Notes and Computations on Forbidden Differences

In this paper, we study the problem of designing in-place algorithms for finding the maximum clique in the intersection graphs of axis-parallel rectangles and disks in 2D. We first propose O(n 2logn) time in-place algorithms for finding the maximum clique of the intersection graphs of a set of axis-parallel rectangles of arbitrary sizes. For the rectangle intersection graph of fixed height rectangles, the time complexity can be slightly improved to O(nlogn + nK), where K is the size of the maximum clique. For disk graphs, we consider two variations of the maximum clique problem, namely geometric clique and graphical clique. The time complexity of our algorithm for finding the largest geometric clique is O(n 2logn), and it works for disks of arbitrary radii. For graphical clique, our proposed algorithm works for unit disks (i.e., of same radii) and the worst case time complexity is O(n 2 + mK 4); m is the number of edges in the unit disk intersection graph, and K is the size of the largest clique in that graph. It uses O(n 4) time in-place computation of maximum matching in a bipartite graph, which is of independent interest. All these algorithms need O(1) work space in addition to the input array \(\cal R\).KeywordsBipartite GraphUnit DiskBottom BoundaryMaximum CliqueIntersection GraphThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The goore game (GG) is a model for collective decision making under uncertainty, which can be used as a tool for stochastic optimization of a discrete variable function. The GG has a fascinating property that can be resolved in a distributed manner with no intercommunication between the players. The game has found applications in many network applications, including sensor networks, quality-of-service routing, and social networks. In this paper, we introduce an extension of GG called cellular goore game (CGG) for the first time. The CGG is a network of GGs. In this network, each node (or subset of nodes in the network) plays the rule of referees, each of which participates in a GG with its neighboring players (voters) at any time. Like in GG, each player independently selects its optimal action between two available actions based on their gains and losses received from its adjacent referee. Players in CGG know nothing about how other players are playing or even how/why they are rewarded/penalized. The potential of the CGG is shown by providing an algorithm for finding a maximum clique in social networks. Our research provides the first-time study of the CGG for finding a maximum clique in graphs. The performance of the CGG-based algorithm for finding maximum clique is studied on the standard clique benchmark called DIMACS by several experiments. The obtained result shows that the CGG-based algorithm is superior to the existing algorithms in terms of finding maximum clique size and time.

The D(-1)-quadruple conjecture states that there does not exist a set of four positive integers such that the product of any two distinct elements is one greater than a perfect square. It is proved that if {; ; ; a ; b ; c ; d}; ; ; is such a set, then max{; ; ; a ; b ; c ; d}; ; ; < 10^{; ; ; 10^23}; ; ; , hence a proof of the conjecture is given up to a completely determined, finite (but currently computationally infeasible large) set of possible exceptions to check.

Subdivisions of digraphs in tournaments

A graph-based approach to finding near-optimal output phase assignments for PLA minimization is presented. A distinctive feature of the approach is that it exploits the necessary and sufficient conditions to reduce the number of product terms needed for PLA implementation and permits the use of existing graph algorithms to solve the PLA output phase optimization problem. The work is based on the transformation of a PLA into a graph whose vertices are the set of conditions required to reduce each product term of the PLA, and whose edges represent the relations between these conditions. Cliques (completely connected subgraphs) in the graph correspond to the output phase assignments required to reduce the product terms represented by the vertices in the cliques. The optimal PLA output phase assignment problem is then formulated as the well-studied problem in graph theory: finding maximum cliques in graphs. Using an existing algorithm for locating cliques in graphs, a modified greedy algorithm is proposed to compute output phase assignments for logic functions in polynomial time. Experimental results using a number of benchmark functions show that the graph-based approach can achieve optimal or near-optimal output phase assignment for PLA minimization and can lead to PLA's with fewer product terms than achieved using existing approaches.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

It is shown that there are infinitely many natural numbers which differ from the next four greater perfect squares by a perfect square. This follows from the determination of certain families of solutions to the diophantine equation 2(b2 + 1) = a2 + c2. However, it is essentially known that any natural number with this property cannot be 1 less than a perfect square. The question whether there exists a number differing from the next five greater squares by squares is open.

Computing maximum k-defective cliques in massive graphs

For any positive integer n, the F.Smarandache LCM function SL(n) is deflned as the smallest positive integer k such that n j (1;2;¢¢¢ ;k), where (1;2;¢¢¢ ;k) denotes the least common multiple of 1;2;¢¢¢ ;k, and let n = p fi 1 1 p fi2 2 ¢¢¢p fis s be the factorization of n into prime powers, then ›(n) = fi1p1 + fi2p2 + ¢¢¢ + fisps. The main purpose of this paper is using the elementary methods to study the mean value properties of ›(n)SL(n), and give a sharper asymptotic formula for it. Keywords F.Smarandache LCM function, ›(n) function, mean value, asymptotic formula.

The intent of maximum clique problem is to discover a clique of largest size or a complete subgraph in a graph; for a graph clique is a subgraph which is complete i.e., every vertex of that subgraph are adjacent to each other. It belongs to the category of the NP-hard problems, for which many solutions exist and many more are continuously being proposed. We are recapitulating the core idea and search strategies of few of the branch and bound based recently proposed algorithms. Intent of this work is to give a comparative study of newly proposed solutions to maximum clique problem which will be helpful to decide which algorithm is efficient for a given graph. It also instigates to design more efficient solutions for maximum clique problem.

In this paper, we give a comprehensive overview of the development of clique algorithms and their use for drug design based on the search for cliques in protein product graphs. The maximum clique problem is a computational problem of finding largest subsets of vertices in a graph that are all pairwise adjacent. A related problem is the maximum weight clique problem and the highest weight k-clique problem, which both extend the algorithm to weighted graphs. The review covers our developed algorithms, starting with our improved branch-and-bound algorithm for finding maximum cliques in undirected graphs from 2007 up to the recent developments of algorithms for weighted graphs in 2024. We show the application of these algorithms to early stages of drug discovery, in particular to protein binding site detection based on protein similarity search in large protein databases and to protein-ligand molecular docking.

&amp;lt;p&amp;gt;Let f(n)=1 if n=1, 2^(2^(n-2)) if n \\in {2,3,4,5}, (2+2^(2^(n-4)))^(2^(n-4)) if n \\in {6,7,8,...}. We conjecture that if a system T \\subseteq {x_i+1=x_k, x_i \\cdot x_j=x_k: i,j,k \\in {1,...,n}} has only finitely many solutions in integers x_1,...,x_n, then each such solution (x_1,...,x_n) satisfies |x_1|,...,|x_n| \\leq f(n). We prove that the function f cannot be decreased and the conjecture implies that there is an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite. We show that the conjecture and Matiyasevich&amp;#39;s conjecture on finite-fold Diophantine representations are jointly inconsistent.&amp;lt;/p&amp;gt;

Let [Formula: see text] be fixed positive integers such that [Formula: see text] is not a perfect square and [Formula: see text] is squarefree, and let [Formula: see text] denote the number of distinct prime divisors of [Formula: see text]. Let [Formula: see text] denote the least solution of Pell equation [Formula: see text]. Further, for any positive integer [Formula: see text], let [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text]. In this paper, using the basic properties of Pell equations and some known results on binary quartic Diophantine equations, a necessary and sufficient condition for the system of equations [Formula: see text] and [Formula: see text] to have positive integer solutions [Formula: see text] is obtained. By this result, we prove that if [Formula: see text] has a positive integer solution [Formula: see text] for [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, then [Formula: see text] and [Formula: see text], where [Formula: see text] is a positive integer, [Formula: see text] or [Formula: see text] and [Formula: see text] or [Formula: see text] according to [Formula: see text] or not, [Formula: see text] is the integer part of [Formula: see text], except for [Formula: see text]

A (p,1)- total labeling of G is an assignment of integers to V(G)∪E(G) such that any two adjacent vertices of G receive distinct integers, any two adjacent edges of G receive distinct integers, and a vertex and its incident edge receive integers that differ by at least p in absolute value. The span of a (p,1)-total labeling is the maximum difference between two labels. The minimum span of a (p,1)-total labeling of G is called the (p,1)-total number and denoted by λ^T_P(G). Let n and k be two positive integers. The graph with vertex sets {u(1),...,u(n)} and {v(1),...,v(n)} and edge sets {u(i)u(i+1)|i=1,2,...,n},{u(i)v(i)|i=1,2,...,n} and {v(i)v(i+k)|i=1,2,...,n;k<n}, where addition is modulo n is called generalized Petersen graph and denoted by P(n,k). In this thesis, we mainly focus on the (2,1)-total labeling of the generalized Petersen graph, and we show that for each pair of positive integer n and k, 1<=n, if n≡0(mod 5) and k≠0(mod 5), then λ^T_2(P(n,k))=5. Moreover, we also prove that λ^T_2(P(n,5))=5 if n≡0(mod 25).

Alon, Angel, Benjamini and Lubetzky [1] recently studied an old problem of Euler on sumsets for which all elements of A+B are integer squares. Improving their result we prove: 1. There exists a set A of 3 positive integers and a corresponding set B⊂[0,N] with |B|≫(logN)15/17, such that all elements of A+B are perfect squares. 2. There exists a set A of 3 integers and a corresponding set B⊂[0,N] with |B|≫(logN)9/11, such that all elements of the sets A, B and A+B are perfect squares. The proofs make use of suitably constructed elliptic curves of high rank.

The well-known Carmichael conjecture concerns the set of positive integers (presumed empty) which occur as a value of the Euler function at one positive integer but at no others. We study the analogous problem for the Carmichael function. We also study the (far more numerous) sets of integers which occur in the image of one of these functions but not of the other, as well as those which occur in the image of both.