error deterministic extractor which may be of independent interest.
    
3. (3)
   We show that the task of constructing an incompressible boolean function \({f:\{0,1\}^n\rightarrow \{0,1\}}\) with negligible error parameter \({\epsilon}\) cannot be achieved by “existing proof techniques”. Namely, nondeterministic reductions (or even \({\Sigma_i}\) reductions) cannot get \({\epsilon=n^{-\omega(1)}}\) for boolean incompressible functions. Our results also apply to constructions of standard Nisan-Wigderson type PRGs and (standard) boolean functions that are hard on average, explaining, in retrospect, the limitations of existing constructions. Our impossibility result builds on an approach of Shaltiel and Viola (STOC 08).
    

     

Polylogarithms and the Asymptotic Formula for the Moments of Lebesgue’s Singular Function

Recall that Lebesgue’s singular function L(t) is defined as the unique solution to the equation L(t) = qL(2t) + pL(2t ? 1), where p, q > 0, q = 1 ? p, p ≠ q. The variables M _n = ∫₀¹t ⁿ dL(t), n = 0,1,… are called the moments of the function The principal result of this work is \({M_n} = {n^{{{\log }_2}p}}{e^{ - \tau (n)}}(1 + O({n^{ - 0.99}}))\), where the function τ(x) is periodic in log₂x with the period 1 and is given as \(\tau (x) = \frac{1}{2}1np + \Gamma '(1)lo{g_2}p + \frac{1}{{1n2}}\frac{\partial }{{\partial z}}L{i_z}( - \frac{q}{p}){|_{z = 1}} + \frac{1}{{1n2}}\sum\nolimits_{k \ne 0} {\Gamma ({z_k})L{i_{{z_k} + 1}}( - \frac{q}{p})} {x^{ - {z_k}}}\), \({z_k} = \frac{{2\pi ik}}{{1n2}}\), k ≠ 0. The proof is based on poissonization and the Mellin transform.     

Entanglement witness criteria of strong-<Emphasis Type="Italic">k</Emphasis>-separability for multipartite quantum states

Let \(H_{1}, H_{2},\ldots ,H_{n}\) be separable complex Hilbert spaces with \(\dim H_{i}\ge 2\) and \(n\ge 2\). Assume that \(\rho \) is a state in \(H=H_1\otimes H_2\otimes \cdots \otimes H_n\). \(\rho \) is called strong-k-separable \((2\le k\le n)\) if \(\rho \) is separable for any k-partite division of H. In this paper, an entanglement witnesses criterion of strong-k-separability is obtained, which says that \(\rho \) is not strong-k-separable if and only if there exist a k-division space \(H_{m_{1}}\otimes \cdots \otimes H_{m_{k}}\) of H, a finite-rank linear elementary operator positive on product states \(\Lambda :\mathcal {B}(H_{m_{2}}\otimes \cdots \otimes H_{m_{k}})\rightarrow \mathcal {B}(H_{m_{1}})\) and a state \(\rho _{0}\in \mathcal {S}(H_{m_{1}}\otimes H_{m_{1}})\), such that \(\mathrm {Tr}(W\rho )<0\), where \(W=(\mathrm{Id}\otimes \Lambda ^{\dagger })\rho _{0}\) is an entanglement witness. In addition, several different methods of constructing entanglement witnesses for multipartite states are also given.     

Patterns and anomalies in <Emphasis Type="Italic">k</Emphasis>-cores of real-world graphs with applications

How do the k-core structures of real-world graphs look like? What are the common patterns and the anomalies? How can we exploit them for applications? A k-core is the maximal subgraph in which all vertices have degree at least k. This concept has been applied to such diverse areas as hierarchical structure analysis, graph visualization, and graph clustering. Here, we explore pervasive patterns related to k-cores and emerging in graphs from diverse domains. Our discoveries are: (1) Mirror Pattern: coreness (i.e., maximum k such that each vertex belongs to the k-core) is strongly correlated with degree. (2) Core-Triangle Pattern: degeneracy (i.e., maximum k such that the k-core exists) obeys a 3-to-1 power-law with respect to the count of triangles. (3) Structured Core Pattern: degeneracy–cores are not cliques but have non-trivial structures such as core–periphery and communities. Our algorithmic contributions show the usefulness of these patterns. (1) Core-A, which measures the deviation from Mirror Pattern, successfully spots anomalies in real-world graphs, (2) Core-D, a single-pass streaming algorithm based on Core-Triangle Pattern, accurately estimates degeneracy up to 12 \(\times \) faster than its competitor. (3) Core-S, inspired by Structured Core Pattern, identifies influential spreaders up to 17 \(\times \) faster than its competitors with comparable accuracy.     

Exact Algorithms for Intervalizing Coloured Graphs

In the Intervalizing Coloured Graphs problem, one must decide for a given graph G = (V, E) with a proper vertex colouring of G whether G is the subgraph of a properly coloured interval graph. For the case that the number of colors is fixed, we give an exact algorithm that uses \(2^{\mathcal {O}(n/\log n)}\) time. We also give an \(\mathcal {O}^{\ast }(2^{n})\) algorithm for the case that the number of colors is not fixed.     

Two Mixed Finite Element Methods for Time-Fractional Diffusion Equations

Based on spatial conforming and nonconforming mixed finite element methods combined with classical L1 time stepping method, two fully-discrete approximate schemes with unconditional stability are first established for the time-fractional diffusion equation with Caputo derivative of order \(0<\alpha <1\). As to the conforming scheme, the spatial global superconvergence and temporal convergence order of \(O(h^2+\tau ^{2-\alpha })\) for both the original variable u in \(H^1\)-norm and the flux \(\vec {p}=\nabla u\) in \(L^2\)-norm are derived by virtue of properties of bilinear element and interpolation postprocessing operator, where h and \(\tau \) are the step sizes in space and time, respectively. At the same time, the optimal convergence rates in time and space for the nonconforming scheme are also investigated by some special characters of \(\textit{EQ}_1^{\textit{rot}}\) nonconforming element, which manifests that convergence orders of \(O(h+\tau ^{2-\alpha })\) and \(O(h^2+\tau ^{2-\alpha })\) for the original variable u in broken \(H^1\)-norm and \(L^2\)-norm, respectively, and approximation for the flux \(\vec {p}\) converging with order \(O(h+\tau ^{2-\alpha })\) in \(L^2\)-norm. Numerical examples are provided to demonstrate the theoretical analysis.     

Subexponential Size Hitting Sets for Bounded Depth Multilinear Formulas

In this paper, we give subexponential size hitting sets for bounded depth multilinear arithmetic formulas. Using the known relation between black-box PIT and lower bounds, we obtain lower bounds for these models.For depth-3 multilinear formulas, of size exp\({(n^\delta)}\), we give a hitting set of size exp\({\left(\tilde{O}\left(n^{2/3 + 2\delta/3}\right) \right)}\). This implies a lower bound of exp\({(\tilde{\Omega}(n^{1/2}))}\) for depth-3 multilinear formulas, for some explicit polynomial.For depth-4 multilinear formulas, of size exp\({(n^\delta)}\), we give a hitting set of size exp\({\left(\tilde{O}\left(n^{2/3 + 4\delta/3}\right) \right)}\). This implies a lower bound of exp\({(\tilde{\Omega}(n^{1/4}))}\) for depth-4 multilinear formulas, for some explicit polynomial.A regular formula consists of alternating layers of \({+,\times}\) gates, where all gates at layer i have the same fan-in. We give a hitting set of size (roughly) exp\({\left(n^{1- \delta}\right)}\), for regular depth-d multilinear formulas with formal degree at most n and size exp\({(n^\delta)}\), where \({\delta = O(1/{\sqrt{5}^d})}\). This result implies a lower bound of roughly exp\({(\tilde{\Omega}(n^{1/{\sqrt{5}^d}}))}\) for such formulas.We note that better lower bounds are known for these models, but also that none of these bounds was achieved via construction of a hitting set. Moreover, no lower bound that implies such PIT results, even in the white-box model, is currently known.Our results are combinatorial in nature and rely on reducing the underlying formula, first to a depth-4 formula, and then to a read-once algebraic branching program (from depth-3 formulas, we go straight to read-once algebraic branching programs).     

Fast arithmetic in algorithmic self-assembly

In this paper we consider the time complexity of adding two n-bit numbers together within the tile self-assembly model. The (abstract) tile assembly model is a mathematical model of self-assembly in which system components are square tiles with different glue types assigned to tile edges. Assembly is driven by the attachment of singleton tiles to a growing seed assembly when the net force of glue attraction for a tile exceeds some fixed threshold. Within this frame work, we examine the time complexity of computing the sum of two n-bit numbers, where the input numbers are encoded in an initial seed assembly, and the output sum is encoded in the final, terminal assembly of the system. We show that this problem, along with multiplication, has a worst case lower bound of \(\varOmega ( \sqrt{n} )\) in 2D assembly, and \(\varOmega (\root 3 \of {n})\) in 3D assembly. We further design algorithms for both 2D and 3D that meet this bound with worst case run times of \(O(\sqrt{n})\) and \(O(\root 3 \of {n})\) respectively, which beats the previous best known upper bound of O(n). Finally, we consider average case complexity of addition over uniformly distributed n-bit strings and show how we can achieve \(O(\log n)\) average case time with a simultaneous \(O(\sqrt{n})\) worst case run time in 2D. As additional evidence for the speed of our algorithms, we implement our algorithms, along with the simpler O(n) time algorithm, into a probabilistic run-time simulator and compare the timing results.     

Close to linear space routing schemes

Let \(G=(V,E)\) be an unweighted undirected graph with n vertices and m edges, and let \(k>2\) be an integer. We present a routing scheme with a poly-logarithmic header size, that given a source s and a destination t at distance \(\varDelta \) from s, routes a message from s to t on a path whose length is \(O(k\varDelta +m^{1/k})\). The total space used by our routing scheme is \(mn^{O(1/\sqrt{\log n})}\), which is almost linear in the number of edges of the graph. We present also a routing scheme with \(n^{O(1/\sqrt{\log n})}\) header size, and the same stretch (up to constant factors). In this routing scheme, the routing table of every \(v\in V\) is at most \(kn^{O(1/\sqrt{\log n})}deg(v)\), where deg(v) is the degree of v in G. Our results are obtained by combining a general technique of Bernstein (2009), that was presented in the context of dynamic graph algorithms, with several new ideas and observations.     

Non-determinism in Gödel’s System <Emphasis Type="Italic">T</Emphasis>

The calculus T^? is a successor-free version of Gödel’s T. It is well known that a number of important complexity classes, like e.g. the classes logspace, \(\textsc{p}\), \(\textsc{linspace}\), \(\textsc{etime}\) and \(\textsc{pspace}\), are captured by natural fragments of T^? and related calculi. We introduce the calculus T^∽, which is a non-deterministic variant of T^?, and compare the computational power of T^∽ and T^?. First, we provide a denotational semantics for T^∽ and prove this semantics to be adequate. Furthermore, we prove that \(\textsc{linspace}\subseteq \mathcal {G}^{\backsim }_{0} \subseteq \textsc{linspace}\) and \(\textsc{etime}\subseteq \mathcal {G}^{\backsim }_{1} \subseteq \textsc{pspace}\) where \(\mathcal {G}^{\backsim }_{0}\) and \(\mathcal {G}^{\backsim }_{1}\) are classes of problems decidable by certain fragments of T^∽. (It is proved elsewhere that the corresponding fragments of T^? equal respectively \(\textsc{linspace}\) and \(\textsc{etime}\).) Finally, we show a way to interpret T^∽ in T^?.     

Exact and approximate flexible aggregate similarity search

Aggregate similarity search, also known as aggregate nearest-neighbor (Ann) query, finds many useful applications in spatial and multimedia databases. Given a group Q of M query objects, it retrieves from a database the objects most similar to Q, where the similarity is an aggregation (e.g., \({{\mathrm{sum}}}\), \(\max \)) of the distances between each retrieved object p and all the objects in Q. In this paper, we propose an added flexibility to the query definition, where the similarity is an aggregation over the distances between p and any subset of \(\phi M\) objects in Q for some support \(0< \phi \le 1\). We call this new definition flexible aggregate similarity search and accordingly refer to a query as a flexible aggregate nearest-neighbor ( Fann ) query. We present algorithms for answering Fann queries exactly and approximately. Our approximation algorithms are especially appealing, which are simple, highly efficient, and work well in both low and high dimensions. They also return near-optimal answers with guaranteed constant-factor approximations in any dimensions. Extensive experiments on large real and synthetic datasets from 2 to 74 dimensions have demonstrated their superior efficiency and high quality.