Many-to-one matchings with lower quotas : algorithms and complexity

We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G = (AUP,E) with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-toone matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-toone matching with lower and upper quotas (wmlq), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of wmlq from the viewpoints of classic polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between NP-hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota umax as basis, and we prove that this dependence is necessary unless FPT = W[1]. Finally, we also present an approximation algorithm for the general case with performance guarantee umax+1, which is asymptotically best possible unless P = NP

On the Trace of the Real Author

In pre-Revival Croatian literature there are works that so far have not been ascribed to any particular author. It is now clear that their real authors can not be identified simplyon the basis of general stylistic impression, as the late 19th century scholar Armin Pavić believed. The approach of Kolendić, who at the start of this century introduced the method of hapaxes (words evidenced only in the corpus of one known author) seemed much more promising. Trying to prove Vetranović‚s authorship of a part of the mythological drama Orfeo, he pointed out several words for which he claimed to be the hapaxes of the said poet. Even if the tenability of his conclusions about the Orfeo can be easily dismissed simply by using the Historical Dictionary of Croatian Language, the national literary historiography has accepted Kolendić‚s attribution. However, another attribution, based on the same method and proposed by the author of the present article, was rejected.Namely, after having found hapaxes of Zoranić‚s Planine in a pastoral eclogue by an unknown author, he attributed the eclogue to the same poet. The conclusion is self-evident. Every new method should be thoroughly tested. But, if no objection is found, in the following period it must be valid for all the cases, wherever it can be competently applied

Approximation algorithms for general cluster routing problem

Graph routing problems have been investigated extensively in operations research, computer science and engineering due to their ubiquity and vast applications. In this paper, we study constant approximation algorithms for some variations of the general cluster routing problem. In this problem, we are given an edge-weighted complete undirected graph $G=(V,E,c),$ whose vertex set is partitioned into clusters $C_{1},\dots ,C_{k}.$ We are also given a subset $V'$ of $V$ and a subset $E'$ of $E.$ The weight function $c$ satisfies the triangle inequality. The goal is to find a minimum cost walk $T$ that visits each vertex in $V'$ only once, traverses every edge in $E'$ at least once and for every $i\in [k]$ all vertices of $C_i$ are traversed consecutively.Comment: In COCOON 202

Travelling on Graphs with Small Highway Dimension

We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter was introduced by Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP and STP naturally occur for various applications in logistics. It was previously shown [Feldmann et al. ICALP 2015] that these problems admit a quasi-polynomial time approximation scheme (QPTAS) on graphs of constant highway dimension. We demonstrate that a significant improvement is possible in the special case when the highway dimension is 1, for which we present a fully-polynomial time approximation scheme (FPTAS). We also prove that STP is weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for graphs of highway dimension 6, which answers an open problem posed in [Feldmann et al. ICALP 2015]

An excluded minor characterization of Seymour graphs,

International audienc

An excluded minor characterization of Seymour graphs,

International audienc