Large Independent Sets on Random d-Regular Graphs with Fixed Degree d

Local algorithms on graphs are algorithms that run in parallel on the nodes of a graph to compute some global structural feature of the graph. Such algorithms use only local information available at nodes to determine local aspects of the global structure, while also potentially using some randomness. Research over the years has shown that such algorithms can be surprisingly powerful in terms of computing structures like large independent sets in graphs locally. These algorithms have also been implicitly considered in the work on graph limits, where a conjecture due to Hatami, Lovász and Szegedy [17] implied that local algorithms may be able to compute near-maximum independent sets in (sparse) random d-regular graphs. In this paper we refute this conjecture and show that every independent set produced by local algorithms is smaller that the largest one by a multiplicative factor of at least 1/2+1/(2√2) ≈ .853, asymptotically as d → ∞.

We present a new approach to showing that random graphs are nearly optimal expanders. This approach is based on recent deep results in combinatorial group theory. It applies to both regular and irregular random graphs. Let G be a random d-regular graph on n vertices, and let \lambda be the largest absolute value of a non-trivial eigenvalue of its adjacency matrix. It was conjectured by Alon [86'] that a random d-regular graph is almost Ramanujan, in the following sense: for every e>0, \lambda<2\sqrt{d-1} + e asymptotically almost surely. Friedman famously presented a proof of this conjecture in [08']. Here we suggest a new, substantially simpler proof of a nearly-optimal result: we show that a random d-regular graph satisfies \lambda < 2\sqrt{d-1} + 1 a.a.s. A main advantage of our approach is that it is applicable to a generalized conjecture: For d even, a d-regular graph on n vertices is an n-covering space of a bouquet of d/2 loops. More generally, fixing an arbitrary base graph H, we study the spectrum of G, a random n-covering of H. Let \lambda be the largest absolute value of a non-trivial eigenvalue of G. Extending Alon's conjecture to this more general model, Friedman [03'] conjectured that for every e>0, a.a.s. \lambda < \rho+e, where \rho is the spectral radius of the universal cover of H. When H is regular we get a bound of \rho+0.84, and for an arbitrary H, we prove a nearly optimal upper bound of \sqrt{3}\rho. This is a substantial improvement upon all known results (by Friedman, Linial-Puder, Lubetzky-Sudakov-Vu and Addario-Berry-Griffiths).

The Maximum Independent Set (MIS) problem is a well-known problem where the aim is to find the maximum cardinality independent set in an associated graph. Map labeling problems can often be modeled as a MIS problem in a conflict graph, where labels are selected to be placed near graphical features not allowing overlaps (conflicts) between labels or between labels and features. However, the MIS problem is NP-hard and exact techniques present difficulties for solving some instances. Thus, this paper presents a Lagrangean decomposition to solve a map labeling problem, the Point-Feature Label Placement problem. We treated the problem by a conflict graph that is partitioned into small sub-problems and copies of some decision variables are done. These copies are used in the sub-problems constraints and in some constraints to ensure the equality between the original variables and their copies. After these steps, we relax these copy constraints in a Lagrangean way. Using instances proposed in the literature, our approach was able to prove the optimality for all of them, except one, and the results were better than the ones provided by a commercial solver.

The Maximal Independent Set problem is a well-studied problem in the distributed community. We study Maximum and Maximal Independent Set problems on two geometric intersection graphs; interval graphs and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in the local communication model. We compute the maximum independent set on interval graphs in O(k) rounds and O(n) messages, where k is the size of the maximum independent set and n is the number of nodes in the graph. We provide a matching lower bound of \(\varOmega (k)\) on the number of rounds whereas \(\varOmega (n)\) is a trivial lower bound on message complexity. Thus our algorithm is both time and message optimal. We also study the maximal independent set problem in bi-interval graphs, a special case of the interval graphs where the intervals have two different lengths. We prove that a maximal independent set can be computed in bi-interval graphs in constant rounds that is \(\frac{1}{6}\)-approximation. For axis-parallel segment intersection graphs, we design an algorithm that finds a maximal independent set in O(D) rounds, where D is the diameter of the graph. We further show that this independent set is a \(\frac{1}{2}\)-approximation. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].

The learning with errors (LWE) problem is a problem derived from machine learning that is believed to be intractable for quantum computers. This paper proposes a method that can reduce an LWE problem to a set of maximum independent set (MIS) problems, which are graph problems that are suitable for a quantum annealing machine to solve. The reduction algorithm can reduce an n-dimensional LWE problem to several small MIS problems with at most 2^{O(sqrt{n})} nodes when the lattice-reduction algorithm used in the LWE-reduction method successfully finds short vectors. The algorithm is useful for solving LWE problems in a quantum-classical hybrid manner by using an existing quantum algorithm to solve the MIS problems. For example, the smallest LWE challenge problem is reduced to MIS problems with about 40,000 vertices. This result means that the smallest LWE challenge problem will be within the scope of a real quantum computer in the near future.

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.

OF THE DISSERTATION Random matrices and random boxes by Linh V. Tran Dissertation Director: Prof. Van Vu This thesis concerns two questions on random structures: the semi-circular law for adjacency matrix of regular random graph and the piercing number for random boxes. • Random matrices We proved in full generality the semi-circular law for random d-regular graph model in the case d tends to infinity as n does. Our result complements the McKay law [19], which applied for the case d is an absolute constant. • Random boxes. Take n random boxes with axis-parallel edges inside the unit cube [0, 1]d, the piercing number τd(n) is the minimum number of points needed to pierce all boxes. Using hypergraph setting, we was able to prove a near sharp estimation for τd(n): for any d ≥ 2 Ωd( √ n(log n)d/2−1) = τd(n) = Od( √ n(log n)d/2−1 log log n). This thesis is based on two papers by the author [31] and [30] (joint work with Van Vu and Ke Wang).

The maximum independent set (MIS) problem is a classical NP-hard problem that appears in diverse applications such as scheduling, wireless sensor networks, resource allocation, and bioinformatics. Metaheuristic approaches, including Chemical Reaction Optimization (CRO), have shown promising results on medium-sized instances; however, plain CRO typically relies on stochastic structural modifications and may require multiple runs to reach optimal or near-optimal independent sets. This paper proposes a hybrid (memetic) variant of CRO for the MIS problem in which each newly generated molecule is passed through a problem-specific local refinement procedure. The local step greedily expands partial independent sets and applies 1-swap/2-swap neighborhood moves to remove conflicts and increase the cardinality of the set. By combining the global exploration capability of CRO with the intensification of local search, the proposed method achieves higher-quality solutions, especially on dense graphs where conflicts are frequent. Experimental results on synthetic graphs of varying sizes and densities, as well as on standard benchmark instances, show that the hybrid CRO consistently outperforms the baseline CRO in terms of best solution, average solution, and number of runs needed to reach the best MIS. These findings indicate that embedding domain knowledge into CRO is an effective way to improve robustness without losing generality.

In this paper, we develop efficient exact and approximate algorithms for computing a maximum independent set in random graphs. In a random graph G, each pair of vertices are joined by an edge with a probability p, where p is a constant between 0 and 1. We show that a maximum independent set in a random graph that contains n vertices can be computed in expected computation time . In addition, we show that, with high probability, the parameterized independent set problem is fixed parameter tractable in random graphs and the maximum independent set in a random graph in n vertices can be approximated within a ratio of in expected polynomial time.

We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016) for the MIS problem have tried to break the long-standing O(log n)-round "barrier" achieved by Luby's algorithm, but these yield o(log n)-round complexity only when the maximum degree Delta is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(log n) (via Luby's algorithm) even for moderately small Delta (i.e., for Delta = Omega(log n) and Delta = o(n)). Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby's algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the Theta(log n) time complexity barrier and the Theta(m) message complexity barrier in the Congest model for MIS or closely-related symmetry breaking problems?&#13;\n&#13;\nThis paper presents progress towards this question for the distributed ruling set problem in the Congest model. A beta-ruling set is an independent set such that every node in the graph is at most beta hops from a node in the independent set. We present the following results:&#13;\n&#13;\n- Time Complexity: We show that we can break the O(log n) "barrier" for 2- and 3-ruling sets. We compute 3-ruling sets in O(log n/log log n) rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O(log Delta (log n)^(1/2 + epsilon) + log n/log log n) rounds for any epsilon &gt; 0, which is o(log n) for a wide range of Delta values (e.g., Delta = 2^(log n)^(1/2-epsilon)). These are the first 2- and 3-ruling set algorithms to improve over the O(log n)-round complexity of Luby's algorithm in the Congest model.&#13;\n&#13;\n- Message Complexity: We show an Omega(n^2) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log^2 n) messages and runs in O(Delta log n) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor).

Let be a graph whose largest independent set has size . A permutation of is an independent set permutation of if where is the number of independent sets of size in . In 1987 Alavi, Malde, Schwenk, and Erdős proved that every permutation of is an independent set permutation of some graph with , that is, with the largest independent set having size . They raised the question of determining, for each , the smallest number such that every permutation of is an independent set permutation of some graph with and with at most vertices, and they gave an upper bound on of roughly . Here we settle the question, determining , and make progress on a related question, that of determining the smallest order such that every permutation of is the unique independent set permutation of some graph of at most that order. More generally we consider an extension of independent set permutations to weak orders, and extend Alavi et al.'s main result to show that every weak order on can be realized by the independent set sequence of some graph with and with at most vertices. Alavi et al. also considered matching permutations, defined analogously to independent set permutations. They observed that not every permutation of is a matching permutation of some graph with the largest matching having size , putting an upper bound of on the number of matching permutations of . Confirming their speculation that this upper bound is not tight, we improve it to .

From matchings to independent sets

We prove that a random d-regular graph, with high probability, is a cut sparsifier of the clique with approximation error at most , where = 1.595 … and on,d(1) denotes an error term that depends on n and d and goes to zero if we first take the limit n → ∞ and then the limit d → ∞. This is established by analyzing linear-size cuts using techniques of Jagannath and Sen [13] derived from ideas in statistical physics, and analyzing small cuts via martingale inequalities. We also prove new lower bounds on spectral sparsification of the clique. If G is a spectral sparsifier of the clique and G has average degree d, we prove that the approximation error is at least the “Ramanujan bound” , which is met by d-regular Ramanujan graphs, provided that either the weighted adjacency matrix of G is a (multiple of) a doubly stochastic matrix, or that G satisfies a certain high “odd pseudo-girth” property. The first case can be seen as an “Alon-Boppana theorem for symmetric doubly stochastic matrices,” showing that a symmetric doubly stochastic matrix with dn non-zero entries has a non-trivial eigenvalue of magnitude at least ; the second case generalizes a lower bound of Srivastava and Trevisan [23], which requires a large girth assumption. Together, these results imply a separation between spectral sparsification and cut sparsification. If G is a random log n-regular graph on n vertices (this is to ensure that G, and consequently any d-regular subgraph, has high pseudogirth), we show that, with high probability, G admits a (weighted subgraph) cut sparsifier of average degree d and approximation error at most , while every (weighted subgraph) spectral sparsifier of G having average degree d has approximation error at least .

Maximum independent set problem is the core issue in NP-complete problems, many problems can be transformed into the maximum independent set problems to solve, so design a good algorithm for solving the problem is very necessary. Molecular beacon is a hairpin-shaped fluorescent probe, which can hybridize with target sequence that is complement to its loop sequence. The specificity of molecular beacon is as high as single base mismatch detection. In this paper, molecular beacon is as for probe, to separation and detection all of the independent sets, finally by observing the fluorescence to determine whether or not the solution is existence. At the same time, in the establishment of the initial data pool, only one type of DNA molecules was synthesized. In the calculation process, the model does not require the restriction enzyme digestion, gel electrophoresis etc.. These steps may avoid possible computational errors and data loss; at the same time, the algorithm also has easy code and reading solution.

The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on G(n,d), a random d-regular graph on n vertices. It is well known that almost every such graph for d≥3 is an expander, and even essentially Ramanujan, implying a mixing time of O(logn). According to a conjecture of Peres, the simple random walk on G(n,d) for such d should then exhibit cutoff with high probability (whp). As a special case of this, Durrett conjectured that the mixing time of the lazy random walk on a random 3-regular graph is whp (6+o(1))log2n. In this work we confirm the above conjectures and establish cutoff in total-variation, its location, and its optimal window, both for simple and for non-backtracking random walks on G(n,d). Namely, for any fixed d≥3, the simple random walk on G(n,d) whp has cutoff at (d/(d−2))logd−1n with window order logn. Surprisingly, the non-backtracking random walk on G(n,d) whp has cutoff already at logd−1n with constant window order. We further extend these results to G(n,d) for any d=no(1) that grows with n (beyond which the mixing time is O(1)), where we establish concentration of the mixing time on one of two consecutive integers