421 research outputs found

    A Local-to-Global Theorem for Congested Shortest Paths

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    Amiri and Wargalla (2020) proved the following local-to-global theorem in directed acyclic graphs (DAGs): if GG is a weighted DAG such that for each subset SS of 3 nodes there is a shortest path containing every node in SS, then there exists a pair (s,t)(s,t) of nodes such that there is a shortest stst-path containing every node in GG. We extend this theorem to general graphs. For undirected graphs, we prove that the same theorem holds (up to a difference in the constant 3). For directed graphs, we provide a counterexample to the theorem (for any constant), and prove a roundtrip analogue of the theorem which shows there exists a pair (s,t)(s,t) of nodes such that every node in GG is contained in the union of a shortest stst-path and a shortest tsts-path. The original theorem for DAGs has an application to the kk-Shortest Paths with Congestion cc ((k,ck,c)-SPC) problem. In this problem, we are given a weighted graph GG, together with kk node pairs (s1,t1),,(sk,tk)(s_1,t_1),\dots,(s_k,t_k), and a positive integer ckc\leq k. We are tasked with finding paths P1,,PkP_1,\dots, P_k such that each PiP_i is a shortest path from sis_i to tit_i, and every node in the graph is on at most cc paths PiP_i, or reporting that no such collection of paths exists. When c=kc=k the problem is easily solved by finding shortest paths for each pair (si,ti)(s_i,t_i) independently. When c=1c=1, the (k,c)(k,c)-SPC problem recovers the kk-Disjoint Shortest Paths (kk-DSP) problem, where the collection of shortest paths must be node-disjoint. For fixed kk, kk-DSP can be solved in polynomial time on DAGs and undirected graphs. Previous work shows that the local-to-global theorem for DAGs implies that (k,c)(k,c)-SPC on DAGs whenever kck-c is constant. In the same way, our work implies that (k,c)(k,c)-SPC can be solved in polynomial time on undirected graphs whenever kck-c is constant.Comment: Updated to reflect reviewer comment

    How finance shapes careers of highly skilled individuals

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    This dissertation analyzes how finance shapes careers of highly skilled individuals. In three chapters, I analyze the role of financial education, labor regulation, and angel investments on choices and careers of individuals

    Planar Disjoint Paths, Treewidth, and Kernels

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    In the Planar Disjoint Paths problem, one is given an undirected planar graph with a set of kk vertex pairs (si,ti)(s_i,t_i) and the task is to find kk pairwise vertex-disjoint paths such that the ii-th path connects sis_i to tit_i. We study the problem through the lens of kernelization, aiming at efficiently reducing the input size in terms of a parameter. We show that Planar Disjoint Paths does not admit a polynomial kernel when parameterized by kk unless coNP \subseteq NP/poly, resolving an open problem by [Bodlaender, Thomass{\'e}, Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel unless the WK-hierarchy collapses. Our reduction carries over to the setting of edge-disjoint paths, where the kernelization status remained open even in general graphs. On the positive side, we present a polynomial kernel for Planar Disjoint Paths parameterized by k+twk + tw, where twtw denotes the treewidth of the input graph. As a consequence of both our results, we rule out the possibility of a polynomial-time (Turing) treewidth reduction to tw=kO(1)tw= k^{O(1)} under the same assumptions. To the best of our knowledge, this is the first hardness result of this kind. Finally, combining our kernel with the known techniques [Adler, Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver, SICOMP'94] yields an alternative (and arguably simpler) proof that Planar Disjoint Paths can be solved in time 2O(k2)nO(1)2^{O(k^2)}\cdot n^{O(1)}, matching the result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].Comment: To appear at FOCS'23, 82 pages, 30 figure

    Online Learning of Energy Consumption for Navigation of Electric Vehicles

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    Energy efficient navigation constitutes an important challenge in electric vehicles, due to their limited battery capacity. We employ a Bayesian approach to model the energy consumption at road segments for efficient navigation. In order to learn the model parameters, we develop an online learning framework and investigate several exploration strategies such as Thompson Sampling and Upper Confidence Bound. We then extend our online learning framework to the multi-agent setting, where multiple vehicles adaptively navigate and learn the parameters of the energy model. We analyze Thompson Sampling and establish rigorous regret bounds on its performance in the single-agent and multi-agent settings, through an analysis of the algorithm under batched feedback. Finally, we demonstrate the performance of our methods via experiments on several real-world city road networks

    Longest Path and Cycle Transversal and Gallai Families

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    A longest path transversal in a graph G is a set of vertices S of G such that every longest path in G has a vertex in S. The longest path transversal number of a graph G is the size of a smallest longest path transversal in G and is denoted lpt(G). Similarly, a longest cycle transversal is a set of vertices S in a graph G such that every longest cycle in G has a vertex in S. The longest cycle transversal number of a graph G is the size of a smallest longest cycle transversal in G and is denoted lct(G). A Gallai family is a family of graphs whose connected members have longest path transversal number 1. In this paper we find several Gallai families and give upper bounds on lpt(G) and lct(G) for general graphs and chordal graphs in terms of |V(G)|

    Designing Equitable Transit Networks

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    Public transit is an essential infrastructure enabling access to employment, healthcare, education, and recreational facilities. While accessibility to transit is important in general, some sections of the population depend critically on transit. However, existing public transit is often not designed equitably, and often, equity is only considered as an additional objective post hoc, which hampers systemic changes. We present a formulation for transit network design that considers different notions of equity and welfare explicitly. We study the interaction between network design and various concepts of equity and present trade-offs and results based on real-world data from a large metropolitan area in the United States of America.Comment: Accepted in the non-archival track at the ACM Conference on Equity and Access in Algorithms, Mechanisms, and Optimization (EAAMO), 202

    End-of-Horizon Load Balancing Problems: Algorithms and Insights

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    Effective load balancing is at the heart of many applications in operations. Often tackled via the balls-into-bins paradigm, seminal results have shown that a limited amount of flexibility goes a long way in order to maintain (approximately) balanced loads throughout the decision-making horizon. This paper is motivated by the fact that balance across time is too stringent a requirement for some applications; rather, the only desideratum is approximate balance at the end of the horizon. In this work we design ``limited-flexibility'' algorithms for three instantiations of the end-of-horizon balance problem: the balls-into-bins problem, opaque selling strategies for inventory management, and parcel delivery for e-commerce fulfillment. For the balls-into-bins model, we show that a simple policy which begins exerting flexibility toward the end of the time horizon (i.e., when Θ(TlogT)\Theta\left(\sqrt{T\log T}\right) periods remain), suffices to achieve an approximately balanced load (i.e., a maximum load within O(1){O}(1) of the average load). Moreover, with just a small amount of adaptivity, a threshold policy achieves the same result, while only exerting flexibility in O(T){O}\left(\sqrt{T}\right) periods, matching a natural lower bound. We then adapt these algorithms to develop order-wise optimal policies for the opaque selling problem. Finally, we show via a data-driven case study that the adaptive policy designed for the balls-into-bins model can be modified to (i) achieve approximate balance at the end of the horizon and (ii) yield significant cost savings relative to policies which either never exert flexibility, or exert flexibility aggressively enough to achieve anytime balance. The unifying motivation behind our algorithms is the observation that exerting flexibility at the beginning of the horizon is likely wasted when system balance is only evaluated at the end

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Odd Paths, Cycles and TT-joins: Connections and Algorithms

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    Minimizing the weight of an edge set satisfying parity constraints is a challenging branch of combinatorial optimization as witnessed by the binary hypergraph chapter of Alexander Schrijver's book ``Combinatorial Optimization'' (Chapter 80). This area contains relevant graph theory problems including open cases of the Max Cut problem, or some multiflow problems. We clarify the interconnections of some problems and establish three levels of difficulties. On the one hand, we prove that the Shortest Odd Path problem in an undirected graph without cycles of negative total weight and several related problems are NP-hard, settling a long-standing open question asked by Lov\'asz (Open Problem 27 in Schrijver's book ``Combinatorial Optimization''. On the other hand, we provide a polynomial-time algorithm to the closely related and well-studied Minimum-weight Odd {s,t}\{s,t\}-Join problem for non-negative weights, whose complexity, however, was not known; more generally, we solve the Minimum-weight Odd TT-Join problem in FPT time when parameterized by T|T|. If negative weights are also allowed, then finding a minimum-weight odd {s,t}\{s,t\}-join is equivalent to the Minimum-weight Odd TT-Join problem for arbitrary weights, whose complexity is only conjectured to be polynomially solvable. The analogous problems for digraphs are also considered.Comment: 24 pages, 2 figure
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