Blockers and transversals

Given an undirected graph G=(V,E) with matching number \nu(G), we define d- blockers as subsets of edges B such that \nu(G=(V,E\B))\leq \nu(G)-d. We define d- transversals T as subsets of edges such that every maximum matching M has |M\cap T|\geq d. We explore connections between d-blockers and d-transversals. Special classes of graphs are examined which include complete graphs, regular bipartite graphs, chains and cycles and we construct minimum d-transversals and d-blockers in these special graphs. We also study the complexity status of finding minimum transversals and blockers in arbitrary graphs

Santa Claus, Machine Scheduling and Bipartite Hypergraphs

Here we discuss two related discrete optimization problems, a prominent problem in scheduling theory, makespan minimization on unrelated parallel machines, and the other a fair allocation problem, the Santa Claus problem. In each case the objective is to make the least well off participant as well off as possible, and in each case we have complexity results that bound how close we may estimate optimal values of worst-case instances in polytime. We explore some of the techniques that have been used in obtaining approximation algorithms or optimal value guarantees for these problems, as well as those involved in getting hardness results, emphasizing the relationships between the problems. A framework for decisional variants of approximation and optimal value estimation for optimization problems is introduced to clarify the discussion. Also discussed are bipartite hypergraphs, which correspond naturally to these problems, including a discussion of Haxell's Theorem for bipartite hypergraphs. Conditions for edge covering in bipartite hypergraphs are introduced and their implications investigated. The conditions are motivated by analogy to Haxell's Theorem and from generalizing conditions that arose from bipartite hypergraphs associated with machine scheduling

Generating clause sequences of a CNF formula

Given a CNF formula $\Phi$ with clauses $C_1,\ldots,C_m$ and variables $V=\{x_1,\ldots,x_n\}$, a truth assignment $a:V\rightarrow\{0,1\}$ of $\Phi$ leads to a clause sequence $\sigma_\Phi(a)=(C_1(a),\ldots,C_m(a))\in\{0,1\}^m$ where $C_i(a) = 1$ if clause $C_i$ evaluates to $1$ under assignment $a$, otherwise $C_i(a) = 0$. The set of all possible clause sequences carries a lot of information on the formula, e.g. SAT, MAX-SAT and MIN-SAT can be encoded in terms of finding a clause sequence with extremal properties. We consider a problem posed at Dagstuhl Seminar 19211 "Enumeration in Data Management" (2019) about the generation of all possible clause sequences of a given CNF with bounded dimension. We prove that the problem can be solved in incremental polynomial time. We further give an algorithm with polynomial delay for the class of tractable CNF formulas. We also consider the generation of maximal and minimal clause sequences, and show that generating maximal clause sequences is NP-hard, while minimal clause sequences can be generated with polynomial delay.Comment: 9 page

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum