# Seminář z aproximačních a online algoritmů Seminar on Approximation and Online Algorithms

## Andreas Emil Feldmann, Petr Kolman, Jiří Sgall

### Doba a místo konání [Time and place]

Seminář se koná v úterý od 12:20 v S10. [Seminar takes place on Tuesday 12:20 in S10]

Oznámení o seminářích se distribuuje pomocí mailing listu, do kterého se můžete zapsat na adrese https://kam.mff.cuni.cz/mailman/listinfo/algo-seminar-l.
You can subscribe to the mailing list with the seminar anouncement at https://kam.mff.cuni.cz/mailman/listinfo/algo-seminar-l.

### Kontakt

https://sites.google.com/site/aefeldmann/, tel. 951 554 372;
http://kam.mff.cuni.cz/~kolman/, tel. 951 554 155;
http://iuuk.mff.cuni.cz/~sgall/, tel. 951 554 293.

### Nejbližší program semináře [Next program]

Nov 20, 2018 (úterý [Tuesday], 12:20, MFF UK Malá Strana S10)

Uriel Feige, Mohammad Mahdian: Finding small balanced separators. STOC 2006.
(presented by Matěj Lieskovský)

Abstract: Let G be an n-vertex graph that has a vertex separator of size k that partitions the graph into connected components of size smaller than α n, for some fixed 2/3 ≤ α < 1. Such a separator is called an α-separator. Finding an α-separator of size at most k is NP-hard. Moreover, under reasonable complexity theoretic assumptions, it is shown that this problem is not polynomially solvable even when k=O(log n). In this paper, we give a randomized algorithm that finds an α-separator of size k in the given graph, unless the graph contains an (α+ε)-separator of size strictly less than k, in which case our algorithm finds one such separator. For fixed ε, the running time of our algorithm is nO(1)2O(k), which is polynomial for k = O(log n). For bounded degree graphs (as well as for the case of finding balanced edge separators), we present a deterministic algorithm with similar running time.Our algorithm involves (among other things) a new concept that we call (ε,k)-samples. This is related to the notion of detection sets for network failures, introduced by Kleinberg [FOCS 2000]. Our proofs adapt and simplify techniques that were introduced by Kleinberg. As a by-product, our proof improves the known bounds on the size of detection sets. We also show applications of (ε,k)-samples to problems in approximation algorithms and rigorous analysis of heuristics.

### Předběžný další program [Preliminary future program]

Nov 27, 2018 (úterý [Tuesday], 12:20, MFF UK Malá Strana S10)

Zachary Friggstad (U. Alberta): Approximation Algorithms for Orienteering Problems

Abstract: In the Orienteering problem, the goal is to find a length-bounded path starting from some depot that visits the maximum number of clients possible. For example, how many packages can be delivered by a delivery truck in a single day?

We survey some approximations for this problem. Rather than focusing on a single result, this talk will present a range of approximations including the original approaches based on dynamic programming and more recent approaches based on linear programming. Our discussion will mostly cover constant-factor approximations for Orienteering in symmetric metrics but sometime will be spent on poly-logarithmic approximations for Orienteering in asymmetric metrics.

Specific results to be discussed are found in the following papers.
https://www.cs.cmu.edu/~avrim/Papers/orienteering-sicomp.pdf
https://dl.acm.org/citation.cfm?id=1007385
http://chekuri.cs.illinois.edu/papers/orienteering-journal.pdf
http://www.contrib.andrew.cmu.edu/~ravi/algorithmica11.pdf
https://arxiv.org/abs/1708.01335

Michael Lampis: Parameterized Approximation Schemes using Graph Widths. https://arxiv.org/abs/1311.2466.
(presented by Michal Berg?)

Andreas Wiese: A (1+eps)-approximation for Unsplittable Flow on a Path in fixed-parameter running time. ICALP 2017.
(presented by Lukáš Folwarczný)

Abstract: Unsplittable Flow on a Path (UFP) is a well-studied problem. It arises in many different settings such as bandwidth allocation, scheduling, and caching. We are given a path with capacities on the edges and a set of tasks, each of them is described by a start and an end vertex and a demand. The goal is to select as many tasks as possible such that the demand of the selected tasks using each edge does not exceed the capacity of this edge. The problem admits a QPTAS and the best known polynomial time result is a (2+epsilon)-approximation. As we prove in this paper, the problem is intractable for fixed-parameter algorithms since it is W[1]-hard. A PTAS seems difficult to construct. However, we show that if we combine the paradigms of approximation algorithms and fixed-parameter tractability we can break the mentioned boundaries. We show that on instances with |OPT|=k we can compute a (1+epsilon)-approximation in time 2^O(k log k)n^O_epsilon(1) log(u_max) (where u_max is the maximum edge capacity). To obtain this algorithm we develop new insights for UFP and enrich a recent dynamic programming framework for the problem. Our results yield a PTAS for (unweighted) UFP instances where |OPT| is at most O(log n/log log n) and they imply that the problem does not admit an EPTAS, unless W[1]=FPT.

Hans-Joachim Bockenhauer, Juraj Hromkovic, Joachim Kneis, Joachim Kupke: The Parameterized Approximability of TSP with Deadlines. Theory of Computing Systems 41:431–444, 2007.
(presented by Petr Vincena)

### Key papers (which may not be too easy to read):

Sanjeev Arora: Polynomial Time Approximation Schemes for Euclidean
Traveling Salesman and Other Geometric Problems
JACM 45:753–782, 1998.

Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, Luca Trevisan: From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More. https://arxiv.org/abs/1708.04218.

### Some (hopefully) easier papers:

Anupam Gupta, Euiwoong Lee, Jason Li: An FPT Algorithm Beating 2-Approximation for k-Cut.
https://arxiv.org/abs/1710.08488.

Samozřejmě, jako vždy, jsou vítany jsou i další náměty, zejména pak prezentace vlastních výsledků účastníků semináře.

### Další články, o kterých jsme uvažovali, zbylé z minulého semestru atd. [Additional proposed papers, leftovers from the last semester]

Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, Luca Trevisan: From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More. FOCS 2017. Also https://arxiv.org/abs/1708.04218.

Michael Lampis: Parameterized Approximation Schemes using Graph Widths. ICALP 2014. Also https://arxiv.org/abs/1311.2466.

Alice Paul, Daniel Freund, Aaron Ferber, David Shmoys and David Williamson: Prize-Collecting TSP with a Budget Constraint. ESA 2017.

Klaus Jansen and Lars Rohwedder: On the Configuration-LP of the Restricted Assignment Problem. SODA 2017: 2670-2678, also https://arxiv.org/abs/1611.01934.

Alantha Newman, Heiko Röglin, and Johanna Seif: The Alternating Stock Size Problem and the Gasoline Puzzle. ESA 2016. Also https://arxiv.org/abs/1511.09259.

János Balogh, József Békési, György Dósa, Leah Epstein, Asaf Levin: Online bin packing with cardinality constraints resolved. At https://arxiv.org/abs/1608.06415.

Harald Räcke: Optimal hierarchical decompositions for congestion minimization in networks. STOC 2008:255-264. Also here.

Jittat Fakcheroenphol, Kunal Talwar and Satish Rao: A tight bound on approximating arbitrary metrics by tree metrics. STOC 2003, J. Comput. Syst. Sci. 69(3): 485-497 (2004).