Chance-Constrained Programming Models and Approximations for General Stochastic Bottleneck Spanning Tree Problems

A linear time algorithm for the maximum capacity path problem

A spanner of a graph G is a subgraph H that approximately preserves shortest path distances in G. Spanners are commonly applied to compress computation on metric spaces corresponding to weighted input graphs. Classic spanner constructions can seamlessly handle edge weights, so long as error is measured multiplicatively. In this work, we investigate whether one can similarly extend constructions of spanners with purely additive error to weighted graphs. These extensions are not immediate, due to a key lemma about the size of shortest path neighborhoods that fails for weighted graphs. Despite this, we recover a suitable amortized version, which lets us prove direct extensions of classic \(+2\) and \(+4\) unweighted spanners (both all-pairs and pairwise) to \(+2W\) and \(+4W\) weighted spanners, where W is the maximum edge weight. Specifically, we show that a weighted graph G contains all-pairs (pairwise) \(+2W\) and \(+4W\) weighted spanners of size \(O(n^{3/2})\) and \(O(n^{7/5})\) (\(O(np^{1/3})\) and \(O(np^{2/7})\)) respectively. For a technical reason, the \(+6\) unweighted spanner becomes a \(+8W\) weighted spanner; closing this error gap is an interesting remaining open problem. That is, we show that G contains all-pairs (pairwise) \(+8W\) weighted spanners of size \(O(n^{4/3})\) (\(O(np^{1/4})\)).

We develop an FPT algorithm and a compression for the Weighted Edge Clique Partition (WECP) problem, where a graph with n vertices and integer edge weights is given together with an integer k, and the aim is to find k cliques, such that every edge appears in exactly as many cliques as its weight. The problem has been previously only studied in the unweighted version called Edge Clique Partition (ECP), where the edges need to be partitioned into k cliques. It was shown that ECP admits a kernel with k² vertices [Mujuni and Rosamond, 2008], but this kernel does not extend to WECP. The previously fastest algorithm known for ECP has a runtime of 2^𝒪(k²)n^O(1) [Issac, 2019]. For WECP we develop a compression (to a slightly more general problem) with 4^k vertices, and an algorithm with runtime 2^𝒪(k^(3/2)w^(1/2)log(k/w))n^O(1), where w is the maximum edge weight. The latter in particular improves the runtime for ECP to 2^𝒪(k^(3/2)log k)n^O(1).

The inverse spanning-tree problem is to modify edge weights in a graph so that a given tree T is a minimum spanning tree. The objective is to minimize the cost of the deviation. The cost of deviation is typically a convex function. We propose algorithms here that are faster than all known algorithms for the problem. Our algorithm's run time for any convex inverse spanning-tree problem is O(nm log2 n) for a graph on n nodes and m edges plus the time required to find the minima of the n convex deviation functions. This not only improves on the complexity of previously known algorithms for the general problem, but also for algorithms devised for special cases. For the case when the objective is weighted absolute deviation, Sokkalingam et al. (1999) devised an algorithm with run time O(n 2 m log(nC)) for C the maximum edge weight. For the sum of absolute deviations our algorithm runs in time O(n 2 log n), matching the run time of a recent (Ahuja and Orlin 2000) improvement for this case. A new algorithm for bipartite matching in path graphs is reported here with complexity of O(n 1.5 log n). This leads to a second algorithm for the sum of absolute deviations running in O(n 1.5 log n log C) steps.

An approximation algorithm is suggested for the problem of finding a d-regular spanning connected subgraph of maximum weight in a complete undirected weighted n-vertex graph. Probabilistic analysis of the algorithm is carried out for the problem with random input data (some weights of edges) in the case of a uniform distribution of the weights of edges and in the case of a minorized type distribution. It is shown that the algorithm finds an asymptotically optimal solution with time complexity O(n 2) when d = o(n). For the minimization version of the problem, an additional restriction on the dispersion of weights of the graph edges is added to the condition of the asymptotical optimality of the modified algorithm.

Blocking flows were introduced by Dinic (Soviet Math Doklady 11: 1277–1280, 1970) to speed up the computation of maximum network flows. They have been used in algorithms for problems such as maximum cardinality matching of bipartite graphs Hopcroft and Karp (SIAM J Comput 2(4), 225–231, 1973) and general graphs Micali and Vazirani (in: Proceedings of the 21st Annual Symposium on Foundations of Computer Science, 17–27, 1980), maximum weight matching of general graphs Gabow and Tarjan (J ACM 38(4), 815–853, 1991), and many others. The blocking algorithm of Gabow and Tarjan (1991) for matching is based on depth-first search. We extend the depth-first search approach to find f-factors of general multigraphs. Here f is an arbitrary integral-valued function on vertices, an f-matching is a subgraph where every vertex x has degree $$\le f(x)$$ , an f-factor has equality in every degree bound. A set of blocking trails for an f-matching M is a maximal collection $$\mathcal{A}$$ of edge-disjoint augmenting trails such that $$M\bigoplus _{A\in {\mathcal{A}}} A $$ is a valid f-matching. Blocking trails are needed in efficient algorithms for maximum cardinality f-matching Huang and Pettie (Algorithmica 84(7): 1952–1992, 2022), maximum weight f-factors/matchings by scaling Duan et al. (In: Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), Vol. 168 of LIPIcs, 41:1-41:17, 2020; Gabow (A weight-scaling algorithm for f-factors of multigraphs. arXiv:2010.01102 , 2020), and approximate maximum weight f-factors and f-edge covers Huang and Pettie (2022). Since these algorithms find many sets of blocking trails, the time to find blocking trails is a dominant factor in the running time. Our blocking trail algorithm runs in linear time O(m). In independent work and using a different approach, Huang and Pettie (2022) present a blocking trails algorithm using time $$O(m\alpha (m,n))$$ . As examples of the time bounds for the above applications, an approximate maximum weight f-factor is found in time $$O(m\,\alpha (m,n))$$ using Huang and Pettie (2022), and our algorithm eliminates the factor $$\alpha (m,n)$$ . Similarly a maximum weight f-factor is found in time $$O(\sqrt{\Phi \log \Phi }\, m\,\alpha (m,n)\, \log (\Phi W))\,$$ using Huang and Pettie (2022) , ( $$\Phi =\sum _{v\in V} f(v)$$ , W the maximum edge weight) and our algorithm eliminates the $$\alpha (m,n)$$ factor, making the time within a factor $$\sqrt{\log {\Phi }}$$ of the bound for bipartite multigraphs. The technical difficulty for this work stems from the fact that a fixed vertex can occur many times in a given search. This does not occur in ordinary matching or in algorithms for maximum cardinality or maximum weight f-matching. These multiple occurrences can create a new variant of blossom, the “skew blossom”. Also they can make blossoms become “incomplete”, i.e., partially processed yet still relevant in future searches.

We present an algorithm which calculates a minimum cut and its weight in an undirected graph with nonnegative real edge weights, n vertices and m edges, in time O(max(log n, min(m/n,δG/e)) n2), where e is the minimal edge weight, and δG is the minimal weighted degree. For integer edge weights this time is further improved to O(δG n2) and O(λG n2). In both cases these bounds are improvements of the previously known best bounds of deterministic algorithms. These were O(nm + n2 log n) for real edge weights and O(nM + n2) and O(M + λG n2) for integer weights, where M is the sum of all edge weights.

In this paper, a new structure for cooperative learning automata called extended learning automata (eDLA) is introduced. Based on the new structure, an iterative randomized heuristic algorithm using sampling is proposed for finding an optimal subgraph in a stochastic edge-weighted graph. Stochastic graphs are graphs in which the weights of edges have an unknown probability distribution. The proposed algorithm uses an eDLA to find a policy that leads to a subgraph that satisfy some restrictions such as minimum or maximum weight (length). At each stage of the proposed algorithm, the eDLA determines which edges should be sampled. The proposed eDLA-based sampling method may reduce unnecessary samples and hence decrease the time required for finding an optimal subgraph. It is shown that the proposed method converges to an optimal solution, the probability of which can be made arbitrarily close to 1 by using a sufficiently small learning parameter. A new variance-aware threshold value is also proposed that can significantly improve the convergence rate of the proposed eDLA-based algorithm. It is further shown that our algorithm is competitive in terms of the quality of the solution.

A novel chance constrained joint optimization method under uncertainties in distribution networks

We present a new scaling algorithm for maximum (or minimum) weight perfect matching on general, edge weighted graphs. Our algorithm runs in [EQUATION] time, [EQUATION] per scale, which matches the running time of the best cardinality matching algorithms on sparse graphs [29, 18]. Here m, n, and N bound the number of edges, vertices, and magnitude of any integer edge weight. Our result improves on a 25-year old algorithm of Gabow and Tarjan, which runs in [EQUATION] log(nN)) time.

In this paper we consider multicriteria formulations of classical online problems where an algorithm must simultaneously perform well with respect to two different cost measures. The performance of the algorithm is compared with that of an adversary that serves the sequence of requests selecting one of the possible optimal offline strategies according to a given selection function. We consider a parametric family of functions based on their monotonicity properties which covers all the possible selections. Then, we provide a universal multicriteria algorithm that can be applied to different online problems. For the multicriteria k-server formulation, for each function class, such an algorithm achieves competitive ratios that are only an O(log W) multiplicative factor away from the corresponding lower bounds that we determine for the class, where W is the maximum edge weight. We then show how to extend our results to other multicriteria online problems sharing similar properties.KeywordsCompetitive RatioOnline AlgorithmNondominated SolutionCost MeasureCompetitive AlgorithmThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Exact and approximation algorithms for the min–max k-traveling salesmen problem on a tree

Fleet deployment and demand fulfillment for container shipping liners

Given a network G=(V,E), edge weights w(.), and a set of terminals S⊆V, the minimum-weight Steiner tree problem is to find a tree in G that spans S with minimum weight. Most provable heuristics treat the network G is a metric; This assumption, in a distributed setting, cannot be easily achieved without a subtle overhead. We give a simple distributed algorithm based on a minimum spanning tree heuristic that returns a solution whose cost is within a factor of two of the optimal. The algorithm runs in time O(|V|log|V|) on a synchronous network. We also show that another heuristic based on iteratively finding shortest paths gives a Θ(log |V|)-approximation using a novel charging scheme based on low-congestion routing on trees. Both algorithms work for unit-cost and general cost cases. The algorithms also have applications in finding multicast trees in wireless ad hoc networks.

Exact and heuristic algorithms for the weighted total domination problem