Distance Labeling for Permutation Graphs

The Symposium on Theoretical Aspects of Computer Science (STACS) is held annually, alternating between France and Germany, and has become one of the most important European annual meetings for the theoretical computer science community covering a wide range of topics in the area of foundations of computer science: algorithms and data structures, automata and formal languages, computational and structural complexity, logic, verification, and current challenges. The 24th conference of the series, STACS 2007, was held at the University of Aachen on February 22–24. Its call for papers led to approximately 450 submissions from all over the world. The scientific quality of the submissions made selecting the best ones a very difficult task. During the two-day, very intense meeting of the Program Committee, 54 papers (only 12% of the submissions) could be chosen for presentation at the conference. Due to this highly selective process, many good and relevant papers with positive reviews could not be included in the final program. On the other hand such a rigorous selection process, which is usual for STACS, is the guarantee that this conference series is a scientifically rich and very fruitful forum for theoretical computer science. The articles in this special issue of Theory of Computing Systems are a further selection from the best results presented at STACS 2007. The full versions of these papers underwent an additional strict refereeing process as is standard for a high-level scientific journal. The contributions illustrate very well the wide range of STACS. We express our gratitude to all authors for their contributions and to the referees for their helpful reports. We also thank Alan Selman for the smooth cooperation.

This special issue contains nine articles which are based on extended abstracts that were presented at the 28th Symposium on Theoretical Aspects of Computer Science (STACS), which was held at Technische Universitat Dortmund, Germany, March 10– 12, 2011. These extended abstracts were among the top papers of those that were chosen for presentation at STACS 2011 in a highly competitive peer-review process (after which only 54 papers out of 271 submission were accepted). Compared with the original extended abstracts the articles have been strongly revised and extended by full proofs and additional results. They underwent a further rigorous reviewing process, following the TOCS standard, completely independent from the selection process of STACS 2011. Three articles are about formal languages. The first two investigate size issues for automata and (extended) regular expressions, respectively, while the third one investigates the syntactic monoids of regular languages over infinite alphabets. The article Series Parallel Digraphs with Loops by Stefan Gulan studies the question which kinds of finite automata can be translated into regular expressions of linear size. It gives a very comprehensive answer to this question by showing that such linear translations are possible from automata with a certain underlying graph structure, series parallel loop graphs, into regular expressions and vice versa. Furthermore it characterizes this graph class (within the class of hammocks, i.e., all graphs with dis-

In the last years, kernelization with structural parameters has been an active area of research within the field of parameterized complexity. As a relevant example, Gajarský et al. (J Comput Syst Sci 84:219–242, 2017) proved that every graph problem satisfying a property called finite integer index admits a linear kernel on graphs of bounded expansion and an almost linear kernel on nowhere dense graphs, parameterized by the size of a c-treedepth modulator, which is a vertex set whose removal results in a graph of treedepth at most c, where $$c \ge 1$$ is a fixed integer. The authors left as further research to investigate this parameter on general graphs, and in particular to find problems that, while admitting polynomial kernels on sparse graphs, behave differently on general graphs. In this article we answer this question by finding two very natural such problems: we prove that Vertex Cover admits a polynomial kernel on general graphs for any integer $$c \ge 1$$ , and that Dominating Set does not for any integer $$c \ge 2$$ even on degenerate graphs, unless $$\text {NP} \subseteq \text {coNP}/\text {poly}$$ . For the positive result, we build on the techniques of Jansen and Bodlaender (Proceedings of the 28th symposium on theoretical aspects of computer science (STACS), volume 9 of LIPIcs, pp 177–188, 2011), and for the negative result we use a polynomial parameter transformation for $$c\ge 3$$ and an or-cross-composition for $$c = 2$$ . As existing results imply that Dominating Set admits a polynomial kernel on degenerate graphs for $$c = 1$$ , our result provides a dichotomy about the existence of polynomial kernels for Dominating Set on degenerate graphs with this parameter.

Deterministic monotone algorithms for scheduling on related machines

The minimum broadcast range assignment problem on linear multi-hop wireless networks

The complexity of shortest path and dilation bounded interval routing

We investigate the power of counting in Group Isomorphism. We first leverage the count-free variant of the Weisfeiler–Leman Version I algorithm for groups [J. Brachter and P. Schweitzer, On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786] in tandem with bounded non-determinism and limited counting to improve the parallel complexity of isomorphism testing for several families of groups. These families include: • Direct products of non-Abelian simple groups. • Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an [Formula: see text]-generated solvable group with solvability class [Formula: see text]. This notably includes instances where the complement is an [Formula: see text]-generated nilpotent group. This problem was previously known to be in [Formula: see text] [Y. Qiao, J. M. N. Sarma and B. Tang, On isomorphism testing of groups with normal Hall subgroups, in Proc. 28th Symp. Theoretical Aspects of Computer Science, Dagstuhl Castle, Leibniz Center for Informatics, 2011), pp. 567–578, doi:10.4230/LIPIcs. STACS.2011.567], and the complexity was recently improved to [Formula: see text] [J. A. Grochow and M. Levet, On the parallel complexity of group isomorphism via Weisfeiler–Leman, in 24th Int. Symp. Fundamentals of Computation Theory, eds. H. Fernau and K. Jansen, Lecture Notes in Computer Science, Vol. 14292, September 18–21, 2023, Trier, Germany (Springer, 2023), pp. 234–247]. • Graphical groups of class 2 and exponent [Formula: see text] [A. H. Mekler, Stability of nilpotent groups of class 2 and prime exponent, J. Symb. Logic 46(4) (1981) 781–788] arising from the CFI and twisted CFI graphs [J.-Y. Cai, M. Fürer and N. Immerman, An optimal lower bound on the number of variables for graph identification, Combinatorica 12(4) (1992) 389–410], respectively. In particular, our work improves upon previous results of Brachter and Schweitzer [On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786]. Notably, each of these families was previously known to be identified by the counting variant of the more powerful Weisfeiler–Leman Version II algorithm. We finally show that the q-ary count-free pebble game is unable to even distinguish Abelian groups. This extends the result of Grochow and Levet (ibid), who established the result in the case of [Formula: see text]. The general theme is that some counting appears necessary to place [Formula: see text] into [Formula: see text].

Given a geometric graph in d-dimensional space, it is useful to preprocess it so that distance queries, exact or approximate, can be answered efficiently. Algorithms that can report distance queries in constant time are also referred to as “distance oracles”. With unlimited preprocessing time and space, it is clear that exact distance oracles can be easily designed. This entry sheds light on the design of approximate distance oracles with limited preprocessing time and space for the family of geometric graphs with constant dilation.

Shortest distance queries are essential not only in graph analysis and graph mining tasks but also in database applications, when a large graph needs to be dealt with. Such shortest distance queries are frequently issued by end-users or requested as a subroutine in real applications. For intensive queries on large graphs, it is impractical to compute shortest distances on-line from scratch, and impractical to materialize all-pairs shortest distances. In the literature, 2-hop distance labeling is proposed to index the all-pairs shortest distances. It assigns distance labels to vertices in a large graph in a pre-computing step off-line and then answers shortest distance queries on-line by making use of such distance labels, which avoids exhaustively traversing the large graph when answering queries. However, the existing algorithms to generate 2-hop distance labels are not scalable to large graphs. Finding an optimal 2-hop distance labeling is NP-hard, and heuristic algorithms may generate large size distance labels while still needing to pre-compute all-pairs shortest paths. In this paper, we propose a multi-hop distance labeling approach, which generates a subset of the 2-hop distance labels as index off-line. We can compute the multi-hop distance labels efficiently by avoiding pre-computing all-pairs shortest paths. In addition, our multi-hop distance labeling is small in size to be stored. To answer a shortest distance query between two vertices, we first generate the query-specific small set of 2-hop distance labels for the two vertices based on our multi-hop distance labels stored and compute the shortest distance between the two vertices based on the 2-hop distance labels generated on-line. We conducted extensive performance studies on large real graphs and confirmed the efficiency of our multi-hop distance labeling scheme.

A simple polygon P with two distinguished vertices, s and t, is called a street if the two boundary chains from s to t are mutually weakly visible. We present an on-line strategy that walks from s to t, in any unknown street, on a path at most $\sqrt{2}$ times longer than the shortest path. This matches the best lower bound previously known and settles an open problem in the area of competitive path planning. (The result was simultaneously and independently obtained by the first three authors and by the last two authors. Both papers, [C. Icking, R. Klein, and E. Langetepe, Proceedings of the 16th Symposium on Theoretical Aspects in Computer Science, Lecture Notes in Comput. Sci. 1563, Springer-Verlag, Berlin, 1999, pp. 110--120] and [S. Schuierer and I. Semrau, Proceedings of the 16th Symposium on Theoretical Aspects of Computer Science, pp. 121--131], were presented at STACS'99. The present paper contains a joint full version.)

Selfish unsplittable flows

STACS '93 : 10th Symposium on Theoretical Aspects of Computer Science '93, February 25–27, 1993

Given a set S of n (distinct) keys from key space [U], each associated with a value from Σ, the static dictionary problem asks to preprocess these (key, value) pairs into a data structure, supporting value-retrieval queries: for any given x∈ [U], valRet(x) must return the value associated with x if x∈ S, or return ⊥ if x∉ S. The special case where |Σ|=1 is called the membership problem. The “textbook” solution is to use a hash table, which occupies linear space and answers each query in constant time. On the other hand, the minimum possible space to encode all (key, value) pairs is only OPT:= ⌈lg2( U n )+nlg2|Σ|⌉ bits, which could be much less. In this paper, we design a randomized dictionary data structure using OPT+lgn+O(lglglglglgU) bits of space, and it has expected constant query time, assuming the query algorithm can access an external lookup table of size n 0.001. The lookup table depends only on U, n and |Σ|, and not the input. Previously, even for membership queries and U≤ n O(1), the best known data structure with constant query time requires OPT+n/lgn bits of space by Pagh (SIAM J. Comput. 2001) and Pǎtrascu (FOCS 2008); the best known using OPT+n 0.999 space has query time O(lgn); the only known non-trivial data structure with OPT+n 0.001 space has O(lgn) query time and requires a lookup table of size ≥ n 2.99 (!). Our new data structure answers open questions by Pǎtrascu and Thorup.

We consider distributed deterministic broadcasting in synchronous radio networks. A node receives a message in a given round if and only if exactly one of its neighbors transmits. The source message has to reach all nodes. We assume that nodes do not know the network topology or even their immediate neighborhood. (Such networks are called ad hoc.) We are concerned with two efficiency measures of broadcasting algorithms: their execution time (number of rounds) and their cost (number of transmissions). We focus our study on the execution time of algorithms which have cost close to minimum. We consider two scenarios depending on whether nodes know or do not know global parameters of the network: the number n of nodes and the eccentricity D of the source. Our main contribution is proving tight lower bounds on the time of low-cost broadcasting which show sharp differences between these scenarios. In each case, we also give broadcasting algorithms whose performance matches these lower bounds. © 2002 Wiley Periodicals, Inc.

We study the maximum weight matching problem in the semi-streaming model, and improve on the currently best one-pass algorithm due to Zelke [Proceedings of the 25th Annual Symposium on Theoretical Aspects of Computer Science, 2008, pp. 669–680] by devising a deterministic approach whose performance guarantee is . In addition, we study preemptive online algorithms, a class of algorithms related to one-pass semi-streaming algorithms, where we are allowed to maintain only a feasible matching in memory at any point in time. We provide a lower bound of 4.967 on the competitive ratio of any such deterministic algorithm, and hence show that future improvements will have to store in memory a set of edges that is not necessarily a feasible matching. We conclude by presenting an empirical study, conducted in order to compare the practical performance of our approach to that of previously suggested algorithms.