On Approximability of Bounded Degree Instances of Selected Optimization Problems

In order to cope with the approximation hardness of an underlying optimization problem, it is advantageous to consider specific families of instances with properties that can be exploited to obtain efficient approximation algorithms for the restricted version of the problem with improved performance guarantees. In this thesis, we investigate the approximation complexity of selected NP-hard optimization problems restricted to instances with bounded degree, occurrence or weight parameter. Specifically, we consider the family of dense instances, where typically the average degree is bounded from below by some function of the size of the instance. Complementarily, we examine the family of sparse instances, in which the average degree is bounded from above by some fixed constant. We focus on developing new methods for proving explicit approximation hardness results for general as well as for restricted instances. The fist part of the thesis contributes to the systematic investigation of the VERTEX COVER problem in k-hypergraphs and k-partite k-hypergraphs with density and regularity constraints. We design efficient approximation algorithms for the problems with improved performance guarantees as compared to the general case. On the other hand, we prove the optimality of our approximation upper bounds under the Unique Games Conjecture or a variant. In the second part of the thesis, we study mainly the approximation hardness of restricted instances of selected global optimization problems. We establish improved or in some cases the first inapproximability thresholds for the problems considered in this thesis such as the METRIC DIMENSION problem restricted to graphs with maximum degree 3 and the (1,2)-STEINER TREE problem. We introduce a new reductions method for proving explicit approximation lower bounds for problems that are related to the TRAVELING SALESPERSON (TSP) problem. In particular, we prove the best up to now inapproximability thresholds for the general METRIC TSP problem, the ASYMMETRIC TSP problem, the SHORTEST SUPERSTRING problem, the MAXIMUM TSP problem and TSP problems with bounded metrics

Combinatorics

Combinatorics is a fundamental mathematical discipline which focuses on the study of discrete objects and their properties. The current workshop brought together researchers from diverse fields such as Extremal and Probabilistic Combinatorics, Discrete Geometry, Graph theory, Combinatorial Optimization and Algebraic Combinatorics for a fruitful interaction. New results, methods and developments and future challenges were discussed. This is a report on the meeting containing abstracts of the presentations and a summary of the problem session

Towards Ryser\u27s Conjecture: Bounds on the Cardinality of Partitioned Intersecting Hypergraphs

This work is motivated by the open conjecture concerning the size of a minimum vertex cover in a partitioned hypergraph. In an r-uniform r-partite hypergraph, the size of the minimum vertex cover C is conjectured to be related to the size of its maximum matching M by the relation (|C|\u3c= (r-1)|M|). In fact it is not known whether this conjecture holds when |M| = 1. We consider r-partite hypergraphs with maximal matching size |M| = 1, and pose a novel algorithmic approach to finding a vertex cover of size (r - 1) in this case. We define a reactive hypergraph to be a back-and-forth algorithm for a hypergraph which chooses new edges in response to a choice of vertex cover, and prove that this algorithm terminates for all hypergraphs of orders r = 3 and 4. We introduce the idea of optimizing the size of the reactive hypergraph and find that the reactive hypergraph terminates for r = 5...20. We then consider the case where the intersection of any two edges is exactly 1. We prove bounds on the size of this 1-intersecting hypergraph and relate the 1-intersecting hypergraph maximization problem to mutually orthogonal Latin squares. We propose a generative algorithm for 1-intersecting hypergraphs of maximal size for prime powers r-1 = pd under the constraint pd+1 is also a prime power of the same form, and therefore pose a new generating algorithm for MOLS based upon intersecting hypergraphs. We prove this algorithm generates a valid set of mutually orthogonal Latin squares and prove the construction guarantees certain symmetric properties. We conclude that a conjecture by Lovasz, that the inequality in Ryser\u27s Conjecture cannot be improved when (r-1) is a prime power, is correct for the 1-intersecting hypergraph of prime power orders

Combinatorics and Probability

The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. Both themes were richly represented at the workshop, with many recent exciting results presented by the lecturers

Graph removal lemmas

The graph removal lemma states that any graph on n vertices with o(n^{v(H)}) copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.Comment: 35 page

Hypergraph matchings and designs

We survey some aspects of the perfect matching problem in hypergraphs, with particular emphasis on structural characterisation of the existence problem in dense hypergraphs and the existence of designs.Comment: 19 pages, for the 2018 IC

Hardness of Rainbow Coloring Hypergraphs

A hypergraph is k-rainbow colorable if there exists a vertex coloring using k colors such that each hyperedge has all the k colors. Unlike usual hypergraph coloring, rainbow coloring becomes harder as the number of colors increases. This work studies the rainbow colorability of hypergraphs which are guaranteed to be nearly balanced rainbow colorable. Specifically, we show that for any Q,k >= 2 and ell <= k/2, given a Qk-uniform hypergraph which admits a k-rainbow coloring satisfying: - in each hyperedge e, for some ell_e <= ell all but 2ell_e colors occur exactly Q times and the rest (Q +/- 1) times, it is NP-hard to compute an independent set of (1 - (ell+1)/k + eps)-fraction of vertices, for any constant eps > 0. In particular, this implies the hardness of even (k/ell)-rainbow coloring such hypergraphs. The result is based on a novel long code PCP test that ensures the strong balancedness property desired of the k-rainbow coloring in the completeness case. The soundness analysis relies on a mixing bound based on uniform reverse hypercontractivity due to Mossel, Oleszkiewicz, and Sen, which was also used in earlier proofs of the hardness of omega(1)-coloring 2-colorable 4-uniform hypergraphs due to Saket, and k-rainbow colorable 2k-uniform hypergraphs due to Guruswami and Lee

Improved Bounds for Point Selections and Halving Hyperplanes in Higher Dimensions

Let $(P,E)$ be a $(d+1)$-uniform geometric hypergraph, where $P$ is an $n$-point set in general position in $\mathbb{R}^d$ and $E\subseteq {P\choose d+1}$ is a collection of $\epsilon{n\choose d+1}$ $d$-dimensional simplices with vertices in $P$, for $0<\epsilon\leq 1$. We show that there is a point $x\in {\mathbb R}^d$ that pierces $\displaystyle \Omega\left(\epsilon^{(d^4+d)(d+1)+\delta}{n\choose d+1}\right)$ simplices in $E$, for any fixed $\delta>0$. This is a dramatic improvement in all dimensions $d\geq 3$, over the previous lower bounds of the general form $\displaystyle \epsilon^{(cd)^{d+1}}n^{d+1}$, which date back to the seminal 1991 work of Alon, B\'{a}r\'{a}ny, F\"{u}redi and Kleitman. As a result, any $n$-point set in general position in $\mathbb{R}^d$ admits only $\displaystyle O\left(n^{d-\frac{1}{d(d-1)^4+d(d-1)}+\delta}\right)$ halving hyperplanes, for any $\delta>0$, which is a significant improvement over the previously best known bound $\displaystyle O\left(n^{d-\frac{1}{(2d)^{d}}}\right)$ in all dimensions $d\geq 5$. An essential ingredient of our proof is the following semi-algebraic Tur\'an-type result of independent interest: Let $(V_1,\ldots,V_k,E)$ be a hypergraph of bounded semi-algebraic description complexity in ${\mathbb R}^d$ that satisfies $|E|\geq \varepsilon |V_1|\cdot\ldots \cdot |V_k|$ for some $\varepsilon>0$. Then there exist subsets $W_i\subseteq V_i$ that satisfy $W_1\times W_2\times\ldots\times W_k\subseteq E$, and $|W_1|\cdot\ldots\cdots|W_k|=\Omega\left(\varepsilon^{d(k-1)+1}|V_1|\cdot |V_2|\cdot\ldots\cdot|V_k|\right)$.Comment: A preliminary version has appeared in the Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA