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1. Parameterized Complexity of Fair Bisection: (FPT-Approximation meets Unbreakability)

   https://doi.org/10.4230/LIPIcs.ESA.2023.63  

   Inamdar, Tanmay; Lokshtanov, Daniel; Saurabh, Saket; Surianarayanan, Vaishali (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Gørtz, Inge Li; Farach-Colton, Martin; Puglisi, Simon J; Herman, Grzegorz (Ed.)

   In the Minimum Bisection problem input is a graph G and the goal is to partition the vertex set into two parts A and B, such that ||A|-|B|| ≤ 1 and the number k of edges between A and B is minimized. The problem is known to be NP-hard, and assuming the Unique Games Conjecture even NP-hard to approximate within a constant factor [Khot and Vishnoi, J.ACM'15]. On the other hand, a 𝒪(log n)-approximation algorithm [Räcke, STOC'08] and a parameterized algorithm [Cygan et al., ACM Transactions on Algorithms'20] running in time k^𝒪(k) n^𝒪(1) is known. The Minimum Bisection problem can be viewed as a clustering problem where edges represent similarity and the task is to partition the vertices into two equally sized clusters while minimizing the number of pairs of similar objects that end up in different clusters. Motivated by a number of egregious examples of unfair bias in AI systems, many fundamental clustering problems have been revisited and re-formulated to incorporate fairness constraints. In this paper we initiate the study of the Minimum Bisection problem with fairness constraints. Here the input is a graph G, positive integers c and k, a function χ:V(G) → {1, …, c} that assigns a color χ(v) to each vertex v in G, and c integers r_1,r_2,⋯,r_c. The goal is to partition the vertex set of G into two almost-equal sized parts A and B with at most k edges between them, such that for each color i ∈ {1, …, c}, A has exactly r_i vertices of color i. Each color class corresponds to a group which we require the partition (A, B) to treat fairly, and the constraints that A has exactly r_i vertices of color i can be used to encode that no group is over- or under-represented in either of the two clusters. We first show that introducing fairness constraints appears to make the Minimum Bisection problem qualitatively harder. Specifically we show that unless FPT=W[1] the problem admits no f(c)n^𝒪(1) time algorithm even when k = 0. On the other hand, our main technical contribution shows that is that this hardness result is simply a consequence of the very strict requirement that each color class i has exactly r_i vertices in A. In particular we give an f(k,c,ε)n^𝒪(1) time algorithm that finds a balanced partition (A, B) with at most k edges between them, such that for each color i ∈ [c], there are at most (1±ε)r_i vertices of color i in A. Our approximation algorithm is best viewed as a proof of concept that the technique introduced by [Lampis, ICALP'18] for obtaining FPT-approximation algorithms for problems of bounded tree-width or clique-width can be efficiently exploited even on graphs of unbounded width. The key insight is that the technique of Lampis is applicable on tree decompositions with unbreakable bags (as introduced in [Cygan et al., SIAM Journal on Computing'14]). An important ingredient of our approximation scheme is a combinatorial result that may be of independent interest, namely that for every k, every graph G admits a tree decomposition with adhesions of size at most 𝒪(k), unbreakable bags, and logarithmic depth. 

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2. Sharp bounds for decomposing graphs into edges and triangles

   https://doi.org/10.1017/S0963548320000358  

   Blumenthal, Adam; Lidický, Bernard; Pehova, Yanitsa; Pfender, Florian; Pikhurko, Oleg; Volec, Jan (March 2021, Combinatorics, Probability and Computing)

   null (Ed.)

   Abstract For a real constant α , let $$\pi _3^\alpha (G)$$ be the minimum of twice the number of K 2 ’s plus α times the number of K 3 ’s over all edge decompositions of G into copies of K 2 and K 3 , where K r denotes the complete graph on r vertices. Let $$\pi _3^\alpha (n)$$ be the maximum of $$\pi _3^\alpha (G)$$ over all graphs G with n vertices. The extremal function $$\pi _3^3(n)$$ was first studied by Győri and Tuza ( Studia Sci. Math. Hungar. 22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova ( Combin. Probab. Comput. 28 (2019) 465–472) proved via flag algebras that $$\pi _3^3(n) \le (1/2 + o(1)){n^2}$$ . We extend their result by determining the exact value of $$\pi _3^\alpha (n)$$ and the set of extremal graphs for all α and sufficiently large n . In particular, we show for α = 3 that K n and the complete bipartite graph $${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$$ are the only possible extremal examples for large n . 

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3. Hermite Rational Function Interpolation with Error Correction

   https://doi.org/10.1007/978-3-030-60026-6_19  

   Kaltofen, Erich L.; Pernet, Clement; Yang, Zhi-Hong (September 2020, Proc. Computer Algebra in Scientific Computing (CASC) 2020)

   We generalize Hermite interpolation with error correction, which is the methodology for multiplicity algebraic error correction codes, to Hermite interpolation of a rational function over a field K from function and function derivative values. We present an interpolation algorithm that can locate and correct <= E errors at distinct arguments y in K where at least one of the values or values of a derivative is incorrect. The upper bound E for the number of such y is input. Our algorithm sufficiently oversamples the rational function to guarantee a unique interpolant. We sample (f/g)^(j)(y[i]) for 0 <= j <= L[i], 1 <= i <= n, y[i] distinct, where (f/g)^(j) is the j-th derivative of the rational function f/g, f, g in K[x], GCD(f,g)=1, g <= 0, and where N = (L[1]+1)+...+(L[n]+1) >= C + D + 1 + 2(L[1]+1) + ... + 2(L[E]+1) where C is an upper bound for deg(f) and D an upper bound for deg(g), which are input to our algorithm. The arguments y[i] can be poles, which is truly or falsely indicated by a function value infinity with the corresponding L[i]=0. Our results remain valid for fields K of characteristic >= 1 + max L[i]. Our algorithm has the same asymptotic arithmetic complexity as that for classical Hermite interpolation, namely soft-O(N). For polynomials, that is, g=1, and a uniform derivative profile L[1] = ... = L[n], our algorithm specializes to the univariate multiplicity code decoder that is based on the 1986 Welch-Berlekamp algorithm. 

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4. Making K _r+1 -free graphs r -partite

   https://doi.org/10.1017/S0963548320000590  

   Balogh, József; Clemen, Felix Christian; Lavrov, Mikhail; Lidický, Bernard; Pfender, Florian (July 2021, Combinatorics, Probability and Computing)

   null (Ed.)

   Abstract The Erdős–Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if G is a K r+ 1 -free graph on n vertices with e ( G ) > ex( n , K r +1 )– α n 2 , then one can remove εn 2 edges from G to obtain an r -partite graph. Füredi gave a short proof that one can choose α = ε . We give a bound for the relationship of α and ε which is asymptotically sharp as ε → 0. 

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5. Tight paths in convex geometric hypergraphs

   https://doi.org/10.19086/aic.12044  

   Mubayi, Dhruv; Füredi, Zoltán; Verstraëte, Jacques; Kostochka, Alexandr; Jiang, Tao (February 2020, Advances in Combinatorics)

   One of the most intruguing conjectures in extremal graph theory is the conjecture of Erdős and Sós from 1962, which asserts that every $$n$$-vertex graph with more than $$\frac{k-1}{2}n$$ edges contains any $$k$$-edge tree as a subgraph. Kalai proposed a generalization of this conjecture to hypergraphs. To explain the generalization, we need to define the concept of a tight tree in an $$r$$-uniform hypergraph, i.e., a hypergraph where each edge contains $$r$$ vertices. A tight tree is an $$r$$-uniform hypergraph such that there is an ordering $$v_1,\ldots,v_n$$ of its its vertices with the following property: the vertices $$v_1,\ldots,v_r$$ form an edge and for every $i>r$, there is a single edge $$e$$ containing the vertex $$v_i$$ and $r-1$ of the vertices $$v_1,\ldots,v_{i-1}$$, and $$e\setminus\{v_i\}$$ is a subset of one of the edges consisting only of vertices from $$v_1,\ldots,v_{i-1}$$. The conjecture of Kalai asserts that every $$n$$-vertex $$r$$-uniform hypergraph with more than $$\frac{k-1}{r}\binom{n}{r-1}$$ edges contains every $$k$$-edge tight tree as a subhypergraph. The recent breakthrough results on the existence of combinatorial designs by Keevash and by Glock, Kühn, Lo and Osthus show that this conjecture, if true, would be tight for infinitely many values of $$n$$ for every $$r$$ and $$k$$.The article deals with the special case of the conjecture when the sought tight tree is a path, i.e., the edges are the $$r$$-tuples of consecutive vertices in the above ordering. The case $r=2$ is the famous Erdős-Gallai theorem on the existence of paths in graphs. The case $r=3$ and $k=4$ follows from an earlier work of the authors on the conjecture of Kalai. The main result of the article is the first non-trivial upper bound valid for all $$r$$ and $$k$$. The proof is based on techniques developed for a closely related problem where a hypergraph comes with a geometric structure: the vertices are points in the plane in a strictly convex position and the sought path has to zigzag beetwen the vertices. 

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