Search

Let B = (X , Y , E) be a bipartite graph. A half-square of B has one color class of B as vertex set, say X; two vertices are adjacent whenever they have a common neighbor in Y. Every planar graph is a half-square of a planar bipartite graph, namely of its subdivision. Until recently, only half-squares of planar bipartite graphs, also known as map graphs (Chen et al., in: Proceedings of the thirtieth annual ACM symposium on the theory of computing, STOC '98, pp 514–523. 10.1145/276698.276865, 1998; J ACM 49(2):127–138. 10.1145/506147.506148, 2002), have been investigated, and the most discussed problem is whether it is possible to recognize these graphs faster and simpler than Thorup's O (n 120) -time algorithm (Thorup, in: Proceedings of the 39th IEEE symposium on foundations of computer science (FOCS), pp 396–405. 10.1109/SFCS.1998.743490, 1998). In this paper, we identify the first hardness case, namely that deciding if a graph is a half-square of a balanced bisplit graph is NP-complete. (Balanced bisplit graphs form a proper subclass of star convex bipartite graphs). For classical subclasses of tree convex bipartite graphs such as biconvex, convex, and chordal bipartite graphs, we give good structural characterizations of their half-squares that imply efficient recognition algorithms. As a by-product, we obtain new characterizations of unit interval graphs, interval graphs, and of strongly chordal graphs in terms of half-squares of biconvex bipartite, convex bipartite, and of chordal bipartite graphs, respectively. Good characterizations of half-squares of star convex and star biconvex bipartite graphs are also given, giving linear-time recognition algorithms for these half-squares. [ABSTRACT FROM AUTHOR]

In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and an integer r > 1 , and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of r for maximum independent set in O ∗ (exp (O ~ (n / r log 2 r + r log 2 r))) time, r for chromatic number in O ∗ (exp (O ~ (n / r log r + r log 2 r))) time, (2 - 1 / r) for minimum vertex cover in O ∗ (exp (n / r Ω (r))) time, and (k - 1 / r) for minimum k-hypergraph vertex cover in O ∗ (exp (n / (k r) Ω (k r))) time. (Throughout, O ~ and O ∗ omit polyloglog (r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were O ∗ (2 n / r) (Bourgeois et al. in Discret Appl Math 159(17):1954–1970, 2011; Cygan et al. in Exponential-time approximation of hard problems, 2008). For maximum independent set and chromatic number, these bounds were complemented by exp (n 1 - o (1) / r 1 + o (1)) lower bounds (under the Exponential Time Hypothesis (ETH)) (Chalermsook et al. in Foundations of computer science, FOCS, pp. 370–379, 2013; Laekhanukit in Inapproximability of combinatorial problems in subexponential-time. Ph.D. thesis, 2014). Our results show that the naturally-looking O ∗ (2 n / r) bounds are not tight for all these problems. The key to these results is a sparsification procedure that reduces a problem to a bounded-degree variant, allowing the use of approximation algorithms for bounded-degree graphs. To obtain the first two results, we introduce a new randomized branching rule. Finally, we show a connection between PCP parameters and exponential-time approximation algorithms. This connection together with our independent set algorithm refute the possibility to overly reduce the size of Chan's PCP (Chan in J. ACM 63(3):27:1–27:32, 2016). It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture (Dinur in Electron Colloq Comput Complex (ECCC) 23:128, 2016; Manurangsi and Raghavendra in A birthday repetition theorem and complexity of approximating dense CSPs, 2016). [ABSTRACT FROM AUTHOR]

Online social networks have continued to attract increased attention since the introduction of this concept nearly three decades ago. Consequently, a study about the workings of online social networks may help in understanding the structure of human society and the characteristics of generic complex networks. Over the past few years, interest in neighboring nodes, which are the number of nodes between any two nodes, the number of neighbors and how many triangles are present in a social network have produced the concepts of a six-degree separation (Milgram, Psychol Today 2(1):60–67, 1967; Backstrom et al., Proceedings of the 4th annual ACM web science conference, p 33–42, 2012), a heavy tail in the degree distribution (Barabási, Albert, Science 286(5439):509–512, 1999) and a high clustering coefficient (Luce, Perry, Psychometrika 14(2):95–116, 1949; Watts, Strogatz, Nature 393(6684):440–442, 1998; Amaral et al., Proc Natl Acad Sci USA 97(21):11149–11152, 2000). In a similar manner, researchers have also been curious about how many polygons are present in a given online social network. Although much effort has been expended (Dantzig et al., International symposium on theory of graphs, p 77–83, 1967; Kamae, IEEE Trans Circuit Theory 14(2):166–171, 1967; Gotlieb, Corneil, Commun ACM 10(12):780–783, 1967; Welch, J ACM 13(2):205–210, 1966; Tiernan, Commun ACM 13(12):722–726, 1970; Tarjan, SIAM J Comput 2(3):211–216, 1973; Johnson, SIAM J Comput 4:77–84, 1975; Mateti, Deo, SIAM J Comput 5(5):90–99, 1976; Marinari et al., Europhys Lett 73(3):301–307, 2005), studying this subject, the inability to enumerate polygons has stymied an in depth understanding of the properties of polygons in an online social network. In the study described in this paper, the estimated number of polygons in an online social network is revealed. It was found that in the current widely used online social networks, e.g., Facebook, Twitter, the number of polygons increases drastically when the length of a polygon is below a set value and then it decreases rapidly. The average length of the network polygons was calculated and it was found that online social networks contain a relatively large average length of polygons. Based on this perspective, a massive labyrinth of polygons would make the online social networks appear to be very complicated. To further investigate this area, a generalized clustering coefficient was explored. Results showed that the generalized clustering coefficient appeared to descend exponentially with the length of polygon and expeditiously approached zero. This result suggested that the polygons with large lengths should be ignored in many scenarios. Since the polygons with lengths greater than five appeared to have little impact on the network, the online social networks appeared to be less complex than anticipated. The polygon is one of the fundamental problems in graph theory and complex networks, so that the work reported here may be beneficial for many disciplines, including transportation [7], engineering [8], computer science [9–16], physics (Birmelé et al., Proceedings of the 24th annual ACM–SIAM symposium on discrete algorithms, p 1884–1896, 2013), sociology (Motter, Albert, Phys Today 65:43, 2012), epidemiology (Feld, Am J Sociol 96(6):1464–1477, 1991; Cohen et al., Phys Rev Lett 91(24):12343, 2002), psychology (Sun et al., Sci Rep 4(6188):5099–5099, 2014), biology (Feiler, Kleinbaum, Psychol Sci, 2015), medicine (Kincaid, Pilette, Comput Appl Biosci Cabios 8:267–273, 1992), geography (Kim et al., Lancet 386(9989):145–153, 2015), etc. [ABSTRACT FROM AUTHOR]

Within the context of Rogers theory of perceived attributes, authors propose a framework that can predict students’ motivation to adopt programming contest like ACM (Association of Computing Machinery) International Collegiate Programming Contest (ICPC). In this paper we investigate the attributes for adoption of programming contest in a social group comprising of (N = 1245) undergraduate engineering students from the regional finals of the contest held in India over a period of 3 years. The results revealed that student motivations are strongly associated with attributes like relative advantage, compatibility, ease of use, peer influence, perceived enjoyment and perceived usefulness. Overall, students expressed positive attitude towards adopting programming contests as it helped improve their problem solving and programming skills and overall employability. Both gender was in agreement that joining and winning programming contests is a status symbol. [ABSTRACT FROM AUTHOR]

Lifted Reed–Solomon codes, introduced by Guo et al. (in: Kleinberg (ed) Proceedings of the 4th conference on innovations in theoretical computer science, ITCS'13, ACM, New York, 2013), are known as one of the few families of high-rate locally correctable codes. They are built through the evaluation over the affine space of multivariate polynomials whose restriction along any affine line can be interpolated as a low degree univariate polynomial. In this work, we give a formal definition of their analogues over projective spaces, and we study some of their parameters and features. Local correcting algorithms are first derived from the very nature of these codes, generalizing the well-known local correcting algorithms for Reed–Muller codes. We also prove that the lifting of both Reed–Solomon and projective Reed–Solomon codes are deeply linked through shortening and puncturing operations. It leads to recursive formulae on their dimension and their monomial bases. We finally emphasize the practicality of lifted projective Reed–Solomon codes by computing their information sets and by providing an implementation of the codes and their local correcting algorithms. [ABSTRACT FROM AUTHOR]