An Improved FPT Algorithm and a Quadratic Kernel for Pathwidth One Vertex Deletion

The Pathwidth One Vertex Deletion (POVD) problem asks whether, given an undirected graph?G and an integer k, one can delete at most k vertices from?G so that the remaining graph has pathwidth at most 1. The question can be considered as a natural variation of the extensively studied Feedback Vertex Set (FVS) problem, where the deletion of at most k vertices has to result in the remaining graph having treewidth at most 1 (i.e., being a forest). Recently Philip et?al. (WG, Lecture Notes in Computer Science, vol.?6410, pp.?196?C207, 2010) initiated the study of the parameterized complexity of POVD, showing a quartic kernel and an algorithm which runs in time 7 ^k n ^O(1). In this article we improve these results by showing a quadratic kernel and an algorithm with time complexity 4.65 ^k n ^O(1), thus obtaining almost tight kernelization bounds when compared to the general result of Dell and van Melkebeek (STOC, pp.?251?C260, ACM, New York, 2010). Techniques used in the kernelization are based on the quadratic kernel for FVS, due to Thomassé (ACM Trans. Algorithms 6(2), 2010).     

The Parameterized Complexity of Unique Coverage and Its Variants

In this paper we study the parameterized complexity of the Unique Coverage problem, a variant of the classic Set Cover problem. This problem admits several parameterizations and we show that all, except the standard parameterization and a generalization of it, are unlikely to be fixed-parameter tractable. We use results from extremal combinatorics to obtain the best-known kernel for Unique Coverage and the well-known color-coding technique of Alon et al. (J. ACM 42(4), 844–856, 1995) to show that a weighted version of this problem is fixed-parameter tractable. Our application of color-coding uses an interesting variation of s-perfect hash families called (k,s)-hash families which were studied by Alon et al. (J. Comb. Theory Ser. A 104(1), 207–215, 2003) in the context of a class of codes called parent identifying codes (Barg et al. in SIAM J. Discrete Math. 14(3), 423–431, 2001). To the best of our knowledge, this is the first application of (k,s)-hash families outside the domain of coding theory. We prove the existence of such families of size smaller than the best-known s-perfect hash families using the probabilistic method (Alon and Spencer in The Probabilistic Method, Wiley, New York, 2000). Explicit constructions of such families of size promised by the probabilistic method is open.     

Better and Simpler Approximation Algorithms for?the?Stable Marriage Problem

We first consider the problem of finding a maximum size stable matching if incomplete lists and ties are both allowed, but ties are on one side only. For this problem we give a simple, linear time 3/2-approximation algorithm, improving on the best known approximation factor 5/3 of Irving and Manlove (J. Comb. Optim., doi:10.1007/s10878-007-9133-x, 2007). Next, we show how this extends to the Hospitals/Residents problem with the same ratio if the residents have strict orders. We also give a simple linear time algorithm for the general problem with approximation factor 5/3, improving the best known 15/8-approximation algorithm of Iwama, Miyazaki and Yamauchi (SODA ??07: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp.?288?C297, 2007). For the cases considered in this paper it is NP-hard to approximate within a factor of 21/19 by the result of Halldórsson et?al. (ACM Transactions on Algorithms 3(3):30, 2007). Our algorithms not only give better approximation ratios than the cited ones, but are much simpler and run significantly faster. Also we may drop a restriction used in (J. Comb. Optim., doi:10.1007/s10878-007-9133-x, 2007) and the analysis is substantially more moderate. Preliminary versions of this paper appeared in (Király, Egres Technical Report TR-2008-04, www.cs.elte.hu/egres/, 2008; Király in Proceedings of MATCH-UP 2008: Matching Under Preferences??Algorithms and Complexity, Satellite Workshop of ICALP, July 6, 2008, Reykjavík, Iceland, pp.?36?C45, 2008; Király in ESA 2008, Lecture Notes in Computer Science, vol.?5193, pp.?623?C634, 2008). For the related results obtained thenceforth see Sect.?5.     

On Finding Min-Min Disjoint Paths

The Min-Min problem of finding a disjoint-path pair with the length of the shorter path minimized is known to be NP-hard and admits no K-approximation for any K>1 in the general case (Xu et al. in IEEE/ACM Trans. Netw. 14:147–158, 2006). In this paper, we first show that Bhatia et al.’s NP-hardness proof (Bhatia et al. in J. Comb. Optim. 12:83–96, 2006), a claim of correction to Xu et al.’s proof (Xu et al. in IEEE/ACM Trans. Netw. 14:147–158, 2006), for the edge-disjoint Min-Min problem in the general undirected graphs is incorrect by giving a counter example that is an unsatisfiable 3SAT instance but classified as a satisfiable 3SAT instance in the proof of Bhatia et al. (J. Comb. Optim. 12:83–96, 2006). We then gave a correct proof of NP-hardness of this problem in undirected graphs. Finally we give a polynomial-time algorithm for the vertex disjoint Min-Min problem in planar graphs by showing that the vertex disjoint Min-Min problem is polynomially solvable in st-planar graph G=(V,E) whose corresponding auxiliary graph G(V,E∪{e(st)}) can be embedded into a plane, and a planar graph can be decomposed into several st-planar graphs whose Min-Min paths collectively contain a Min-Min disjoint-path pair between s and t in the original graph G. To the best of our knowledge, these are the first polynomial algorithms for the Min-Min problems in planar graphs.     

Log-Space Algorithms for Paths and Matchings in k-Trees

Reachability and shortest path problems are NL-complete for general graphs. They are known to be in L for graphs of tree-width 2 (Jakoby and Tantau in Proceedings of FSTTCS’07: The 27th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 216–227, 2007). In this paper, we improve these bounds for k-trees, where k is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed k-trees, and for computation of shortest and longest paths in directed acyclic k-trees. Besides the path problems mentioned above, we also consider the problem of deciding whether a k-tree has a perfect matching (decision version), and if so, finding a perfect matching (search version), and prove that these two problems are L-complete. These problems are known to be in P and in RNC for general graphs, and in SPL for planar bipartite graphs, as shown in Datta et al. (Theory Comput. Syst. 47:737–757, 2010). Our results settle the complexity of these problems for the class of k-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique central to our algorithms is a careful implementation of the divide-and-conquer approach in log-space, along with some ideas from Jakoby and Tantau (Proceedings of FSTTCS’07: The 27th Annual Conference on Foundations of Software Technology and Theoretical Computer Science, pp. 216–227, 2007) and Limaye et al. (Theory Comput. Syst. 46(3):499–522, 2010).     

WENO Scheme with Subcell Resolution for Computing Nonconservative Euler Equations with Applications to One-Dimensional Compressible Two-Medium Flows

High order path-conservative schemes have been developed for solving nonconservative hyperbolic systems in (Parés, SIAM J.?Numer. Anal. 44:300?C321, 2006; Castro et al., Math. Comput. 75:1103?C1134, 2006; J.?Sci. Comput. 39:67?C114, 2009). Recently, it has been observed in (Abgrall and Karni, J.?Comput. Phys. 229:2759?C2763, 2010) that this approach may have some computational issues and shortcomings. In this paper, a modification to the high order path-conservative scheme in (Castro et al., Math. Comput. 75:1103?C1134, 2006) is proposed to improve its computational performance and to overcome some of the shortcomings. This modification is based on the high order finite volume WENO scheme with subcell resolution and it uses an exact Riemann solver to catch the right paths at the discontinuities. An application to one-dimensional compressible two-medium flows of nonconservative or primitive Euler equations is carried out to show the effectiveness of this new approach.     

On Cutwidth Parameterized by Vertex Cover

We study the Cutwidth problem, where the input is a graph G, and the objective is find a linear layout of the vertices that minimizes the maximum number of edges intersected by any vertical line inserted between two consecutive vertices. We give an algorithm for Cutwidth with running time O(2 ^k n ^O(1)). Here k is the size of a minimum vertex cover of the input graph G, and n is the number of vertices in G. Our algorithm gives an O(2 ^n/2 n ^O(1)) time algorithm for Cutwidth on bipartite graphs as a corollary. This is the first non-trivial exact exponential time algorithm for Cutwidth on a graph class where the problem remains NP-complete. Additionally, we show that Cutwidth parameterized by the size of the minimum vertex cover of the input graph does not admit a polynomial kernel unless NP?coNP/poly. Our kernelization lower bound contrasts with the recent results of Bodlaender et al. (ICALP, Springer, Berlin, 2011; SWAT, Springer, Berlin, 2012) that both Treewidth and Pathwidth parameterized by vertex cover do admit polynomial kernels.     

On anonymizing transactions with sensitive items

K-anonymity (Samarati and Sweeny 1998; Samarati, IEEE Trans Knowl Data Eng, 13(6):1010–1027, 2001; Sweeny, Int J Uncertain, Fuzziness Knowl-Based Syst, 10(5):557–570, 2002) and its variants, l-diversity (Machanavajjhala et al., ACM TKDD, 2007) and tcloseness (Li et al. 2007) among others are anonymization techniques for relational data and transaction data, which are used to protect privacy against re-identification attacks. A relational dataset D is k-anonymous if every record in D has at least k-1 other records with identical quasi-identifier attribute values. The combination of released data with external data will never allow the recipient to associate each released record with less than k individuals (Samarati, IEEE Trans Knowl Data Eng, 13(6):1010–1027, 2001). However, the current concept of k-anonymity on transaction data treats all items as quasi-identifiers. The anonymized data set has k identical transactions in groups and suffers from lower data utility (He and Naughton 2009; He et al. 2011; Liu and Wang 2010; Terrovitis et al., VLDB J, 20(1):83–106, 2011; Terrovitis et al. 2008). To improve the utility of anonymized transaction data, this work proposes a novel anonymity concept on transaction data that contain both quasi-identifier items (QID) and sensitive items (SI). A transaction that contains sensitive items must have at least k-1 other identical transactions (Ghinita et al. IEEE TKDE, 33(2):161–174, 2011; Xu et al. 2008). For a transaction that does not contain a sensitive item, no anonymization is required. A transaction dataset that satisfies this property is said to be sensitive k-anonymous. Three algorithms, Sensitive Transaction Neighbors (STN) Gray Sort Clustering (GSC) and Nearest Neighbors for K-anonymization (K-NN), are developed. These algorithms use adding/deleting QID items and only adding SI to achieve sensitive k-anonymity on transaction data. Additionally, a simple “privacy value” is proposed to evaluate the degree of privacy for different types of k-anonymity on transaction data. Extensive numerical simulations were carried out to demonstrate the characteristics of the proposed algorithms and also compared to other types of k-anonymity approaches. The results show that each technique possesses its own advantage under different criteria such as running time, operation, and information loss. The results obtained here can be used as a guideline of the selection of anonymization technique on different data sets and for different applications.     

Improved Approximation Algorithms for the Min-max Tree Cover and Bounded Tree Cover Problems

In this paper we provide improved approximation algorithms for the Min-Max Tree Cover and Bounded Tree Cover problems. Given a graph G=(V,E) with weights w:E→?⁺, a set T ₁,T ₂,…,T _k of subtrees of G is called a tree cover of G if $V=\bigcup_{i=1}^{k} V(T_{i})$ . In the Min-Max k-tree Cover problem we are given graph G and a positive integer k and the goal is to find a tree cover with k trees, such that the weight of the largest tree in the cover is minimized. We present a 3-approximation algorithm for this improving the two different approximation algorithms presented in Arkin et al. (J. Algorithms 59:1–18, 2006) and Even et al. (Oper. Res. Lett. 32(4):309–315, 2004) with ratio 4. The problem is known to have an APX-hardness lower bound of $\frac{3}{2}$ (Xu and Wen in Oper. Res. Lett. 38:169–173, 2010). In the Bounded Tree Cover problem we are given graph G and a bound λ and the goal is to find a tree cover with minimum number of trees such that each tree has weight at most λ. We present a 2.5-approximation algorithm for this, improving the 3-approximation bound in Arkin et al. (J. Algorithms 59:1–18, 2006).     

QUBLE: towards blending interactive visual subgraph search queries on large networks

In a previous paper, we laid out the vision of a novel graph query processing paradigm where instead of processing a visual query graph after its construction, it interleaves visual query formulation and processing by exploiting the latency offered by the gui to filter irrelevant matches and prefetch partial query results [8]. Our recent attempts at implementing this vision [8, 9] show significant improvement in system response time (srt) for subgraph queries. However, these efforts are designed specifically for graph databases containing a large collection of small or medium-sized graphs. In this paper, we propose a novel algorithm called quble (QUery Blender for Large nEtworks) to realize this visual subgraph querying paradigm on very large networks (e.g., protein interaction networks, social networks). First, it decomposes a large network into a set of graphlets and supergraphlets using a minimum cut-based graph partitioning technique. Next, it mines approximate frequent and small infrequent fragments (sifs) from them and identifies their occurrences in these graphlets and supergraphlets. Then, the indexing framework of [9] is enhanced so that the mined fragments can be exploited to index graphlets for efficient blending of visual subgraph query formulation and query processing. Extensive experiments on large networks demonstrate effectiveness of quble.     

Functional un|unparsing

Danvy??s functional unparsing problem (Danvy in J. Funct. Program. 8(6), 621?C625, 1998) is to implement a type-safe ??printf?? function, which converts a sequence of heterogeneous arguments to a string according to a given format. The dual problem is to implement a type-safe ??scanf?? function, which extracts a sequence of heterogeneous arguments from a string by interpreting (Friedman and Wand in LFP, pp. 348?C355, 1984 and in Essentials of Programming Languages, MIT Press, 2008) the same format as an equally heterogeneous sequence of patterns that binds zero or more variables. We derive multiple solutions to both problems (Wand in J. ACM 27(1), 164?C180, 1980) from their formal specifications (Wand in Theor. Comput. Sci. 20(1), 3?C32, 1982). On one hand, our solutions show how the Hindley-Milner type system, unextended, permits accessing heterogeneous sequences with the static assurance of type safety. On the other hand, our solutions demonstrate the use of control operators (Felleisen et al. in Proceedings of the 1988 ACM Conference on Lisp and Functional Programming, pp. 52?C62, ACM Press, New York, 1988; Wand in POPL 85: Conference Record of the Annual ACM Symposium on Principles of Programming Languages, vol. 16, ACM Press, New York, 1985; Meyer and Wand in Logics of Programs, Lecture Notes in Computer Science, vol. 193, pp. 219?C224, Springer, Berlin, 1985) to communicate with formats as coroutines (Wand in Proceedings of the 1980 ACM Conference on Lisp and Functional Programming, vol. 12, pp. 285?C299, ACM Press, New York, 1980 and Haynes et al. in LFP, pp. 293?C298, 1984).     

Generalising Unit-Refutation Completeness and SLUR via Nested Input Resolution

The class ${\mathcal{SLUR}}$ (Single Lookahead Unit Resolution) was introduced in Schlipf et al. (Inf Process Lett 54:133–137, 1995) as an umbrella class for efficient (poly-time) SAT solving, with linear-time SAT decision, while the recognition problem was not considered. ?epek et al. (2012) and Balyo et al. (2012) extended this class in various ways to hierarchies covering all of CNF (all clause-sets). We introduce a hierarchy ${\mathcal{SLUR}}_k$ which we argue is the natural “limit” of such approaches. The second source for our investigations is the class ${\mathcal{UC}}$ of unit-refutation complete clause-sets, introduced in del Val (1994) as a target class for knowledge compilation. Via the theory of “hardness” of clause-sets as developed in Kullmann (1999), Kullmann (Ann Math Artif Intell 40(3–4):303–352, 2004) and Ansótegui et al. (2008) we obtain a natural generalisation ${\mathcal{UC}}_k$ , containing those clause-sets which are “unit-refutation complete of level k”, which is the same as having hardness at most k. Utilising the strong connections to (tree-)resolution complexity and (nested) input resolution, we develop basic methods for the determination of hardness (the level k in ${\mathcal{UC}}_k$ ). A fundamental insight now is that ${\mathcal{SLUR}}_k = {\mathcal{UC}}_k$ holds for all k. We can thus exploit both streams of intuitions and methods for the investigations of these hierarchies. As an application we can easily show that the hierarchies from ?epek et al. (2012) and Balyo et al. (2012) are strongly subsumed by ${\mathcal{SLUR}}_k$ . Finally we consider the problem of “irredundant” clause-sets in ${\mathcal{UC}}_k$ . For 2-CNF we show that strong minimisations are possible in polynomial time, while already for (very special) Horn clause-sets minimisation is NP-complete. We conclude with an extensive discussion of open problems and future directions. We envisage the concepts investigated here to be the starting point for a theory of good SAT translations, which brings together the good SAT-solving aspects from ${\mathcal{SLUR}}$ together with the knowledge-representation aspects from ${\mathcal{UC}}$ , and expands this combination via notions of “hardness”.     

On the modelling of time-interleaved sequential lamination micromixers

We develop a novel and simple theoretical model of time-interleaved sequential lamination micromixers that improves the model proposed by Nguyen and coworkers (Microfluid Nanofluid 1:373–375, 2005a, Lab Chip 5:1320–1326, b, J Phys Conf Ser 34:136–141, 2006) based on the Taylor–Aris dispersion theory. The Nguyen model takes into account the non uniform structure of the velocity profile through an effective dispersion coefficient. However, it is essentially a one-dimensional model that is not suitable to describe (i) neither the behavior of mixing occurring at short length-scales, and characterized by the growth of a mixing boundary layer near the channel walls, (ii) nor the exponential decay of the concentration field occurring at larger length-scales. The model we propose, which is based upon the theory of imaginary potential developed by Giona et?al. (J Fluid Mech 513:221–237, 2004, Europhys Lett 83:34001, 2008, J Fluid Mech 639:291–341, 2009a), is able to provide quantitative predictions on the evolution of the L ²-norm of the concentration fields as function of the axial coordinate ξ,?both for short and asymptotic lengthscales. The quantitative comparison with respect to the Nguyen model is illustrated and discussed. Finally, the coupling between parallel lamination and sequential segmentation is analyzed, and leads to unexpected and apparently counter-intuitive findings.     

On sunflowers and matrix multiplication

We present several variants of the sunflower conjecture of Erd?s & Rado (J Lond Math Soc 35:85–90, 1960) and discuss the relations among them. We then show that two of these conjectures (if true) imply negative answers to the questions of Coppersmith & Winograd (J Symb Comput 9:251–280, 1990) and Cohn et al. (2005) regarding possible approaches for obtaining fast matrix-multiplication algorithms. Specifically, we show that the Erd?s–Rado sunflower conjecture (if true) implies a negative answer to the “no three disjoint equivoluminous subsets” question of Coppersmith & Winograd (J Symb Comput 9:251–280, 1990); we also formulate a “multicolored” sunflower conjecture in ${\mathbb{Z}_3^n}$ and show that (if true) it implies a negative answer to the “strong USP” conjecture of Cohn et al. (2005) (although it does not seem to impact a second conjecture in Cohn et al. (2005) or the viability of the general group-theoretic approach). A surprising consequence of our results is that the Coppersmith–Winograd conjecture actually implies the Cohn et al. conjecture. The multicolored sunflower conjecture in ${\mathbb{Z}_3^n}$ is a strengthening of the well-known (ordinary) sunflower conjecture in ${\mathbb{Z}_3^n}$ , and we show via our connection that a construction from Cohn et al. (2005) yields a lower bound of (2.51 . . .) ⁿ on the size of the largest multicolored 3-sunflower-free set, which beats the current best-known lower bound of (2.21 . . . ) ⁿ Edel (2004) on the size of the largest 3-sunflower-free set in ${\mathbb{Z}_3^n}$ .     

On the (Non-)Existence of Polynomial Kernels for P l -Free Edge Modification Problems

Given a graph G=(V,E) and a positive integer k, an edge modification problem for a graph property Π consists in deciding whether there exists a set F of pairs of V of size at most k such that the graph $H=(V,E\vartriangle F)$ satisfies the property Π. In the Π edge-completion problem, the set F is constrained to be disjoint from E; in the Π edge-deletion problem, F is a subset of E; no constraint is imposed on F in the Π edge-editing problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity (Cai in Inf. Process. Lett. 58:171–176, 1996; Fellows et al. in FCT, pp. 312–321, 2007; Heggernes et al. in STOC, pp. 374–381, 2007). When parameterized by the size k of the set F, it has been proved that if Π is a hereditary property characterized by a finite set of forbidden induced subgraphs, then the three Π edge-modification problems are FPT (Cai in Inf. Process. Lett. 58:171–176, 1996). It was then natural to ask (Bodlaender et al. in IWPEC, 2006) whether these problems also admit a polynomial kernel. in polynomial time to an equivalent instance (G′,k′) with size bounded by a polynomial in k). Using recent lower bound techniques, Kratsch and Wahlström answered this question negatively (Kratsch and Wahlström in IWPEC, pp. 264–275, 2009). However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. question to characterize for which type of graph properties, the parameterized edge-modification problems have polynomial kernels. Kratsch and Wahlström asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of P ₄-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that Parameterized cograph edge-modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the P _l -free edge-deletion and the C _l -free edge-deletion problems for l?7 and l≥4 respectively. Indeed, if they exist, then NP?coNP/poly.     

A Self-stabilizing Algorithm for the Median Problem in?Partial Rectangular Grids and Their Relatives

Given a graph G=(V,E), a vertex v of G is a median vertex if it minimizes the sum of the distances to all other vertices of G. The median problem consists of finding the set of all median vertices of G. In this note, we present self-stabilizing algorithms for the median problem in partial rectangular grids and relatives. Our algorithms are based on the fact that partial rectangular grids can be isometrically embedded into the Cartesian product of two trees, to which we apply the algorithm proposed by Antonoiu and Srimani (J. Comput. Syst. Sci. 58:215–221, 1999) and Bruell et al. (SIAM J. Comput. 29:600–614, 1999) for computing the medians in trees. Then we extend our approach from partial rectangular grids to a more general class of plane quadrangulations. We also show that the characterization of medians of trees given by Gerstel and Zaks (Networks 24:23–29, 1994) extends to cube-free median graphs, a class of graphs which includes these quadrangulations.     

The Duo-Item Bisection Auction

This paper proposes an iterative sealed-bid auction for selling multiple heterogeneous items to bidders interested in buying at most one item. It generalizes the single item bisection auction (Grigorieva et al. Econ Theory, 30:107–118, 2007) to the environment with multiple heterogeneous items. We focus on the case with two items for sale. We show that the auction elicits a minimal amount of information on preferences required to find the Vickrey–Clark–Groves outcome (Clarke, Public Choice, XI:17–33, 1971; Groves, Econometrica, 61:617–631, 1973; Vickrey, J Finance, 16:8–37, 1961), when there are two items for sale and an arbitrary number of bidders.     

Preprocess, Set, Query!

Thorup and Zwick (J. ACM 52(1):1–24, 2005 and STOC’01) in their seminal work introduced the notion of distance oracles. Given an n-vertex weighted undirected graph with m edges, they show that for any integer k≥1 it is possible to preprocess the graph in $\tilde {O}(mn^{1/k})$ time and generate a compact data structure of size O(kn ^1+1/k ). For each pair of vertices, it is then possible to retrieve an estimated distance with multiplicative stretch 2k?1 in O(k) time. For k=2 this gives an oracle of O(n ^1.5) size that produces in constant time estimated distances with stretch 3. Recently, Pǎtra?cu and Roditty (In: Proc. of 51st FOCS, 2010) broke the theoretical status-quo in the field of distance oracles and obtained a distance oracle for sparse unweighted graphs of O(n ^5/3) size that produces in constant time estimated distances with stretch 2. In this paper we show that it is possible to break the stretch 2 barrier at the price of non-constant query time in unweighted undirected graphs. We present a data structure that produces estimated distances with 1+ε stretch. The size of the data structure is O(nm ^1?ε′) and the query time is $\tilde {O}(m^{1-\varepsilon '})$ . Using it for sparse unweighted graphs we can get a data structure of size O(n ^1.87) that can supply in O(n ^0.87) time estimated distances with multiplicative stretch 1.75.     

Scheduling of pipelined operator graphs

We investigate a class of scheduling problems that arise in the optimization of SQL queries for parallel machines (Chekuri et al. in PODS??95, pp.?255?C265, 1995). In these problems, an undirected graph is used to represent communication and inter-operator parallelism. The goal is to minimize the global response time of the system. We provide a polynomial time approximation scheme for the special cases where the operator graph is a tree, thereby improving on a polynomial time 2.87-approximation algorithm by Chekuri et al. The approximation scheme is generalized to the case where the operator graph has treewidth bounded by a constant. We analyze instances with small response times for general operator graphs: Deciding whether a response time of four time units can be reached is easy, but deciding whether a response time of six time units can be reached is NP-hard. Finally, we prove that for general operator graphs the existence of a polynomial time approximation algorithm with worst case performance guarantee better than 4/3 would imply P=NP.     

Algebraic flux correction for nonconforming finite element discretizations of scalar transport problems

This paper is concerned with the extension of the algebraic flux-correction (AFC) approach (Kuzmin in Computational fluid and solid mechanics, Elsevier, Amsterdam, pp 887–888, 2001; J Comput Phys 219:513–531, 2006; Comput Appl Math 218:79–87, 2008; J Comput Phys 228:2517–2534, 2009; Flux-corrected transport: principles, algorithms, and applications, 2nd edn. Springer, Berlin, pp 145–192, 2012; J Comput Appl Math 236:2317–2337, 2012; Kuzmin et al. in Comput Methods Appl Mech Eng 193:4915–4946, 2004; Int J Numer Methods Fluids 42:265–295, 2003; Kuzmin and Möller in Flux-corrected transport: principles, algorithms, and applications. Springer, Berlin, 2005; Kuzmin and Turek in J Comput Phys 175:525–558, 2002; J Comput Phys 198:131–158, 2004) to nonconforming finite element methods for the linear transport equation. Accurate nonoscillatory approximations to convection-dominated flows are obtained by stabilizing the continuous Galerkin method by solution-dependent artificial diffusion. Its magnitude is controlled by a flux limiter. This concept dates back to flux-corrected transport schemes. The unique feature of AFC is that all information is extracted from the system matrices which are manipulated to satisfy certain mathematical constraints. AFC schemes have been devised with conforming $P_1$ and $Q_1$ finite elements in mind but this is not a prerequisite. Here, we consider their extension to the nonconforming Crouzeix–Raviart element (Crouzeix and Raviart in RAIRO R3 7:33–76, 1973) on triangular meshes and its quadrilateral counterpart, the class of rotated bilinear Rannacher–Turek elements (Rannacher and Turek in Numer Methods PDEs 8:97–111, 1992). The underlying design principles of AFC schemes are shown to hold for (some variant of) both elements. However, numerical tests for a purely convective flow and a convection–diffusion problem demonstrate that flux-corrected solutions are overdiffusive for the Crouzeix–Raviart element. Good resolution of smooth and discontinuous profiles is attested to $Q_1^\mathrm{nc}$ approximations on quadrilateral meshes. A synthetic benchmark is used to quantify the artificial diffusion present in conforming and nonconforming high-resolution schemes of AFC-type. Finally, the implementation of efficient sparse matrix–vector multiplications is addressed.