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APPROX

**Published in:** LIPIcs, Volume 275, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)

We consider a general p-norm objective for experimental design problems that captures some well-studied objectives (D/A/E-design) as special cases. We prove that a randomized local search approach provides a unified algorithm to solve this problem for all nonnegative integer p. This provides the first approximation algorithm for the general p-norm objective, and a nice interpolation of the best known bounds of the special cases.

Lap Chi Lau, Robert Wang, and Hong Zhou. Experimental Design for Any p-Norm. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 4:1-4:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{lau_et_al:LIPIcs.APPROX/RANDOM.2023.4, author = {Lau, Lap Chi and Wang, Robert and Zhou, Hong}, title = {{Experimental Design for Any p-Norm}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)}, pages = {4:1--4:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-296-9}, ISSN = {1868-8969}, year = {2023}, volume = {275}, editor = {Megow, Nicole and Smith, Adam}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2023.4}, URN = {urn:nbn:de:0030-drops-188292}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2023.4}, annote = {Keywords: Approximation Algorithm, Optimal Experimental Design, Randomized Local Search} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

We design a polynomial time algorithm that for any weighted undirected graph G = (V, E, w) and sufficiently large \delta > 1, partitions V into subsets V(1),..., V(h) for some h>= 1, such that at most \delta^{-1} fraction of the weights are between clusters, i.e.
sum(i < j) |E(V(i), V(j)| < w(E)/\delta
and the effective resistance diameter of each of the induced subgraphs
G[V(i)] is at most \delta^3 times the inverse of the average weighted degree, i.e.
max{ Reff(u, v) : u, v \in V(i)} < \delta^3 · |V|/w(E)
for all i = 1,..., h. In particular, it is possible to remove one
percent of weight of edges of any given graph such that each of the
resulting connected components has effective resistance diameter at
most the inverse of the average weighted degree. Our proof is based
on a new connection between effective resistance and low conductance
sets. We show that if the effective resistance between two vertices u and v is large, then there must be a low conductance cut separating u from v. This implies that very mildly expanding graphs have constant effective resistance diameter. We believe that this connection could be of independent interest in algorithm design.

Vedat Levi Alev, Nima Anari, Lap Chi Lau, and Shayan Oveis Gharan. Graph Clustering using Effective Resistance. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{alev_et_al:LIPIcs.ITCS.2018.41, author = {Alev, Vedat Levi and Anari, Nima and Lau, Lap Chi and Oveis Gharan, Shayan}, title = {{Graph Clustering using Effective Resistance}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {41:1--41:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.41}, URN = {urn:nbn:de:0030-drops-83696}, doi = {10.4230/LIPIcs.ITCS.2018.41}, annote = {Keywords: Electrical Flows, Effective Resistance, Conductance, Graph Partitioning} }

Document

**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

We design approximation algorithms for Unique Gmeas when the constraint graph admits good low diameter graph decomposition. For the M2Lin(k) problem in K(r)-minor free graphs, when there is an assignment satisfying 1-eps fraction of constraints, we present an algorithm that produces an assignment satisfying 1-O(r*eps) fraction of constraints, with the approximation ratio independent of the alphabet size. A corollary is an improved approximation algorithm for the Min-UnCut problem for K(r)-minor free graphs. For general Unique Games in K(r)-minor free graphs, we provide another algorithm that produces an assignment satisfying 1-O(r *sqrt(eps)) fraction of constraints.
Our approach is to round a linear programming relaxation to find a minimum subset of edges that intersects all the inconsistent cycles. We show that it is possible to apply the low diameter graph decomposition technique on the constraint graph directly, rather than to work on the label extended graph as in previous algorithms for Unique Games. The same approach applies when the constraint graph is of genus g, and we get similar results with r replaced by log g in the M2Lin(k) problem and by sqrt(log g) in the general problem. The former result generalizes the result of Gupta-Talwar for Unique Games in the M2Lin(k) case, and the latter result generalizes the result of Trevisan for general Unique Games.

Vedat Levi Alev and Lap Chi Lau. Approximating Unique Games Using Low Diameter Graph Decomposition. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{levialev_et_al:LIPIcs.APPROX-RANDOM.2017.18, author = {Levi Alev, Vedat and Lau, Lap Chi}, title = {{Approximating Unique Games Using Low Diameter Graph Decomposition}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {18:1--18:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.18}, URN = {urn:nbn:de:0030-drops-75676}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.18}, annote = {Keywords: Unique Games, Low Diameter Graph Decomposition, Bounded Genus Graphs, Fixed Minor Free Graphs, Approximation Algorithms, Linear Programming} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

Given a lazy regular graph G, we prove that the expansion of G^t is at least sqrt(t) times the expansion of G. This bound is tight and can be generalized to small set expansion. We show some applications of this result.

Tsz Chiu Kwok and Lap Chi Lau. Lower Bounds on Expansions of Graph Powers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 313-324, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{kwok_et_al:LIPIcs.APPROX-RANDOM.2014.313, author = {Kwok, Tsz Chiu and Lau, Lap Chi}, title = {{Lower Bounds on Expansions of Graph Powers}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {313--324}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.313}, URN = {urn:nbn:de:0030-drops-47057}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.313}, annote = {Keywords: Conductance, Expansion, Graph power, Random walk} }

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

Graph $G$ is the square of graph $H$ if two vertices $x,y$ have an edge in $G$ if and only if $x,y$ are of distance at most two in $H$. Given $H$ it is easy to compute its square $H^2$, however Motwani and Sudan proved that it is NP-complete to determine if a given graph $G$ is the square of some graph $H$ (of girth $3$). In this paper we consider the characterization and recognition problems of graphs that are squares of graphs of small girth, i.e. to determine if $G=H^2$ for some graph $H$ of small girth. The main results are the following.
\begin{itemize}
\item There is a graph theoretical characterization for graphs that are squares of some graph of girth at least $7$. A corollary is that if a graph $G$ has a square root $H$ of girth at least $7$ then $H$ is unique up to isomorphism.
\item There is a polynomial time algorithm to recognize if $G=H^2$ for some graph $H$ of girth at least $6$.
\item It is NP-complete to recognize if $G=H^2$ for some graph $H$ of girth $4$.
\end{itemize}
These results almost provide a dichotomy theorem for the complexity of the recognition problem in terms of girth of the square roots. The algorithmic and graph theoretical results generalize previous results on tree square roots, and provide polynomial time algorithms to compute a graph square root of small girth if it exists. Some open questions and conjectures will also be discussed.

Babak Farzad, Lap Chi Lau, Van Bang Le, and Nguyen Ngoc Tuy. Computing Graph Roots Without Short Cycles. In 26th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 3, pp. 397-408, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{farzad_et_al:LIPIcs.STACS.2009.1827, author = {Farzad, Babak and Lau, Lap Chi and Le, Van Bang and Tuy, Nguyen Ngoc}, title = {{Computing Graph Roots Without Short Cycles}}, booktitle = {26th International Symposium on Theoretical Aspects of Computer Science}, pages = {397--408}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2009.1827}, URN = {urn:nbn:de:0030-drops-18279}, doi = {10.4230/LIPIcs.STACS.2009.1827}, annote = {Keywords: Graph roots, Graph powers, Recognition algorithms, NP-completeness} }

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