Document

**Published in:** LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

The original Specker-Blatter Theorem (1983) was formulated for classes of structures 𝒞 of one or several binary relations definable in Monadic Second Order Logic MSOL. It states that the number of such structures on the set [n] is modularly C-finite (MC-finite). In previous work we extended this to structures definable in CMSOL, MSOL extended with modular counting quantifiers. The first author also showed that the Specker-Blatter Theorem does not hold for one quaternary relation (2003).
If the vocabulary allows a constant symbol c, there are n possible interpretations on [n] for c. We say that a constant c is hard-wired if c is always interpreted by the same element j ∈ [n]. In this paper we show:
(i) The Specker-Blatter Theorem also holds for CMSOL when hard-wired constants are allowed. The proof method of Specker and Blatter does not work in this case.
(ii) The Specker-Blatter Theorem does not hold already for 𝒞 with one ternary relation definable in First Order Logic FOL. This was left open since 1983.
Using hard-wired constants allows us to show MC-finiteness of counting functions of various restricted partition functions which were not known to be MC-finite till now. Among them we have the restricted Bell numbers B_{r,A}, restricted Stirling numbers of the second kind S_{r,A} or restricted Lah-numbers L_{r,A}. Here r is an non-negative integer and A is an ultimately periodic set of non-negative integers.

Eldar Fischer and Johann A. Makowsky. Extensions and Limits of the Specker-Blatter Theorem. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 26:1-26:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fischer_et_al:LIPIcs.CSL.2024.26, author = {Fischer, Eldar and Makowsky, Johann A.}, title = {{Extensions and Limits of the Specker-Blatter Theorem}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {26:1--26:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.26}, URN = {urn:nbn:de:0030-drops-196694}, doi = {10.4230/LIPIcs.CSL.2024.26}, annote = {Keywords: Specker-Blatter Theorem, Monadic Second Order Logic, MC-finiteness} }

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RANDOM

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

The framework of distribution testing is currently ubiquitous in the field of property testing. In this model, the input is a probability distribution accessible via independently drawn samples from an oracle. The testing task is to distinguish a distribution that satisfies some property from a distribution that is far in some distance measure from satisfying it. The task of tolerant testing imposes a further restriction, that distributions close to satisfying the property are also accepted.
This work focuses on the connection between the sample complexities of non-tolerant testing of distributions and their tolerant testing counterparts. When limiting our scope to label-invariant (symmetric) properties of distributions, we prove that the gap is at most quadratic, ignoring poly-logarithmic factors. Conversely, the property of being the uniform distribution is indeed known to have an almost-quadratic gap.
When moving to general, not necessarily label-invariant properties, the situation is more complicated, and we show some partial results. We show that if a property requires the distributions to be non-concentrated, that is, the probability mass of the distribution is sufficiently spread out, then it cannot be non-tolerantly tested with o(√n) many samples, where n denotes the universe size. Clearly, this implies at most a quadratic gap, because a distribution can be learned (and hence tolerantly tested against any property) using 𝒪(n) many samples. Being non-concentrated is a strong requirement on properties, as we also prove a close to linear lower bound against their tolerant tests.
Apart from the case where the distribution is non-concentrated, we also show if an input distribution is very concentrated, in the sense that it is mostly supported on a subset of size s of the universe, then it can be learned using only 𝒪(s) many samples. The learning procedure adapts to the input, and works without knowing s in advance.

Sourav Chakraborty, Eldar Fischer, Arijit Ghosh, Gopinath Mishra, and Sayantan Sen. Exploring the Gap Between Tolerant and Non-Tolerant Distribution Testing. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 27:1-27:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chakraborty_et_al:LIPIcs.APPROX/RANDOM.2022.27, author = {Chakraborty, Sourav and Fischer, Eldar and Ghosh, Arijit and Mishra, Gopinath and Sen, Sayantan}, title = {{Exploring the Gap Between Tolerant and Non-Tolerant Distribution Testing}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {27:1--27:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.27}, URN = {urn:nbn:de:0030-drops-171497}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.27}, annote = {Keywords: Distribution Testing, Tolerant Testing, Non-tolerant Testing, Sample Complexity} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally (and sometimes proved more easily) in analytic language. We extend the theory of graph limits to the ordered setting, presenting a limit object for dense vertex-ordered graphs, which we call an orderon. As a special case, this yields limit objects for matrices whose rows and columns are ordered, and for dynamic graphs that expand (via vertex insertions) over time. Along the way, we devise an ordered locality-preserving variant of the cut distance between ordered graphs, showing that two graphs are close with respect to this distance if and only if they are similar in terms of their ordered subgraph frequencies. We show that the space of orderons is compact with respect to this distance notion, which is key to a successful analysis of combinatorial objects through their limits. For the proof we combine techniques used in the unordered setting with several new techniques specifically designed to overcome the challenges arising in the ordered setting.
We derive several applications of the ordered limit theory in extremal combinatorics, sampling, and property testing in ordered graphs. In particular, we prove a new ordered analogue of the well-known result by Alon and Stav [RS&A'08] on the furthest graph from a hereditary property; this is the first known result of this type in the ordered setting. Unlike the unordered regime, here the Erdős–Rényi random graph 𝐆(n, p) with an ordering over the vertices is not always asymptotically the furthest from the property for some p. However, using our ordered limit theory, we show that random graphs generated by a stochastic block model, where the blocks are consecutive in the vertex ordering, are (approximately) the furthest. Additionally, we describe an alternative analytic proof of the ordered graph removal lemma [Alon et al., FOCS'17].

Omri Ben-Eliezer, Eldar Fischer, Amit Levi, and Yuichi Yoshida. Ordered Graph Limits and Their Applications. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 42:1-42:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{beneliezer_et_al:LIPIcs.ITCS.2021.42, author = {Ben-Eliezer, Omri and Fischer, Eldar and Levi, Amit and Yoshida, Yuichi}, title = {{Ordered Graph Limits and Their Applications}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {42:1--42:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.42}, URN = {urn:nbn:de:0030-drops-135815}, doi = {10.4230/LIPIcs.ITCS.2021.42}, annote = {Keywords: graph limits, ordered graph, graphon, cut distance, removal lemma} }

Document

**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed ℓ, we construct a property P^(ℓ)⊆ {0,1}^n satisfying the following: Any testing algorithm for P^(ℓ) requires Ω(n) many queries, and yet P^(ℓ) has a constant query PCPP whose proof size is O(n⋅log^(ℓ)n), where log^(ℓ) denotes the ℓ times iterated log function (e.g., log^(2)n = log log n). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was O(n⋅polylog(n)).
As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed ℓ, we construct a property that has a constant-query tester, but requires Ω(n/log^(ℓ)(n)) queries for every tolerant or erasure-resilient tester.

Omri Ben-Eliezer, Eldar Fischer, Amit Levi, and Ron D. Rothblum. Hard Properties with (Very) Short PCPPs and Their Applications. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 9:1-9:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{beneliezer_et_al:LIPIcs.ITCS.2020.9, author = {Ben-Eliezer, Omri and Fischer, Eldar and Levi, Amit and Rothblum, Ron D.}, title = {{Hard Properties with (Very) Short PCPPs and Their Applications}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {9:1--9:27}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.9}, URN = {urn:nbn:de:0030-drops-116949}, doi = {10.4230/LIPIcs.ITCS.2020.9}, annote = {Keywords: PCPP, Property testing, Tolerant testing, Erasure resilient testing, Randomized encoding, Coding theory} }

Document

**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

One of the main challenges in property testing is to characterize those properties that are testable with a constant number of queries. For unordered structures such as graphs and hypergraphs this task has been mostly settled. However, for ordered structures such as strings, images, and ordered graphs, the characterization problem seems very difficult in general.
In this paper, we identify a wide class of properties of ordered structures - the earthmover resilient (ER) properties - and show that the "good behavior" of such properties allows us to obtain general testability results that are similar to (and more general than) those of unordered graphs. A property P is ER if, roughly speaking, slight changes in the order of the elements in an object satisfying P cannot make this object far from P. The class of ER properties includes, e.g., all unordered graph properties, many natural visual properties of images, such as convexity, and all hereditary properties of ordered graphs and images.
A special case of our results implies, building on a recent result of Alon and the authors, that the distance of a given image or ordered graph from any hereditary property can be estimated (with good probability) up to a constant additive error, using a constant number of queries.

Omri Ben-Eliezer and Eldar Fischer. Earthmover Resilience and Testing in Ordered Structures. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 18:1-18:35, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{beneliezer_et_al:LIPIcs.CCC.2018.18, author = {Ben-Eliezer, Omri and Fischer, Eldar}, title = {{Earthmover Resilience and Testing in Ordered Structures}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {18:1--18:35}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.18}, URN = {urn:nbn:de:0030-drops-88718}, doi = {10.4230/LIPIcs.CCC.2018.18}, annote = {Keywords: characterizations of testability, distance estimation, earthmover resilient, ordered structures, property testing} }

Document

**Published in:** LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Distribution testing deals with what information can be deduced about an unknown distribution over {1,...,n}, where the algorithm is only allowed to obtain a relatively small number of independent samples from the distribution. In the extended conditional sampling model, the algorithm is also allowed to obtain samples from the restriction of the original distribution on subsets of {1,...,n}.
In 2015, Canonne, Diakonikolas, Gouleakis and Rubinfeld unified several previous results, and showed that for any property of distributions satisfying a "decomposability" criterion, there exists an algorithm (in the basic model) that can distinguish with high probability distributions satisfying the property from distributions that are far from it in variation distance.
We present here a more efficient yet simpler algorithm for the basic model, as well as very efficient algorithms for the conditional model, which until now was not investigated under the umbrella of decomposable properties. Additionally, we provide an algorithm for the conditional model that handles a much larger class of properties.
Our core mechanism is a way of efficiently producing an interval-partition of {1,...,n} that satisfies a "fine-grain" quality. We show that with such a partition at hand we can directly move forward with testing individual intervals, instead of first searching for the "correct" partition of {1,...,n}.

Eldar Fischer, Oded Lachish, and Yadu Vasudev. Improving and Extending the Testing of Distributions for Shape-Restricted Properties. In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, pp. 31:1-31:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{fischer_et_al:LIPIcs.STACS.2017.31, author = {Fischer, Eldar and Lachish, Oded and Vasudev, Yadu}, title = {{Improving and Extending the Testing of Distributions for Shape-Restricted Properties}}, booktitle = {34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)}, pages = {31:1--31:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-028-6}, ISSN = {1868-8969}, year = {2017}, volume = {66}, editor = {Vollmer, Heribert and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.31}, URN = {urn:nbn:de:0030-drops-70024}, doi = {10.4230/LIPIcs.STACS.2017.31}, annote = {Keywords: conditional sampling, distribution testing, property testing, statistics} }

Document

**Published in:** LIPIcs, Volume 8, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)

We present several new examples of speed-ups obtainable by quantum algorithms in the context of property testing.
First, motivated by sampling algorithms, we consider probability distributions given in the form of an oracle $f:[n]\to[m]$. Here the probability $P_f(j)$ of an outcome $j$ in $[m]$ is the fraction of its domain that $f$ maps to $j$. We give quantum algorithms for testing whether two such distributions are identical or $epsilon$-far in $L_1$-norm. Recently, Bravyi, Hassidim, and Harrow showed that if
$P_f$ and $P_g$ are both unknown (i.e., given by oracles $f$ and $g$), then this testing can be done in roughly $sqrt{m}$ quantum queries to the functions. We consider the case where the second distribution is known, and show that testing can be done with roughly $m^{1/3}$ quantum queries, which we prove to be essentially optimal. In contrast, it is known that classical testing algorithms need about $m^{2/3}$ queries in the unknown-unknown case and about $sqrt{m}$ queries in the known-unknown case. Based on this result, we also reduce the query complexity of graph isomorphism testers with quantum oracle access.
While those examples provide polynomial quantum speed-ups, our third example gives a much larger improvement (constant quantum queries vs polynomial classical queries) for the problem of testing periodicity, based on Shor's algorithm and a modification of a classical lower bound by Lachish and Newman. This provides an alternative to a recent constant-vs-polynomial speed-up due to Aaronson.

Sourav Chakraborty, Eldar Fischer, Arie Matsliah, and Ronald de Wolf. New Results on Quantum Property Testing. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010). Leibniz International Proceedings in Informatics (LIPIcs), Volume 8, pp. 145-156, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{chakraborty_et_al:LIPIcs.FSTTCS.2010.145, author = {Chakraborty, Sourav and Fischer, Eldar and Matsliah, Arie and de Wolf, Ronald}, title = {{New Results on Quantum Property Testing}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2010)}, pages = {145--156}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-23-1}, ISSN = {1868-8969}, year = {2010}, volume = {8}, editor = {Lodaya, Kamal and Mahajan, Meena}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2010.145}, URN = {urn:nbn:de:0030-drops-28603}, doi = {10.4230/LIPIcs.FSTTCS.2010.145}, annote = {Keywords: quantum algorithm, property testing} }

Document

**Published in:** LIPIcs, Volume 5, 27th International Symposium on Theoretical Aspects of Computer Science (2010)

A {\em parametric weighted graph} is a graph whose edges are labeled with continuous real functions of a single common variable. For any instantiation of the variable, one obtains a standard edge-weighted graph. Parametric weighted graph problems are generalizations of weighted graph problems, and arise in various natural scenarios. Parametric weighted graph algorithms consist of two phases. A {\em preprocessing phase} whose input is a parametric weighted graph, and whose output is a data structure, the advice, that is later used by the {\em instantiation phase}, where a specific value for the variable is given. The instantiation phase outputs the solution to the (standard) weighted graph problem that arises from the instantiation. The goal is to have the running time of the instantiation phase supersede the running time of any algorithm that solves the weighted graph problem from scratch, by taking advantage of the advice.
In this paper we construct several parametric algorithms for the
shortest path problem. For the case of linear function weights we
present an algorithm for the single source shortest path problem. Its
preprocessing phase runs in $\tilde{O}(V^4)$ time, while its instantiation phase runs in only $O(E+V \log V)$ time. The fastest standard algorithm for single source shortest path runs in $O(VE)$ time. For the case of weight functions defined by degree $d$ polynomials, we present an algorithm with quasi-polynomial preprocessing time $O(V^{(1 + \log f(d))\log V})$ and instantiation time only $\tilde{O}(V)$. In fact, for any pair of vertices $u,v$, the instantiation phase computes the distance from $u$ to $v$ in only $O(\log^2 V)$ time. Finally, for linear function weights, we present
a randomized algorithm whose preprocessing time is $\tilde{O (V^{3.5})$ and so that for any pair of vertices $u,v$ and any instantiation variable, the instantiation phase computes, in $O(1)$ time, a length of a path from $u$ to $v$ that is at most (additively) $\epsilon$ larger than the length of a shortest path. In particular, an all-pairs shortest path solution, up to an additive constant error, can be computed in $O(V^2)$ time.

Sourav Chakraborty, Eldar Fischer, Oded Lachish, and Raphael Yuster. Two-phase Algorithms for the Parametric Shortest Path Problem. In 27th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 5, pp. 167-178, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{chakraborty_et_al:LIPIcs.STACS.2010.2452, author = {Chakraborty, Sourav and Fischer, Eldar and Lachish, Oded and Yuster, Raphael}, title = {{Two-phase Algorithms for the Parametric Shortest Path Problem}}, booktitle = {27th International Symposium on Theoretical Aspects of Computer Science}, pages = {167--178}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-16-3}, ISSN = {1868-8969}, year = {2010}, volume = {5}, editor = {Marion, Jean-Yves and Schwentick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2010.2452}, URN = {urn:nbn:de:0030-drops-24523}, doi = {10.4230/LIPIcs.STACS.2010.2452}, annote = {Keywords: Parametric Algorithms, Shortest path problem} }

Document

**Published in:** LIPIcs, Volume 3, 26th International Symposium on Theoretical Aspects of Computer Science (2009)

An edge-colored graph $G$ is {\em rainbow connected} if any two vertices are connected by a path whose edges have distinct colors. The {\em rainbow connectivity} of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in order to make $G$ rainbow connected. In addition to being a natural combinatorial problem, the rainbow connectivity problem is motivated by applications in cellular networks. In this paper we give the first proof that computing $rc(G)$ is NP-Hard. In fact, we prove that it is already NP-Complete to decide if $rc(G)=2$, and also that it is NP-Complete to decide whether a given edge-colored (with an unbounded number of colors) graph is rainbow connected. On the positive side, we prove that for every $\epsilon >0$, a connected graph with minimum degree at least $\epsilon n$ has bounded rainbow connectivity, where the bound depends only on $\epsilon$, and the corresponding coloring can be constructed in polynomial time. Additional non-trivial upper bounds, as well as open problems and conjectures are also presented.

Sourav Chakraborty, Eldar Fischer, Arie Matsliah, and Raphael Yuster. Hardness and Algorithms for Rainbow Connectivity. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 243-254, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{chakraborty_et_al:LIPIcs.STACS.2009.1811, author = {Chakraborty, Sourav and Fischer, Eldar and Matsliah, Arie and Yuster, Raphael}, title = {{Hardness and Algorithms for Rainbow Connectivity}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {243--254}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-09-5}, ISSN = {1868-8969}, year = {2009}, volume = {3}, editor = {Albers, Susanne and Marion, Jean-Yves}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1811}, URN = {urn:nbn:de:0030-drops-18115}, doi = {10.4230/LIPIcs.STACS.2009.1811}, annote = {Keywords: } }

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