Computer Science

Given a graph G=(V,E) with vertex weights w(v) and a desired number of parts k , the goal in graph partitioning problems is to partition the vertex set V into parts V 1 ,?? V k . Metrics for compactness, contiguity, and balance of the parts V i are frequent objectives, with much existing literature focusing on compactness and balance. Revisiting an old method known as striping, we give the first polynomial-time algorithms with guaranteed contiguity and provable bicriteria approximations for compactness and balance for planar grid graphs. We consider several types of graph partitioning, including when vertex weights vary smoothly or are stochastic, reflecting concerns in various real-world instances. We show significant improvements in experiments for balancing workloads for the fire department and reducing over-policing using 911 call data from South Fulton, GA.

A mixed-integer linear programming (MILP) formulation is presented for parameter estimation of the Potts model. Two algorithms are developed; the first method estimates the parameters such that the set of ground states replicate the user-prescribed data set; the second method allows the user to prescribe the ground states multiplicity. In both instances, the optimization process ensures that the bandgap is maximized. Consequently, the model parameter efficiently describes the user data for a broad range of temperatures. This is useful in the development of energy-based graph models to be simulated on Quantum annealing hardware where the exact simulation temperature is unknown. Computationally, the memory requirement in this method grows exponentially with the graph size. Therefore, this method can only be practically applied to small graphs. Such applications include learning of small generative classifiers and spin-lattice model with energy described by Ising hamiltonian. Learning large data sets poses no extra cost to this method; however, applications involving the learning of high dimensional data are out of scope.

We study the maximum matching problem in the random-order semi-streaming setting. In this problem, the edges of an arbitrary n -vertex graph G=(V,E) arrive in a stream one by one and in a random order. The goal is to have a single pass over the stream, use n?�poly(logn) space, and output a large matching of G . We prove that for an absolute constant ϵ 0 >0 , one can find a (2/3+ ϵ 0 ) -approximate maximum matching of G using O(nlogn) space with high probability. This breaks the natural boundary of 2/3 for this problem prevalent in the prior work and resolves an open problem of Bernstein [ICALP'20] on whether a (2/3+Ω(1)) -approximation is achievable.

One of the primary goals of the mathematical analysis of algorithms is to provide guidance about which algorithm is the "best" for solving a given computational problem. Worst-case analysis summarizes the performance profile of an algorithm by its worst performance on any input of a given size, implicitly advocating for the algorithm with the best-possible worst-case performance. Strong worst-case guarantees are the holy grail of algorithm design, providing an application-agnostic certification of an algorithm's robustly good performance. However, for many fundamental problems and performance measures, such guarantees are impossible and a more nuanced analysis approach is called for. This chapter surveys several alternatives to worst-case analysis that are discussed in detail later in the book.

Dynamic time warping distance (DTW) is a widely used distance measure between time series x,y??Σ n . It was shown by Abboud, Backurs, and Williams that in the \emph{binary case}, where |Σ|=2 , DTW can be computed in time O( n 1.87 ) . We improve this running time O(n) . Moreover, if x and y are run-length encoded, then there is an algorithm running in time O ~ (k+?? , where k and ??are the number of runs in x and y , respectively. This improves on the previous best bound of O(k?? due to Dupont and Marteau.

We present an O ~ (m+ n 1.5 ) -time randomized algorithm for maximum cardinality bipartite matching and related problems (e.g. transshipment, negative-weight shortest paths, and optimal transport) on m -edge, n -node graphs. For maximum cardinality bipartite matching on moderately dense graphs, i.e. m=Ω( n 1.5 ) , our algorithm runs in time nearly linear in the input size and constitutes the first improvement over the classic O(m n − − √ ) -time [Dinic 1970; Hopcroft-Karp 1971; Karzanov 1973] and O ~ ( n ω ) -time algorithms [Ibarra-Moran 1981] (where currently ω≈2.373 ). On sparser graphs, i.e. when m= n 9/8+δ for any constant δ>0 , our result improves upon the recent advances of [Madry 2013] and [Liu-Sidford 2020b, 2020a] which achieve an O ~ ( m 4/3+o(1) ) runtime. We obtain these results by combining and advancing recent lines of research in interior point methods (IPMs) and dynamic graph algorithms. First, we simplify and improve the IPM of [v.d.Brand-Lee-Sidford-Song 2020], providing a general primal-dual IPM framework and new sampling-based techniques for handling infeasibility induced by approximate linear system solvers. Second, we provide a simple sublinear-time algorithm for detecting and sampling high-energy edges in electric flows on expanders and show that when combined with recent advances in dynamic expander decompositions, this yields efficient data structures for maintaining the iterates of both [v.d.Brand et al.] and our new IPMs. Combining this general machinery yields a simpler O ~ (n m − − √ ) time algorithm for matching based on the logarithmic barrier function, and our state-of-the-art O ~ (m+ n 1.5 ) time algorithm for matching based on the [Lee-Sidford 2014] barrier (as regularized in [v.d.Brand et al.]).

Given a vertex-weighted graph, the maximum weight independent set problem asks for a pair-wise non-adjacent set of vertices such that the sum of their weights is maximum. The branch-and-reduce paradigm is the de facto standard approach to solve the problem to optimality in practice. In this paradigm, data reduction rules are applied to decrease the problem size. These data reduction rules ensure that given an optimum solution on the new (smaller) input, one can quickly construct an optimum solution on the original input. We introduce new generalized data reduction and transformation rules for the problem. A key feature of our work is that some transformation rules can increase the size of the input. Surprisingly, these so-called increasing transformations can simplify the problem and also open up the reduction space to yield even smaller irreducible graphs later throughout the algorithm. In experiments, our algorithm computes significantly smaller irreducible graphs on all except one instance, solves more instances to optimality than previously possible, is up to two orders of magnitude faster than the best state-of-the-art solver, and finds higher-quality solutions than heuristic solvers DynWVC and HILS on many instances. While the increasing transformations are only efficient enough for preprocessing at this time, we see this as a critical initial step towards a new branch-and-transform paradigm.

We say that a square real matrix M is \emph{off-diagonal nonnegative} if and only if all entries outside its diagonal are nonnegative real numbers. In this note we show that for any off-diagonal nonnegative symmetric matrix M , there exists a nonnegative symmetric matrix M ˆ which is sparse and close in spectrum to M .

The coloring problem (i.e., computing the chromatic number of a graph) can be solved in O ∗ ( 2 n ) time, as shown by Björklund, Husfeldt and Koivisto in 2009. For k=3,4 , better algorithms are known for the k -coloring problem. 3 -coloring can be solved in O( 1.33 n ) time (Beigel and Eppstein, 2005) and 4 -coloring can be solved in O( 1.73 n ) time (Fomin, Gaspers and Saurabh, 2007). Surprisingly, for k>4 no improvements over the general O ∗ ( 2 n ) are known. We show that both 5 -coloring and 6 -coloring can also be solved in O( (2−ε) n ) time for some ε>0 . As a crucial step, we obtain an exponential improvement for computing the chromatic number of a very large family of graphs. In particular, for any constants Δ,α>0 , the chromatic number of graphs with at least α⋅n vertices of degree at most Δ can be computed in O( (2−ε) n ) time, for some ε= ε Δ,α >0 . This statement generalizes previous results for bounded-degree graphs (Björklund, Husfeldt, Kaski, and Koivisto, 2010) and graphs with bounded average degree (Golovnev, Kulikov and Mihajilin, 2016). We generalize the aforementioned statement to List Coloring, for which no previous improvements are known even for the case bounded-degree graphs.

Given a weighted bipartite graph with n vertices and m edges, the maximum weight bipartite matching problem is to find a set of vertex-disjoint edges with the maximum weight. This classic problem has been extensively studied for over a century. In this paper, we present a new streaming algorithm for the maximum weight bipartite matching problem that uses O ˜ (n) space and O ˜ ( m − − √ ) passes, which breaks the n -pass barrier. All the previous algorithms either require Ω(nlogn) passes or only find an approximate solution. To achieve this pass bound, our algorithm combines a number of techniques from different fields such as the interior point method (IPM), symmetric diagonally dominant (SDD) system solving, the isolation lemma, and LP duality. To the best of our knowledge, this is the first work that implements the SDD solver and IPM in the streaming model in O ˜ (n) spaces for graph matrix. All the previous IPMs only focus on optimizing the running time, regardless of the space usage. The LP solver for general matrix is impossible to be implemented in O ˜ (n) spaces.