Computer Science

Comparing alternatives in pairs is a very well known technique of ranking creation. The answer to how reliable and trustworthy ranking is depends on the inconsistency of the data from which it was created. There are many indices used for determining the level of inconsistency among compared alternatives. Unfortunately, most of them assume that the set of comparisons is complete, i.e. every single alternative is compared to each other. This is not true and the ranking must sometimes be made based on incomplete data. In order to fill this gap, this work aims to adapt the selected twelve existing inconsistency indices for the purpose of analyzing incomplete data sets. The modified indices are subjected to Monte Carlo experiments. Those of them that achieved the best results in the experiments carried out are recommended for use in practice.

This paper aims at comparing two coupling approaches as basic layers for building clustering criteria, suited for modularizing and clustering very large networks. We briefly use "optimal transport theory" as a starting point, and a way as well, to derive two canonical couplings: "statistical independence" and "logical indetermination". A symmetric list of properties is provided and notably the so called "Monge's properties", applied to contingency matrices, and justifying the ⊗ versus ⊕ notation. A study is proposed, highlighting "logical indetermination", because it is, by far, lesser known. Eventually we estimate the average difference between both couplings as the key explanation of their usually close results in network clustering.

A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S . If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G . In 2013 Goddard and Henning [Discrete Math 313 (2013), 839--854] conjectured that if G is a connected cubic graph of order n , then i(G)≤ 3 8 n , except if G is the complete bipartite graph K 3,3 or the 5 -prism C 5 □ K 2 . Further they construct two infinite families of connected cubic graphs with independent domination three-eighths their order. They remark that perhaps it is even true that for n>10 these two families are only families for which equality holds. In this paper, we provide a new family of connected cubic graphs G of order n such that i(G)= 3 8 n . We also show that if G is a subcubic graph of order n with no isolated vertex, then i(G)≤ 1 2 n , and we characterize the graphs achieving equality in this bound.

The Maximum Weight Independent Set Problem (WIS) is a well-known NP-hard problem. A popular way to study WIS is to detect graph classes for which WIS can be solved in polynomial time, with particular reference to hereditary graph classes, i.e., defined by a hereditary graph property or equivalently by forbidding one or more induced subgraphs. Given two graphs G and H , G+H denotes the disjoint union of G and H . This manuscript shows that (i) WIS can be solved for ( P 4 + P 4 , Triangle)-free graphs in polynomial time, where a P 4 is an induced path of four vertices and a Triangle is a cycle of three vertices, and that in particular it turns out that (ii) for every ( P 4 + P 4 , Triangle)-free graph G there is a family S of subsets of V(G) inducing (complete) bipartite subgraphs of G , which contains polynomially many members and can be computed in polynomial time, such that every maximal independent set of G is contained in some member of S . These results seem to be harmonic with respect to other polynomial results for WIS on certain [subclasses of] S i,j,k -free graphs and to other structure results on [subclasses of] Triangle-free graphs.

In this paper independent sets of closure operations are introduced. We characterize minimal keys and antikeys of closure operations in terms of independent sets. We establish an expression on the connection between minimal keys and antikeys of closure operations based on independent sets. We construct two combinatorial algorithms for finding all minimal keys and all antikeys of a given closure operation based on independent sets. We estimate the time complexity of these algorithms. Finally, we give an NP-complete problem concerning nonkeys of closure operations.

The \emph{induced odd cycle packing number} iocp(G) of a graph G is the maximum integer k such that G contains an induced subgraph consisting of k pairwise vertex-disjoint odd cycles. Motivated by applications to geometric graphs, Bonamy et al. proved that graphs of bounded induced odd cycle packing number, bounded VC dimension, and linear independence number admit a randomized EPTAS for the independence number. We show that the assumption of bounded VC dimension is not necessary, exhibiting a randomized algorithm that for any integers k≥0 and t≥1 and any n -vertex graph G of induced odd cycle packing number returns in time O k,t ( n k+4 ) an independent set of G whose size is at least α(G)−n/t with high probability. In addition, we present χ -boundedness results for graphs with bounded odd cycle packing number, and use them to design a QPTAS for the independence number only assuming bounded induced odd cycle packing number.

Random factor graphs provide a powerful framework for the study of inference problems such as decoding problems or the stochastic block model. Information-theoretically the key quantity of interest is the mutual information between the observed factor graph and the underlying ground truth around which the factor graph was created; in the stochastic block model, this would be the planted partition. The mutual information gauges whether and how well the ground truth can be inferred from the observable data. For a very general model of random factor graphs we verify a formula for the mutual information predicted by physics techniques. As an application we prove a conjecture about low-density generator matrix codes from [Montanari: IEEE Transactions on Information Theory 2005]. Further applications include phase transitions of the stochastic block model and the mixed k -spin model from physics.

In the group testing problem we aim to identify a small number of infected individuals within a large population. We avail ourselves to a procedure that can test a group of multiple individuals, with the test result coming out positive iff at least one individual in the group is infected. With all tests conducted in parallel, what is the least number of tests required to identify the status of all individuals? In a recent test design [Aldridge et al.\ 2016] the individuals are assigned to test groups randomly, with every individual joining an equal number of groups. We pinpoint the sharp threshold for the number of tests required in this randomised design so that it is information-theoretically possible to infer the infection status of every individual. Moreover, we analyse two efficient inference algorithms. These results settle conjectures from [Aldridge et al.\ 2014, Johnson et al.\ 2019].

We consider a restricted game on weighted graphs associated with minimum partitions. We replace in the classical definition of Myerson restricted game the connected components of any subgraph by the sub-components obtained with a specific partition P ~ min . This partition relies on the same principle as the partition P min introduced by Grabisch and Skoda (2012) but restricted to connected coalitions. More precisely, this new partition P ~ min is induced by the deletion of the minimum weight edges in each connected component associated with a coalition. We provide a characterization of the graphs satisfying inheritance of convexity from the underlying game to the restricted game associated with P ~ min .

A graph is Helly if its disks satisfy the Helly property, i.e., every family of pairwise intersecting disks in G has a common intersection. It is known that for every graph G, there exists a unique smallest Helly graph H(G) into which G isometrically embeds; H(G) is called the injective hull of G. Motivated by this, we investigate the structural properties of the injective hulls of various graph classes. We say that a class of graphs C is closed under Hellification if G∈C implies H(G)∈C . We identify several graph classes that are closed under Hellification. We show that permutation graphs are not closed under Hellification, but chordal graphs, square-chordal graphs, and distance-hereditary graphs are. Graphs that have an efficiently computable injective hull are of particular interest. A linear-time algorithm to construct the injective hull of any distance-hereditary graph is provided and we show that the injective hull of several graphs from some other well-known classes of graphs are impossible to compute in subexponential time. In particular, there are split graphs, cocomparability graphs, bipartite graphs G such that H(G) contains Ω( a n ) vertices, where n=|V(G)| and a>1 .