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Documents authored by Wolleb-Graf, Daniel

Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.7
Faster Algorithms for All-Pairs Bounded Min-Cuts

Authors: Amir Abboud, Loukas Georgiadis, Giuseppe F. Italiano, Robert Krauthgamer, Nikos Parotsidis, Ohad Trabelsi, Przemysław Uznański, and Daniel Wolleb-Graf

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

Abstract
The All-Pairs Min-Cut problem (aka All-Pairs Max-Flow) asks to compute a minimum s-t cut (or just its value) for all pairs of vertices s,t. We study this problem in directed graphs with unit edge/vertex capacities (corresponding to edge/vertex connectivity). Our focus is on the k-bounded case, where the algorithm has to find all pairs with min-cut value less than k, and report only those. The most basic case k=1 is the Transitive Closure (TC) problem, which can be solved in graphs with n vertices and m edges in time O(mn) combinatorially, and in time O(n^{omega}) where omega<2.38 is the matrix-multiplication exponent. These time bounds are conjectured to be optimal. We present new algorithms and conditional lower bounds that advance the frontier for larger k, as follows: - A randomized algorithm for vertex capacities that runs in time {O}((nk)^{omega}). This is only a factor k^omega away from the TC bound, and nearly matches it for all k=n^{o(1)}. - Two deterministic algorithms for edge capacities (which is more general) that work in DAGs and further reports a minimum cut for each pair. The first algorithm is combinatorial (does not involve matrix multiplication) and runs in time {O}(2^{{O}(k^2)}* mn). The second algorithm can be faster on dense DAGs and runs in time {O}((k log n)^{4^{k+o(k)}}* n^{omega}). Previously, Georgiadis et al. [ICALP 2017], could match the TC bound (up to n^{o(1)} factors) only when k=2, and now our two algorithms match it for all k=o(sqrt{log n}) and k=o(log log n). - The first super-cubic lower bound of n^{omega-1-o(1)} k^2 time under the 4-Clique conjecture, which holds even in the simplest case of DAGs with unit vertex capacities. It improves on the previous (SETH-based) lower bounds even in the unbounded setting k=n. For combinatorial algorithms, our reduction implies an n^{2-o(1)} k^2 conditional lower bound. Thus, we identify new settings where the complexity of the problem is (conditionally) higher than that of TC. Our three sets of results are obtained via different techniques. The first one adapts the network coding method of Cheung, Lau, and Leung [SICOMP 2013] to vertex-capacitated digraphs. The second set exploits new insights on the structure of latest cuts together with suitable algebraic tools. The lower bounds arise from a novel reduction of a different structure than the SETH-based constructions.
Cite as

Amir Abboud, Loukas Georgiadis, Giuseppe F. Italiano, Robert Krauthgamer, Nikos Parotsidis, Ohad Trabelsi, Przemysław Uznański, and Daniel Wolleb-Graf. Faster Algorithms for All-Pairs Bounded Min-Cuts. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{abboud_et_al:LIPIcs.ICALP.2019.7,
  author =	{Abboud, Amir and Georgiadis, Loukas and Italiano, Giuseppe F. and Krauthgamer, Robert and Parotsidis, Nikos and Trabelsi, Ohad and Uzna\'{n}ski, Przemys{\l}aw and Wolleb-Graf, Daniel},
  title =	{{Faster Algorithms for All-Pairs Bounded Min-Cuts}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{7:1--7:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.7},
  URN =		{urn:nbn:de:0030-drops-105833},
  doi =		{10.4230/LIPIcs.ICALP.2019.7},
  annote =	{Keywords: All-pairs min-cut, k-reachability, network coding, Directed graphs, fine-grained complexity}
}
Document
DOI: 10.4230/LIPIcs.CPM.2019.14
Hamming Distance Completeness

Authors: Karim Labib, Przemysław Uznański, and Daniel Wolleb-Graf

Published in: LIPIcs, Volume 128, 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)

Abstract
We show, given a binary integer function diamond that is piecewise polynomial, that (+,diamond) vector products are equivalent under one-to-polylog reductions to the computation of the Hamming distance. Examples include the dominance and l_{2p+1} distances for constant p. Our results imply equivalence (up to polylog factors) between the complexity of computing All Pairs Hamming Distance, All Pairs l_{2p+1} Distance and Dominance Matrix Product, and equivalence between Hamming Distance Pattern Matching, l_{2p+1} Pattern Matching and Less-Than Pattern Matching. The resulting algorithms for l_{2p+1} Pattern Matching and All Pairs l_{2p+1}, for 2p+1 = 3,5,7,... are likely to be optimal, given lack of progress in improving upper bounds for Hamming distance in the past 30 years. While reductions between selected pairs of products were presented in the past, our work is the first to generalize them to a general class of functions, showing that a wide class of "intermediate" complexity problems are in fact equivalent.
Cite as

Karim Labib, Przemysław Uznański, and Daniel Wolleb-Graf. Hamming Distance Completeness. In 30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 128, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{labib_et_al:LIPIcs.CPM.2019.14,
  author =	{Labib, Karim and Uzna\'{n}ski, Przemys{\l}aw and Wolleb-Graf, Daniel},
  title =	{{Hamming Distance Completeness}},
  booktitle =	{30th Annual Symposium on Combinatorial Pattern Matching (CPM 2019)},
  pages =	{14:1--14:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-103-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{128},
  editor =	{Pisanti, Nadia and P. Pissis, Solon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2019.14},
  URN =		{urn:nbn:de:0030-drops-104853},
  doi =		{10.4230/LIPIcs.CPM.2019.14},
  annote =	{Keywords: fine grained complexity, approximate pattern matching, matrix products}
}
Document
DOI: 10.4230/OASIcs.SOSA.2019.4
A Framework for Searching in Graphs in the Presence of Errors

Authors: Dariusz Dereniowski, Stefan Tiegel, Przemyslaw Uznanski, and Daniel Wolleb-Graf

Published in: OASIcs, Volume 69, 2nd Symposium on Simplicity in Algorithms (SOSA 2019)

Abstract
We consider a problem of searching for an unknown target vertex t in a (possibly edge-weighted) graph. Each vertex-query points to a vertex v and the response either admits that v is the target or provides any neighbor s of v that lies on a shortest path from v to t. This model has been introduced for trees by Onak and Parys [FOCS 2006] and for general graphs by Emamjomeh-Zadeh et al. [STOC 2016]. In the latter, the authors provide algorithms for the error-less case and for the independent noise model (where each query independently receives an erroneous answer with known probability p<1/2 and a correct one with probability 1-p). We study this problem both with adversarial errors and independent noise models. First, we show an algorithm that needs at most (log_2 n)/(1 - H(r)) queries in case of adversarial errors, where the adversary is bounded with its rate of errors by a known constant r<1/2. Our algorithm is in fact a simplification of previous work, and our refinement lies in invoking an amortization argument. We then show that our algorithm coupled with a Chernoff bound argument leads to a simpler algorithm for the independent noise model and has a query complexity that is both simpler and asymptotically better than the one of Emamjomeh-Zadeh et al. [STOC 2016]. Our approach has a wide range of applications. First, it improves and simplifies the Robust Interactive Learning framework proposed by Emamjomeh-Zadeh and Kempe [NIPS 2017]. Secondly, performing analogous analysis for edge-queries (where a query to an edge e returns its endpoint that is closer to the target) we actually recover (as a special case) a noisy binary search algorithm that is asymptotically optimal, matching the complexity of Feige et al. [SIAM J. Comput. 1994]. Thirdly, we improve and simplify upon an algorithm for searching of unbounded domains due to Aslam and Dhagat [STOC 1991].
Cite as

Dariusz Dereniowski, Stefan Tiegel, Przemyslaw Uznanski, and Daniel Wolleb-Graf. A Framework for Searching in Graphs in the Presence of Errors. In 2nd Symposium on Simplicity in Algorithms (SOSA 2019). Open Access Series in Informatics (OASIcs), Volume 69, pp. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dereniowski_et_al:OASIcs.SOSA.2019.4,
  author =	{Dereniowski, Dariusz and Tiegel, Stefan and Uznanski, Przemyslaw and Wolleb-Graf, Daniel},
  title =	{{A Framework for Searching in Graphs in the Presence of Errors}},
  booktitle =	{2nd Symposium on Simplicity in Algorithms (SOSA 2019)},
  pages =	{4:1--4:17},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-099-6},
  ISSN =	{2190-6807},
  year =	{2019},
  volume =	{69},
  editor =	{Fineman, Jeremy T. and Mitzenmacher, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.SOSA.2019.4},
  URN =		{urn:nbn:de:0030-drops-100305},
  doi =		{10.4230/OASIcs.SOSA.2019.4},
  annote =	{Keywords: graph algorithms, noisy binary search, query complexity, reliability}
}
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