Let H=(V,E) be a hypergraph with n vertices and m edges. A subset S⊆V is an eliminating vertex feedback set of H if Image 1 contains no cycles, and we denote its minimum cardinality by Dc(H). Analogously, a subset A⊆E is called an eliminating edge feedback set of H if Image 2 contains no cycles, and its minimum cardinality is denoted by Dc′(H). In this paper, we establish upper bounds for both parameters. For the vertex version, we prove that Dc(H)≤(k−1)n−m+1 for any k-regular connected hypergraph H. For the edge version, we show that: (i) Dc′(H)≤n/2 for any hypergraph H; (ii) Dc′(H)≤n/4 for any 2-regular hypergraph H with girth at least 4; and (iii) Dc′(H)≤n/3 for any 2-regular or 3-regular hypergraph H with girth at least 3. Furthermore, we characterize the extremal regular hypergraphs attaining these upper bounds. Based on our proofs, we also design combinatorial algorithms for computing the eliminating feedback numbers.
We propose and study the diversity-aware l-centrum (div-l-centrum) problem, where we are given a set of clients and a set of facilities in a metric space, integers k and l, and r[i] (i∈[t]) for each facility group. The diversity constraints refer to the requirement that each group must include at least r[i] open facilities. The goal is to find a subset of facilities of size at most k so as to minimize the sum of the l largest distances from a client to its nearest open facility, while satisfying the diversity constraints. As our main contribution, we provide a fixed-parameter tractable (1+2/e+ε)-approximation algorithm that improves upon the previous (3+ε) for the div-l-centrum problem, based on a combination of leader selection, coreset construction and submodular optimization techniques.
For a graph G=(V,E), let Gn denote the graph whose vertex set is Vn in which two distinct vertices (u1,u2,…,un) and (v1,v2,…,vn) are adjacent, if and only if for all 1≤i≤n, either ui=vi or uivi∈E. The Shannon capacity of G is defined to be c(G)=supnα(Gn)1n, where α(Gn) is the maximum size of an independent set of vertices in Gn. In this paper, we prove that α(G⊠H)=α(G)α(H) for any connected graph G and connected bipartite graph H. We prove that for any graph G and bipartite graph H (or point-symmetric H which satisfies that α(H)ω(H)=|V(H)|), if the Shannon capacity of G is finite, then the Shannon capacity of G⊠H is finite. The r-color Ramsey number R(k1,…,kr) is defined as the smallest positive integer N such that in every edge-coloring of KN by r colors there is a monochromatic copy of Kki in color i for some 1≤i≤k. In 2002, Robertson obtained a lower bound on the r-color Ramsey number R(k1,k2,…,kr). We establish a new lower bound on the r-color Ramsey number of graphs. From this, we can improves the theorem of Robertson for some cases. A specific result improves the inequality R(7,9,7,7)≥5384 to that R(7,9,7,7)≥64261.
The d-Hitting Set problem is a fundamental problem in parameterized complexity, which asks whether a given hypergraph contains a vertex subset S of size at most k that intersects every hyperedge (i.e., S∩e≠∅ for each hyperedge e). The best known kernel for this problem, established by Abu-Khzam (JCSS 2010), has (2d−1)kd−1+k vertices. This result has been very widely used in the literature as many problems can be modeled as a special d-Hitting Set problem. In this work, we present a refinement to this result by employing linear programming techniques to construct crown decompositions in hypergraphs. This approach yields a slight but notable improvement, reducing the size to (2d−2)kd−1+k vertices.
We consider a language together with the subword relation, binary relations that relate the lengths of two words, and regular predicates. For such structures, we consider the extension of first-order logic by threshold and modulo counting quantifiers. Depending on the language, the used predicates, and the fragment of the logic, we determine four new combinations that yield decidable theories. These results extend earlier ones where only the language of all words without the length relations and fragments of first-order logic were considered.
