353 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Fast Algorithms for Separable Linear Programs

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    In numerical linear algebra, considerable effort has been devoted to obtaining faster algorithms for linear systems whose underlying matrices exhibit structural properties. A prominent success story is the method of generalized nested dissection~[Lipton-Rose-Tarjan'79] for separable matrices. On the other hand, the majority of recent developments in the design of efficient linear program (LP) solves do not leverage the ideas underlying these faster linear system solvers nor consider the separable structure of the constraint matrix. We give a faster algorithm for separable linear programs. Specifically, we consider LPs of the form minAx=b,lxucx\min_{\mathbf{A}\mathbf{x}=\mathbf{b}, \mathbf{l}\leq\mathbf{x}\leq\mathbf{u}} \mathbf{c}^\top\mathbf{x}, where the graphical support of the constraint matrix ARn×m\mathbf{A} \in \mathbb{R}^{n\times m} is O(nα)O(n^\alpha)-separable. These include flow problems on planar graphs and low treewidth matrices among others. We present an O~((m+m1/2+2α)log(1/ϵ))\tilde{O}((m+m^{1/2 + 2\alpha}) \log(1/\epsilon)) time algorithm for these LPs, where ϵ\epsilon is the relative accuracy of the solution. Our new solver has two important implications: for the kk-multicommodity flow problem on planar graphs, we obtain an algorithm running in O~(k5/2m3/2)\tilde{O}(k^{5/2} m^{3/2}) time in the high accuracy regime; and when the support of A\mathbf{A} is O(nα)O(n^\alpha)-separable with α1/4\alpha \leq 1/4, our algorithm runs in O~(m)\tilde{O}(m) time, which is nearly optimal. The latter significantly improves upon the natural approach of combining interior point methods and nested dissection, whose time complexity is lower bounded by Ω(m(m+mαω))=Ω(m3/2)\Omega(\sqrt{m}(m+m^{\alpha\omega}))=\Omega(m^{3/2}), where ω\omega is the matrix multiplication constant. Lastly, in the setting of low-treewidth LPs, we recover the results of [DLY,STOC21] and [GS,22] with significantly simpler data structure machinery.Comment: 55 pages. To appear at SODA 202

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Dynamic programming on bipartite tree decompositions

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    We revisit a graph width parameter that we dub bipartite treewidth, along with its associated graph decomposition that we call bipartite tree decomposition. Bipartite treewidth can be seen as a common generalization of treewidth and the odd cycle transversal number. Intuitively, a bipartite tree decomposition is a tree decomposition whose bags induce almost bipartite graphs and whose adhesions contain at most one vertex from the bipartite part of any other bag, while the width of such decomposition measures how far the bags are from being bipartite. Adapted from a tree decomposition originally defined by Demaine, Hajiaghayi, and Kawarabayashi [SODA 2010] and explicitly defined by Tazari [Th. Comp. Sci. 2012], bipartite treewidth appears to play a crucial role for solving problems related to odd-minors, which have recently attracted considerable attention. As a first step toward a theory for solving these problems efficiently, the main goal of this paper is to develop dynamic programming techniques to solve problems on graphs of small bipartite treewidth. For such graphs, we provide a number of para-NP-completeness results, FPT-algorithms, and XP-algorithms, as well as several open problems. In particular, we show that KtK_t-Subgraph-Cover, Weighted Vertex Cover/Independent Set, Odd Cycle Transversal, and Maximum Weighted Cut are FPTFPT parameterized by bipartite treewidth. We provide the following complexity dichotomy when HH is a 2-connected graph, for each of HH-Subgraph-Packing, HH-Induced-Packing, HH-Scattered-Packing, and HH-Odd-Minor-Packing problem: if HH is bipartite, then the problem is para-NP-complete parameterized by bipartite treewidth while, if HH is non-bipartite, then it is solvable in XP-time. We define 1-H{\cal H}-treewidth by replacing the bipartite graph class by any class H{\cal H}. Most of the technology developed here works for this more general parameter.Comment: Presented in IPEC 202

    How to assign volunteers to tasks compatibly ? A graph theoretic and parameterized approach

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    In this paper we study a resource allocation problem that encodes correlation between items in terms of \conflict and maximizes the minimum utility of the agents under a conflict free allocation. Admittedly, the problem is computationally hard even under stringent restrictions because it encodes a variant of the {\sc Maximum Weight Independent Set} problem which is one of the canonical hard problems in both classical and parameterized complexity. Recently, this subject was explored by Chiarelli et al.~[Algorithmica'22] from the classical complexity perspective to draw the boundary between {\sf NP}-hardness and tractability for a constant number of agents. The problem was shown to be hard even for small constant number of agents and various other restrictions on the underlying graph. Notwithstanding this computational barrier, we notice that there are several parameters that are worth studying: number of agents, number of items, combinatorial structure that defines the conflict among the items, all of which could well be small under specific circumstancs. Our search rules out several parameters (even when taken together) and takes us towards a characterization of families of input instances that are amenable to polynomial time algorithms when the parameters are constant. In addition to this we give a superior 2^{m}|I|^{\Co{O}(1)} algorithm for our problem where mm denotes the number of items that significantly beats the exhaustive \Oh(m^{m}) algorithm by cleverly using ideas from FFT based fast polynomial multiplication; and we identify simple graph classes relevant to our problem's motivation that admit efficient algorithms

    Thresholds for Latin squares and Steiner triple systems: Bounds within a logarithmic factor

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    We prove that for nNn \in \mathbb N and an absolute constant CC, if pClog2n/np \geq C\log^2 n / n and Li,j[n]L_{i,j} \subseteq [n] is a random subset of [n][n] where each k[n]k\in [n] is included in Li,jL_{i,j} independently with probability pp for each i,j[n]i, j\in [n], then asymptotically almost surely there is an order-nn Latin square in which the entry in the iith row and jjth column lies in Li,jL_{i,j}. The problem of determining the threshold probability for the existence of an order-nn Latin square was raised independently by Johansson, by Luria and Simkin, and by Casselgren and H{\"a}ggkvist; our result provides an upper bound which is tight up to a factor of logn\log n and strengthens the bound recently obtained by Sah, Sawhney, and Simkin. We also prove analogous results for Steiner triple systems and 11-factorizations of complete graphs, and moreover, we show that each of these thresholds is at most the threshold for the existence of a 11-factorization of a nearly complete regular bipartite graph.Comment: 32 pages, 1 figure. Final version, to appear in Transactions of the AM

    Determinantal Sieving

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    We introduce determinantal sieving, a new, remarkably powerful tool in the toolbox of algebraic FPT algorithms. Given a polynomial P(X)P(X) on a set of variables X={x1,,xn}X=\{x_1,\ldots,x_n\} and a linear matroid M=(X,I)M=(X,\mathcal{I}) of rank kk, both over a field F\mathbb{F} of characteristic 2, in 2k2^k evaluations we can sieve for those terms in the monomial expansion of PP which are multilinear and whose support is a basis for MM. Alternatively, using 2k2^k evaluations of PP we can sieve for those monomials whose odd support spans MM. Applying this framework, we improve on a range of algebraic FPT algorithms, such as: 1. Solving qq-Matroid Intersection in time O(2(q2)k)O^*(2^{(q-2)k}) and qq-Matroid Parity in time O(2qk)O^*(2^{qk}), improving on O(4qk)O^*(4^{qk}) (Brand and Pratt, ICALP 2021) 2. TT-Cycle, Colourful (s,t)(s,t)-Path, Colourful (S,T)(S,T)-Linkage in undirected graphs, and the more general Rank kk (S,T)(S,T)-Linkage problem, all in O(2k)O^*(2^k) time, improving on O(2k+S)O^*(2^{k+|S|}) respectively O(2S+O(k2log(k+F)))O^*(2^{|S|+O(k^2 \log(k+|\mathbb{F}|))}) (Fomin et al., SODA 2023) 3. Many instances of the Diverse X paradigm, finding a collection of rr solutions to a problem with a minimum mutual distance of dd in time O(2r(r1)d/2)O^*(2^{r(r-1)d/2}), improving solutions for kk-Distinct Branchings from time 2O(klogk)2^{O(k \log k)} to O(2k)O^*(2^k) (Bang-Jensen et al., ESA 2021), and for Diverse Perfect Matchings from O(22O(rd))O^*(2^{2^{O(rd)}}) to O(2r2d/2)O^*(2^{r^2d/2}) (Fomin et al., STACS 2021) All matroids are assumed to be represented over a field of characteristic 2. Over general fields, we achieve similar results at the cost of using exponential space by working over the exterior algebra. For a class of arithmetic circuits we call strongly monotone, this is even achieved without any loss of running time. However, the odd support sieving result appears to be specific to working over characteristic 2

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum
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