Computer Science

Multidimensional packing problems generalize the classical packing problems such as Bin Packing, Multiprocessor Scheduling by allowing the jobs to be d -dimensional vectors. While the approximability of the scalar problems is well understood, there has been a significant gap between the approximation algorithms and the hardness results for the multidimensional variants. In this paper, we close this gap by giving almost tight hardness results for these problems. 1. We show that Vector Bin Packing has no polynomial time Ω(logd) factor asymptotic approximation algorithm when d is a large constant, assuming P?�NP . This matches the lnd+O(1) factor approximation algorithms (Chekuri, Khanna SICOMP 2004, Bansal, Caprara, Sviridenko SICOMP 2009, Bansal, Eliás, Khan SODA 2016) upto constants. 2. We show that Vector Scheduling has no polynomial time algorithm with an approximation ratio of Ω((logd ) 1?��?) when d is part of the input, assuming NP?�ZPTIME( n (logn ) O(1) ) . This almost matches the O( logd loglogd ) factor algorithms(Harris, Srinivasan JACM 2019, Im, Kell, Kulkarni, Panigrahi SICOMP 2019). We also show that the problem is NP-hard to approximate within (loglogd ) ?(1) . 3. We show that Vector Bin Covering is NP-hard to approximate within Ω( logd loglogd ) when d is part of the input, almost matching the O(logd) factor algorithm (Alon et al., Algorithmica 1998). Previously, no hardness results that grow with d were known for Vector Scheduling and Vector Bin Covering when d is part of the input and for Vector Bin Packing when d is a fixed constant.

We give almost-linear-time algorithms for constructing sparsifiers with n poly(logn) edges that approximately preserve weighted ( ??2 2 + ??p p ) flow or voltage objectives on graphs. For flow objectives, this is the first sparsifier construction for such mixed objectives beyond unit ??p weights, and is based on expander decompositions. For voltage objectives, we give the first sparsifier construction for these objectives, which we build using graph spanners and leverage score sampling. Together with the iterative refinement framework of [Adil et al, SODA 2019], and a new multiplicative-weights based constant-approximation algorithm for mixed-objective flows or voltages, we show how to find (1+ 2 ?�poly(logn) ) approximations for weighted ??p -norm minimizing flows or voltages in p( m 1+o(1) + n 4/3+o(1) ) time for p=?(1), which is almost-linear for graphs that are slightly dense ( m??n 4/3+o(1) ).

In this paper, we study the Maximum-Profit Routing Problem with Variable Supply (MPRP-VS). This is a more general version of the Maximum-Profit Public Transportation Route Planning Problem, or simply Maximum-Profit Routing Problem (MPRP), introduced in \cite{Armaselu-PETRA}. In this new version, the quantity q i (t) supplied at site i is linearly increasing in time t , as opposed to \cite{Armaselu-PETRA}, where the quantity is constant in time. Our main result is a 5.5logT(1+ϵ)(1+ 1 1+ m √ ) 2 approximation algorithm, where T is the latest time window and m is the number of vehicles used. In addition, we improve upon the MPRP algorithm in \cite{Armaselu-PETRA} under certain conditions.

The Maximum Weighted Independent Set (MWIS) problem, which considers a graph with weights assigned to nodes and seeks to discover the "heaviest" independent set, that is, a set of nodes with maximum total weight so that no two nodes in the set are connected by an edge. The MWIS problem arises in many application domains, including the resource-constrained scheduling, error-correcting coding, complex system analysis and optimization, and communication networks. Since solving the MWIS problem is the core function for finding the optimum solution of our novel graph-based formulation of the resource-constrained Process Planning and Scheduling (PPS) problem, it is essential to have "good-performance" algorithms to solve the MWIS problem. In this paper, we propose a Novel Hybrid Heuristic Algorithm (NHHA) framework in a divide-and-conquer structure that yields optimum feasible solutions to the MWIS problem. The NHHA framework is optimized to minimize the recurrence. Using the NHHA framework, we also solve the All Maximal Independent Sets Listing (AMISL) problem, which can be seen as the subproblem of the MWIS problem. Moreover, building composed MWIS algorithms that utilizing fast approximation algorithms with the NHHA framework is an effective way to improve the accuracy of approximation MWIS algorithms (e.g., GWMIN and GWMIN2 (Sakai et al., 2003)). Eight algorithms for the MWIS problem, the exact MWIS algorithm, the AMISL algorithm, two approximation algorithms from the literature, and four composed algorithms, are applied and tested for solving the graph-based formulation of the resource-constrained PPS problem to evaluate the scalability, accuracy, and robustness.

We consider the online vector bin packing problem where n items specified by d -dimensional vectors must be packed in the fewest number of identical d -dimensional bins. Azar et al. (STOC'13) showed that for any online algorithm A , there exist instances I, such that A(I) , the number of bins used by A to pack I , is Ω(d/ log 2 d) times OPT(I) , the minimal number of bins to pack I . However in those instances, OPT(I) was only O(logd) , which left open the possibility of improved algorithms with better asymptotic competitive ratio when OPT(I)≫d . We rule this out by showing that for any arbitrary function q(⋅) and any randomized online algorithm A , there exist instances I such that E[A(I)]≥c⋅d/ log 3 d⋅OPT(I)+q(d) , for some universal constant c .

We present a polynomial space Monte Carlo algorithm that given a directed graph on n vertices and average outdegree δ , detects if the graph has a Hamiltonian cycle in 2 n−Ω( n δ ) time. This asymptotic scaling of the savings in the running time matches the fastest known exponential space algorithm by Björklund and Williams ICALP 2019. By comparison, the previously best polynomial space algorithm by Kowalik and Majewski IPEC 2020 guarantees a 2 n−Ω( n 2 δ ) time bound. Our algorithm combines for the first time the idea of obtaining a fingerprint of the presence of a Hamiltonian cycle through an inclusion--exclusion summation over the Laplacian of the graph from Björklund, Kaski, and Koutis ICALP 2017, with the idea of sieving for the non-zero terms in an inclusion--exclusion summation by listing solutions to systems of linear equations over Z 2 from Björklund and Husfeldt FOCS 2013.

Given a temporal network G(V,E,T) , (X,[ t a , t b ]) (where X⊆V(G) and [ t a , t b ]⊆T ) is said to be a (Δ,γ) \mbox{-}clique of G , if for every pair of vertices in X , there must exist at least γ links in each Δ duration within the time interval [ t a , t b ] . Enumerating such maximal cliques is an important problem in temporal network analysis, as it reveals contact pattern among the nodes of G . In this paper, we study the maximal (Δ,γ) \mbox{-}clique enumeration problem in online setting; i.e.; the entire link set of the network is not known in advance, and the links are coming as a batch in an iterative manner. Suppose, the link set till time stamp T 1 (i.e., E T 1 ), and its corresponding (Δ,γ) -clique set are known. In the next batch (till time T 2 ), a new set of links (denoted as E ( T 1 , T 2 ] ) is arrived.

Given a positive integer d , the d -CUT problem is to decide if an undirected graph G=(V,E) has a non trivial bipartition (A,B) of V such that every vertex in A (resp. B ) has at most d neighbors in B (resp. A ). When d=1 , this is the MATCHING CUT problem. Gomes and Sau, in IPEC 2019, gave the first fixed parameter tractable algorithm for d -CUT, when parameterized by maximum number of the crossing edges in the cut (i.e. the size of edge cut). However, their paper doesn't provide an explicit bound on the running time, as it indirectly relies on a MSOL formulation and Courcelle's Theorem. Motivated by this, we design and present an FPT algorithm for the MATCHING CUT (and more generally for d -CUT) for general graphs with running time 2 O(klogk) n O(1) where k is the maximum size of the edge cut. This is the first FPT algorithm for the MATCHING CUT (and d -CUT) with an explicit dependence on this parameter. We also observe a lower bound of 2 Ω(k) n O(1) with same parameter for MATCHING CUT assuming ETH.

We present a 5 3 -approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. A 7 4 -approximation algorithm for the same problem was presented recently, see Cheriyan, et al., "The matching augmentation problem: a 7 4 -approximation algorithm," {\em Math. Program.}, 182(1):315--354, 2020; arXiv:1810.07816. Our improvement is based on new algorithmic techniques, and some of these may lead to advances on related problems.

We give a randomized 1+ 8lnk k ??????????-approximation algorithm for the minimum k -edge connected spanning multi-subgraph problem, k -ECSM.