Planar Feedback Vertex Set and Face Cover: Combinatorial Bounds and Subexponential Algorithms

The Planar Feedback Vertex Set problem asks whether an n-vertex planar graph contains at most k vertices meeting all its cycles. The Face Cover problem asks whether all vertices of a plane graph G lie on the boundary of at most k faces of G. Standard techniques from parameterized algorithm design indicate that both problems can be solved by sub-exponential parameterized algorithms (where k is the parameter). In this paper we improve the algorithmic analysis of both problems by proving a series of combinatorial results relating the branchwidth of planar graphs with their face cover. Combining this fact with duality properties of branchwidth, allows us to derive analogous results on feedback vertex set. As a consequence, it follows that Planar Feedback Vertex Set and Face Cover can be solved in \(O(2^{15.11\cdot\sqrt{k}}+n^{2})\) and \(O(2^{10.1\cdot\sqrt {k}}+n^{2})\) steps, respectively.     

Lower Bounds for Agnostic Learning via Approximate Rank

We prove that the concept class of disjunctions cannot be pointwise approximated by linear combinations of any small set of arbitrary real-valued functions. That is, suppose that there exist functions f₁, ?, f_r\phi_{1}, \ldots , \phi_{r} : {− 1, 1}ⁿ → \mathbbR{\mathbb{R}} with the property that every disjunction f on n variables has $\|f - \sum\nolimits_{i=1}^{r} \alpha_{i}\phi _{i}\|_{\infty}\leq 1/3$\|f - \sum\nolimits_{i=1}^{r} \alpha_{i}\phi _{i}\|_{\infty}\leq 1/3 for some reals a₁, ?, a_r\alpha_{1}, \ldots , \alpha_{r}. We prove that then $r \geq exp \{\Omega(\sqrt{n})\}$r \geq exp \{\Omega(\sqrt{n})\}, which is tight. We prove an incomparable lower bound for the concept class of decision lists. For the concept class of majority functions, we obtain a lower bound of W(2ⁿ/n)\Omega(2^{n}/n) , which almost meets the trivial upper bound of 2ⁿ for any concept class. These lower bounds substantially strengthen and generalize the polynomial approximation lower bounds of Paturi (1992) and show that the regression-based agnostic learning algorithm of Kalai et al. (2005) is optimal.     

Collective Tree Spanners in Graphs with Bounded Parameters

In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G=(V,E) admits a system of μ collective additive tree r -spanners if there is a system $\mathcal{T}(G)In this paper we study collective additive tree spanners for special families of graphs including planar graphs, graphs with bounded genus, graphs with bounded tree-width, graphs with bounded clique-width, and graphs with bounded chordality. We say that a graph G=(V,E) admits a system of μ collective additive tree r -spanners if there is a system T(G)\mathcal{T}(G) of at most μ spanning trees of G such that for any two vertices x,y of G a spanning tree T ? T(G)T\in\mathcal{T}(G) exists such that d _T (x,y)≤d _G (x,y)+r. We describe a general method for constructing a “small” system of collective additive tree r-spanners with small values of r for “well” decomposable graphs, and as a byproduct show (among other results) that any weighted planar graph admits a system of O(?n)O(\sqrt{n}) collective additive tree 0-spanners, any weighted graph with tree-width at most k−1 admits a system of klog ₂ n collective additive tree 0-spanners, any weighted graph with clique-width at most k admits a system of klog _3/2 n collective additive tree (2w)(2\mathsf{w}) -spanners, and any weighted graph with size of largest induced cycle at most c admits a system of log ₂ n collective additive tree (2?c/2?w)(2\lfloor c/2\rfloor\mathsf{w}) -spanners and a system of 4log ₂ n collective additive tree (2(?c/3?+1)w)(2(\lfloor c/3\rfloor +1)\mathsf {w}) -spanners (here, w\mathsf{w} is the maximum edge weight in G). The latter result is refined for weighted weakly chordal graphs: any such graph admits a system of 4log ₂ n collective additive tree (2w)(2\mathsf{w}) -spanners. Furthermore, based on this collection of trees, we derive a compact and efficient routing scheme for those families of graphs.     

Sub-Constant Error Probabilistically Checkable Proof of Almost-Linear Size

We show a construction of a PCP with both sub-constant error and almost-linear size. Specifically, for some constant 0 < α < 1, we construct a PCP verifier for checking satisfiability of Boolean formulas that on input of size n uses log n+O((log n)^1-a)\log\, n+O((\log\, n)^{1-\alpha}) random bits to make 7 queries to a proof of size n·2^{O((log n)1-a)}n·2^{O((\log\, n)^{1-\alpha})}, where each query is answered by O((log n)^1-a)O((\log\, n)^{1-\alpha}) bit long string, and the verifier has perfect completeness and error 2^{-W((log n)a)}2^{-\Omega((\log\, n)^{\alpha})}.     

Subexponential algorithms for partial cover problems

Partial Cover problems are optimization versions of fundamental and well-studied problems like Vertex Cover and Dominating Set. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number k of vertices, rather than covering all edges (or vertices). In general graphs, these problems are hard for parameterized complexity classes when parameterized by k. It was recently shown by Amini et al. (2008) [1] that Partial Vertex Cover and Partial Dominating Set are fixed parameter tractable on large classes of sparse graphs, namely H-minor-free graphs, which include planar graphs and graphs of bounded genus. In particular, it was shown that on planar graphs both problems can be solved in time ²O(k)nO(1).During the last decade there has been an extensive study on parameterized subexponential algorithms. In particular, it was shown that the classical Vertex Cover and Dominating Set problems can be solved in subexponential time on H-minor-free graphs. The techniques developed to obtain subexponential algorithms for classical problems do not apply to partial cover problems. It was left as an open problem by Amini et al. (2008) [1] whether there is a subexponential algorithm for Partial Vertex Cover and Partial Dominating Set. In this paper, we answer the question affirmatively by solving both problems in time not only on planar graphs but also on much larger classes of graphs, namely, apex-minor-free graphs. Compared to previously known algorithms for these problems our algorithms are significantly faster and simpler.     

Comparing Nontriviality for E and EXP

A set A is nontrivial for the linear-exponential-time class E=DTIME(2 ^lin ) if for any k≥1 there is a set B _k ∈E such that B _k is (p-m-)reducible to A and B_k ? DTIME(2^k·n)B_{k} \not\in \mathrm{DTIME}(2^{k\cdot n}). I.e., intuitively, A is nontrivial for E if there are arbitrarily complex sets in E which can be reduced to A. Similarly, a set A is nontrivial for the polynomial-exponential-time class EXP=DTIME(2 ^poly ) if for any k≥1 there is a set [^(B)]_k ? EXP\hat{B}_{k} \in \mathrm {EXP} such that [^(B)]_k\hat{B}_{k} is reducible to A and [^(B)]_k ? DTIME(2^nk)\hat{B}_{k} \not\in \mathrm{DTIME}(2^{n^{k}}). We show that these notions are independent, namely, there are sets A ₁ and A ₂ in E such that A ₁ is nontrivial for E but trivial for EXP and A ₂ is nontrivial for EXP but trivial for E. In fact, the latter can be strengthened to show that there is a set A∈E which is weakly EXP-hard in the sense of Lutz (SIAM J. Comput. 24:1170–1189, 11) but E-trivial.     

The Gaussian Surface Area and Noise Sensitivity of Degree-d Polynomial Threshold Functions

We prove asymptotically optimal bounds on the Gaussian noise sensitivity and Gaussian surface area of degree-d polynomial threshold functions. In particular, we show that for f a degree-d polynomial threshold function that the Gaussian noise sensitivity of f with parameter e{\epsilon} is at most \fracdarcsin(?{2e-e²})p{\frac{d\arcsin\left(\sqrt{2\epsilon-\epsilon^2}\right)}{\pi}} . This bound translates into an optimal bound on the Gaussian surface area of such functions, namely that the Gaussian surface area is at most \fracd?{2p}{\frac{d}{\sqrt{2\pi}}} . Finally, we note that the later result implies bounds on the runtime of agnostic learning algorithms for polynomial threshold functions.     

Soft set theory applied to p-ideals of BCI-algebras related to fuzzy points

The notions of $(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})The notions of ([`( ? )],[`( ? )] ú[`q])(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})-fuzzy p-ideals and fuzzy p-ideals with thresholds related to soft set theory are discussed. Relations between ([`( ? )],[`( ? )] ú[`q])(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})-fuzzy ideals and ([`( ? )],[`( ? )] ú[`q])(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})-fuzzy p-ideals are investigated. Characterizations of an ([`( ? )],[`( ? )] ú[`q])(\overline{\in}, \overline{\in} \vee \overline{\hbox{q}})-fuzzy p-ideal and a fuzzy p-ideal with thresholds are displayed. Implication-based fuzzy p-ideals are discussed.     

Onn-column 0, 1-matrices with allk-projections surjective

This paper presents a new approach to construct a smalln-column 0, 1-matrix for two given integersn andk(k, such that everyk-column projection contains all 2 ^k possible row vectors, namely surjective on {0, 1} ^k . The number of the matrix's rows does not exceed . This approach has considerable advantage for smallk and practical sizes ofn. It can be applied to the test generation of VLSI circuits, the design of fault tolerant systems and other fields.     

Reoptimization of constraint satisfaction problems with approximation resistant predicates

If k = O(log n) and a predicate P is approximation resistant for the reoptimization of the Max-EkCSP-P problem, then, after inserting a truth-value into the predicate and imposing some constraint, there exists a polynomial algorithm with the approximation ratio q(P) = \frac12 - d(P) q(P) = \frac{1}{{2 - d(P)}} , where d(P) = 2 ^{- k}| P ^{- 1}(1) | d(P) = {2^{ - k}}\left| {{P^{ - 1}}(1)} \right| is a “random” threshold approximation ratio of the predicate P. The ratio q(P) is a threshold approximation ratio.     

Fixed Parameter Algorithms for DOMINATING SET and Related Problems on Planar Graphs

Abstract. We present an algorithm that constructively produces a solution to the k -DOMINATING SET problem for planar graphs in time O(c^ \sqrt k n) , where c=4^ 6\sqrt 34 . To obtain this result, we show that the treewidth of a planar graph with domination number γ (G) is O(\sqrt \rule 0pt 4pt \smash γ (G) ) , and that such a tree decomposition can be found in O(\sqrt \rule 0pt 4pt \smash γ (G) n) time. The same technique can be used to show that the k -FACE COVER problem (find a size k set of faces that cover all vertices of a given plane graph) can be solved in O(c ₁ ^ \sqrt k n) time, where c ₁ =3^ 36\sqrt 34 and k is the size of the face cover set. Similar results can be obtained in the planar case for some variants of k -DOMINATING SET, e.g., k -INDEPENDENT DOMINATING SET and k -WEIGHTED DOMINATING SET.     

Computing Diameter in the Streaming and Sliding-Window Models

We investigate the diameter problem in the streaming and sliding-window models. We show that, for a stream of nn points or a sliding window of size nn, any exact algorithm for diameter requires W(n)\Omega(n) bits of space. We present a simple e\epsilon-approximation algorithm for computing the diameter in the streaming model. Our main result is an e\epsilon-approximation algorithm that maintains the diameter in two dimensions in the sliding-window model using O((1/e^3/2) log³n(logR+loglogn + log(1/e)))O(({1}/{\epsilon^{3/2}}) \log^{3}n(\log R+\log\log n + \log ({1}/{\epsilon}))) bits of space, where RR is the maximum, over all windows, of the ratio of the diameter to the minimum non-zero distance between any two points in the window.     

Quantitative Separation Logic and Programs with Lists

This paper presents an extension of a decidable fragment of Separation Logic for singly-linked lists, defined by Berdine et al. (2004). Our main extension consists in introducing atomic formulae of the form ls ^k (x, y) describing a list segment of length k, stretching from x to y, where k is a logical variable interpreted over positive natural numbers, that may occur further inside Presburger constraints. We study the decidability of the full first-order logic combining unrestricted quantification of arithmetic and location variables. Although the full logic is found to be undecidable, validity of entailments between formulae with the quantifier prefix in the language $^* {$_\mathbbN, "_\mathbbN}^*\exists^* \{\exists_{\bf \mathbb{N}}, \forall_{\bf \mathbb{N}}\}^* is decidable. We provide here a model theoretic method, based on a parametric notion of shape graphs. We have implemented our decision technique, providing a fully automated framework for the verification of quantitative properties expressed as pre- and post-conditions on programs working on lists and integer counters.     

Interpolation of Shifted-Lacunary Polynomials

Given a “black box” function to evaluate an unknown rational polynomial f ? \mathbbQ[x]f \in {\mathbb{Q}}[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity $t \in {\mathbb{Z}}_{>0}$t \in {\mathbb{Z}}_{>0}, the shift a ? \mathbbQ\alpha \in {\mathbb{Q}}, the exponents 0 ￡ e₁ < e₂ < ? < e_t{0 \leq e_{1} < e_{2} < \cdots < e_{t}}, and the coefficients c₁, ?, c_t ? \mathbbQ \{0}c_{1}, \ldots , c_{t} \in {\mathbb{Q}} \setminus \{0\} such that

On the complexity of approximating k-set packing

Given a k-uniform hypergraph, the Maximum k -Set Packing problem is to find the maximum disjoint set of edges. We prove that this problem cannot be efficiently approximated to within a factor of unless P = NP. This improves the previous hardness of approximation factor of by Trevisan. This result extends to the problem of k-Dimensional-Matching.     

An Improved Parameterized Algorithm for the Minimum Node Multiway Cut Problem

The parameterized node multiway cut problem is for a given graph to find a separator of size bounded by k whose removal separates a collection of terminal sets in the graph. In this paper, we develop an O(k4 ^k n ³) time algorithm for this problem, significantly improving the previous algorithm of time for the problem. Our result gives the first polynomial time algorithm for the minimum node multiway cut problem when the separator size is bounded by O(log n). A preliminary version of this paper was presented at The 10th Workshop on Algorithms and Data Structures (WADS 2007). This work was supported in part by the National Science Foundation under the Grants CCR-0311590 and CCF-0430683.     

Efficient Algorithms for <Emphasis Type="Italic">k</Emphasis>-Terminal Cuts on Planar Graphs

The minimum k-terminal cut problem is of considerable theoretical interest and arises in several applied areas such as parallel and distributed computing, VLSI circuit design, and networking. In this paper we present two new approximation and exact algorithms for this problem on an n-vertex undirected weighted planar graph G. For the case when the k terminals are covered by the boundaries of m > 1 faces of G, we give a min{O(n ² log n logm), O(m ² n ^1.5 log² n + k n)} time algorithm with a (2–2/k)-approximation ratio (clearly, m \le k). For the case when all k terminals are covered by the boundary of one face of G, we give an O(n k3 + (n log n)k ²) time exact algorithm, or a linear time exact algorithm if k = 3, for computing an optimal k-terminal cut. Our algorithms are based on interesting observations and improve the previous algorithms when they are applied to planar graphs. To our best knowledge, no previous approximation algorithms specifically for solving the k-terminal cut problem on planar graphs were known before. The (2–2/k)-approximation algorithm of Dahlhaus et al. (for general graphs) takes O(k n ² log n) time when applied to planar graphs. Our approximation algorithm for planar graphs runs faster than that of Dahlhaus et al. by at least an O(k/logm) factor (m \le k).     

Efficient Algorithms for k-Terminal Cuts on Planar Graphs

The minimum k-terminal cut problem is of considerable theoretical interest and arises in several applied areas such as parallel and distributed computing, VLSI circuit design, and networking. In this paper we present two new approximation and exact algorithms for this problem on an n-vertex undirected weighted planar graph G. For the case when the k terminals are covered by the boundaries of m > 1 faces of G, we give a min{O(n ² log n logm), O(m ² n ^1.5 log² n + k n)} time algorithm with a (2–2/k)-approximation ratio (clearly, m \le k). For the case when all k terminals are covered by the boundary of one face of G, we give an O(n k3 + (n log n)k ²) time exact algorithm, or a linear time exact algorithm if k = 3, for computing an optimal k-terminal cut. Our algorithms are based on interesting observations and improve the previous algorithms when they are applied to planar graphs. To our best knowledge, no previous approximation algorithms specifically for solving the k-terminal cut problem on planar graphs were known before. The (2–2/k)-approximation algorithm of Dahlhaus et al. (for general graphs) takes O(k n ² log n) time when applied to planar graphs. Our approximation algorithm for planar graphs runs faster than that of Dahlhaus et al. by at least an O(k/logm) factor (m \le k).     

Redundant modeling in permutation weighted constraint satisfaction problems

In classical constraint satisfaction, redundant modeling has been shown effective in increasing constraint propagation and reducing search space for many problem instances. In this paper, we investigate, for the first time, how to benefit the same from redundant modeling in weighted constraint satisfaction problems (WCSPs), a common soft constraint framework for modeling optimization and over-constrained problems. Our work focuses on a popular and special class of problems, namely, permutation problems. First, we show how to automatically generate a redundant permutation WCSP model from an existing permutation WCSP using generalized model induction. We then uncover why naively combining mutually redundant permutation WCSPs by posting channeling constraints as hard constraints and relying on the standard node consistency (NC*) and arc consistency (AC*) algorithms would miss pruning opportunities, which are available even in a single model. Based on these observations, we suggest two approaches to handle the combined WCSP models. In our first approach, we propose m\text -NC_{\text c}^*m\text {-NC}_{\text c}^* and m\text -AC_{\text c}^*m\text {-AC}_{\text c}^* and their associated algorithms for effectively enforcing node and arc consistencies in a combined model with m sub-models. The two notions are strictly stronger than NC* and AC* respectively. While the first approach specifically refines NC* and AC* so as to apply to combined models, in our second approach, we propose a parameterized local consistency LB(m,Φ). The consistency can be instantiated with any local consistency Φ for single models and applied to a combined model with m sub-models. We also provide a simple algorithm to enforce LB(m,Φ). With the two suggested approaches, we demonstrate their applicabilities on several permutation problems in the experiments. Prototype implementations of our proposed algorithms confirm that applying 2\text -NC_{\text c}^*,  2\text -AC_{\text c}^*2\text {-NC}_{\text c}^*,\;2\text {-AC}_{\text c}^*, and LB(2,Φ) on combined models allow far more constraint propagation than applying the state-of-the-art AC*, FDAC*, and EDAC* algorithms on single models of hard benchmark problems.