Let G be a graph on n t> 3 vertices. Dirac's minimum degree condition is the condition that all vertices of G have degree at least in. This is a well-known sufficient condition for the existence of a Hamilton cycle in G. We give related sufficiency conditions for the existence of a Hamilton cycle or a perfect matching involving a restriction of Dirac's minimum degree condition to certain subset...

This correspondence represents two new soft decision decoding algorithms that promise to reduce complexity and at the same time achieve the maximum likelihood decoding (MLD) performance. The first method is an Adaptive Two-Stage Maximum Likelihood Decoder [1] that first estimates a minimum sufficient set and performs decoding within the smaller set to reduce complexity and at the same time achi...

While every instance of the Hospitals/Residents problem admits a stable matching, the problem with lower quotas (HR-LQ) has instances with no stable matching. For such an instance, we expect the existence of an envy-free matching, which is a relaxation of a stable matching preserving a kind of fairness property. In this paper, we investigate the existence of an envy-free matching in several set...

In recent years, even though Stochastic Gradient Descent (SGD) and its variants are well-known for training neural networks, it suffers from limitations such as the lack of theoretical guarantees, vanishing gradients, excessive sensitivity to input. To overcome these drawbacks, alternating minimization methods have attracted fast-increasing attention recently. As an emerging open domain, howeve...

Is the Pareto optimality of matching mechanisms robust to the introduction of boundedly rational behavior? To address this question I define a restrictive and a permissive notion of Pareto optimality and consider the large set of hierarchical exchange mechanisms which contains serial dictatorship as well as Gale’s top trading cycles. Fix a housing problem with boundedly rational agents and a hi...

We conjecture that the balanced complete bipartite graph K⌊n/2⌋,⌈n/2⌉ contains more cycles than any other n-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent conditions for cycle-maximal triangle-free graphs; show bounds on the numbers of cycles in graphs depending on numbers of vertices and edges, girth, and homomorphisms to small fixed graphs; and u...

If a linear block code C of length n has a Tanner graph without cycles, then maximum-likelihood soft-decision decoding of C can be achieved in time O(n 2). However, we show that cycle-free Tanner graphs cannot support good codes. Specically, let C be an (n;k;d) linear code of rate R = k=n that can be represented by a Tanner graph without cycles. We prove that if R 0:5 then d 2, while if R < 0:5...