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We introduce the complexity class ∃ ℝ $\exists \mathbb {R}$ based on the existential theory of the reals. We show that the definition of ∃ ℝ $\exists \mathbb {R}$ is robust in the sense that even the fragment of the theory expressing solvability of systems of strict polynomial inequalities leads to the same complexity class. Several natural and well-known(More)
Recent inapproximability results of Sly (2010), together with an approximation algorithm presented by Weitz (2006) establish a beautiful picture for the computational complexity of approximating the partition function of the hard-core model. Let λc(T∆) denote the critical activity for the hard-model on the infinite ∆-regular tree. Weitz presented an FPTAS(More)
An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this(More)
We present an improved "cooling schedule" for simulated annealing algorithms for combinatorial counting problems. Under our new schedule the rate of cooling accelerates as the temperature decreases. Thus, fewer intermediate temperatures are needed as the simulated annealing algorithm moves from the high temperature (easy region) to the low temperature(More)
Given $n$ elements with non-negative integer weights $w_1,..., w_n$ and an integer capacity $C$, we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most $C$. We give the first deterministic, fully polynomial-time approximation scheme (FPTAS) for estimating the number of solutions(More)
Given n elements with nonnegative integer weights w 1 ,. .. , wn and an integer capacity C, we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most the given capacity. We give a deterministic algorithm that estimates the number of solutions to within relative error 1 ± ε in time(More)
We show that string graphs can be recognized in nondeterministic exponential time by giving an exponential upper bound on the number of intersections for a drawing realizing the string graph in the plane. This upper bound confirms a conjecture by Kratochv\'{\i}l and Matou\v{s}ek~\cite{KM91} and settles the long-standing open problem of the decidability of(More)
We address phylogenetic reconstruction when the data is generated from a mixture distribution. Such topics have gained considerable attention in the biological community with the clear evidence of heterogeneity of mutation rates. In our work we consider data coming from a mixture of trees which share a common topology, but differ in their edge weights(More)