network design engineers

Abstract: Finding a global minimum cut in an unweighted graph is a fundamental task in the cut-query model. While there exists a randomized algorithm for this task using $\tilde O(n)$ queries, known deterministic algorithms use $\tilde{O}(n^{5/3})$ queries. We improve upon this result and give a simple deterministic algorithm that finds a global minimum cut using $\tilde O(\min\left\{ nδ,n^2/δ\right\})$ cut queries, which equals $\tilde{O}(n^{3/2})$ in the worst case, where $δ$ is the minimum degree of the graph. The key ingredient in our approach is an efficient maximum $s,t$-flow algorithm, for the case when the minimum $s,t$-cut has value at most $δ$.

Abstract: Let f:2^N --> \cZ^+ be a polymatroid (an integer-valued non-decreasing submodular set function with f(emptyset) = 0). A k-polymatroid satisfies that f(e) <= k for all e in N. We call a subset S of N independent if f(S) equals the sum of f(e) over the elements e of S and f(e) > 0 for all e in S. Finding a maximum-size independent set in a 2-polymatroid has been studied and polynomial-time algorithms are known for linear polymatroids. For k >= 3, the problem is NP-hard, and an approximation algorithm with ratio approaching 2/k is known and is obtained by swapping as long as possible a "large" subset from the current solution by a set with one more element. Here we give a simple analysis of the more particular two-for-one repeated swapping heuristic, obtaining a (weaker) 2/(k+1)-approximation.

Abstract: We present a new lower bound on the number of lattice points of all totally unimodular polytopes, generalizing previous lower bounds on contingency tables, integer flows, and beyond. Our bound is based on the Gurvits capacity convex optimization problem, and thus our result implies an efficient deterministic algorithm for approximate counting of the lattice points up to an explicit exponential factor. We achieve our bounds by showing that the associated generating polynomials fit into a new general class of log-concave polynomials called VLC ("variable-wise log-concavity''). This also implies a conjecture of Ferroni and Higashitani on the evaluations of the Ehrhart polynomials of unimodular polytopes. The essential ingredient for these results is the resolution of Barvinok's log-concavity conjecture for contingency tables on lines, which was proven using ChatGPT 6 Astra. We conjecture a generalization of Barvinok's conjecture, which we believe will lead to stronger and more general bounds.

Abstract: The $k$ vertex-disjoint paths problem asks whether, given a graph $G$ and $k$ pairs of vertices $(s_1,t_1)$, \ldots, $(s_k,t_k)$, $G$ has $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all $1\leq i\leq k$. If $G$ is undirected, then this problem is NP-complete, but there exist FPT algorithms parameterized by $k$.Since these algorithms involve an extremely large function on $k$, algorithms for restricted graphs have also been investigated. In particular, a $2^{2tw\log tw+O(tw)}\cdot n$ time algorithm for undirected graphs with $n$ vertices and treewidth $tw$ is proposed by Scheffler (Technical Report 396, TU Berlin, '94), and it is proved by Lokshtanov, Marx, and Saurabh (SIAM J. Comput. '18) that, under the ETH, there exists no $2^{o(pw\log pw)}\cdot n^{O(1)}$ time algorithm for either directed or undirected graphs with pathwidth $pw$ and for $k=Ω(pw^4)$. It has not been known whether the lower bound also holds for a smaller $k$. In this paper, we prove that, for both the directed and undirected cases, there is an algorithm faster than Lokshtanov et al.'s lower bound for $k=tw^{o(1)}$ by proposing a $2^{O((tw+k)\log k)}\cdot n$ time algorithm. Besides, we prove a lower bound that, under the SETH, there exists no $(2-ε)^{pw\log pw}\cdot n^{O(1)}$ time algorithm for directed graphs and for a general $k$. This lower bound is tight because, with slight modifications, Scheffler's algorithm runs in $2^{pw\log pw+O(pw)}\cdot n$ time also for directed graphs.