Algorithmica最新文献

In the spanning tree congestion problem, given a connected graph G, the objective is to compute a spanning tree T in G that minimizes its maximum edge congestion, where the congestion of an edge e of T is the number of edges in G for which the unique path in T between their endpoints traverses e. The problem is known to be (mathbb{N}mathbb{P})-hard, but its approximability is still poorly understood, and it is not even known whether the optimum solution can be efficiently approximated with ratio o(n). In the decision version of this problem, denoted ({varvec{K}-textsf {STC}}), we need to determine if G has a spanning tree with congestion at most K. It is known that ({varvec{K}-textsf {STC}}) is (mathbb{N}mathbb{P})-complete for (Kge 8), and this implies a lower bound of 1.125 on the approximation ratio of minimizing congestion. On the other hand, ({varvec{3}-textsf {STC}}) can be solved in polynomial time, with the complexity status of this problem for (Kin { left{ 4,5,6,7 right} }) remaining an open problem. We substantially improve the earlier hardness results by proving that ({varvec{K}-textsf {STC}}) is (mathbb{N}mathbb{P})-complete for (Kge 5). This leaves only the case (K=4) open, and improves the lower bound on the approximation ratio to 1.2. Motivated by evidence that minimizing congestion is hard even for graphs of small constant radius, we also consider ({varvec{K}-textsf {STC}}) restricted to graphs of radius 2, and we prove that this variant is (mathbb{N}mathbb{P})-complete for all (Kge 6).

Let M be a perfect matching on a set of points in the plane where every edge is a line segment between two points. We say that M is globally maximum if it is a maximum-length matching on all points. We say that M is k-local maximum if for any subset (M'={a_1b_1,dots ,a_kb_k}) of k edges of M it holds that (M') is a maximum-length matching on points ({a_1,b_1,dots ,a_k,b_k}). We show that local maximum matchings are good approximations of global ones. Let (mu _k) be the infimum ratio of the length of any k-local maximum matching to the length of any global maximum matching, over all finite point sets in the Euclidean plane. It is known that (mu _kgeqslant frac{k-1}{k}) for any (kgeqslant 2). We show the following improved bounds for (kin {2,3}): (sqrt{3/7}leqslant mu _2< 0.93 ) and (sqrt{3}/2leqslant mu _3< 0.98). We also show that every pairwise crossing matching is unique and it is globally maximum. Towards our proof of the lower bound for (mu _2) we show the following result which is of independent interest: If we increase the radii of pairwise intersecting disks by factor (2/sqrt{3}), then the resulting disks have a common intersection.

It is natural to generalize the online (k)-Server problem by allowing each request to specify not only a point p, but also a subset S of servers that may serve it. To date, only a few special cases of this problem have been studied. The objective of the work presented in this paper has been to more systematically explore this generalization in the case of uniform and star metrics. For uniform metrics, the problem is equivalent to a generalization of Paging in which each request specifies not only a page p, but also a subset S of cache slots, and is satisfied by having a copy of p in some slot in S. We call this problem Slot-Heterogenous Paging. In realistic settings only certain subsets of cache slots or servers would appear in requests. Therefore we parameterize the problem by specifying a family ({mathcal {S}}subseteq 2^{[k]}) of requestable slot sets, and we establish bounds on the competitive ratio as a function of the cache size k and family ({mathcal {S}}):

• If all request sets are allowed (({mathcal {S}}=2^{[k]}setminus {emptyset })), the optimal deterministic and randomized competitive ratios are exponentially worse than for standard Paging (({mathcal {S}}={[k]})).

• As a function of (|{mathcal {S}}|) and k, the optimal deterministic ratio is polynomial: at most (O(k^2|{mathcal {S}}|)) and at least (Omega (sqrt{|{mathcal {S}}|})).

• For any laminar family ({mathcal {S}}) of height h, the optimal ratios are O(hk) (deterministic) and (O(h^2log k)) (randomized).

• The special case of laminar ({mathcal {S}}) that we call All-or-One Paging extends standard Paging by allowing each request to specify a specific slot to put the requested page in. The optimal deterministic ratio for weighted All-or-One Paging is (Theta (k)). Offline All-or-One Paging is (mathbb{N}mathbb{P})-hard.

Some results for the laminar case are shown via a reduction to the generalization of Paging in which each request specifies a set (P) of pages, and is satisfied by fetching any page from (P) into the cache. The optimal ratios for the latter problem (with laminar family of height h) are at most hk (deterministic) and (hH_k) (randomized).

In the general AntiFactor problem, a graph G and, for every vertex v of G, a set (X_vsubseteq {mathbb {N}}) of forbidden degrees is given. The task is to find a set S of edges such that the degree of v in S is not in the set (X_v). Standard techniques (dynamic programming plus fast convolution) can be used to show that if M is the largest forbidden degree, then the problem can be solved in time ((M+2)^{{operatorname {tw}}}cdot n^{{mathcal {O}}(1)}) if a tree decomposition of width ({operatorname {tw}}) is given. However, significantly faster algorithms are possible if the sets (X_v) are sparse: our main algorithmic result shows that if every vertex has at most (x) forbidden degrees (we call this special case AntiFactor_x), then the problem can be solved in time ((x+1)^{{mathcal {O}}({operatorname {tw}})}cdot n^{{mathcal {O}}(1)}). That is, AntiFactor_x is fixed-parameter tractable parameterized by treewidth ({operatorname {tw}}) and the maximum number (x) of excluded degrees. Our algorithm uses the technique of representative sets, which can be generalized to the optimization version, but (as expected) not to the counting version of the problem. In fact, we show that #AntiFactor₁ is already #W ([1])-hard parameterized by the width of the given decomposition. Moreover, we show that, unlike for the decision version, the standard dynamic programming algorithm is essentially optimal for the counting version. Formally, for a fixed nonempty set (X), we denote by (X)-AntiFactor the special case where every vertex v has the same set (X_v=X) of forbidden degrees. We show the following lower bound for every fixed set (X): if there is an (epsilon >0) such that #(X)-AntiFactor can be solved in time ((max X+2-epsilon )^{{operatorname {tw}}}cdot n^{{mathcal {O}}(1)}) given a tree decomposition of width ({operatorname {tw}}), then the counting strong exponential-time hypothesis (#SETH) fails.

Depth first search is a natural algorithmic technique for constructing a closed route that visits all vertices of a graph. The length of such a route equals, in an edge-weighted tree, twice the total weight of all edges of the tree and this is asymptotically optimal over all exploration strategies. This paper considers a variant of such search strategies where the length of each route is bounded by a positive integer B (e.g. due to limited energy resources of the searcher). The objective is to cover all the edges of a tree T using the minimum number of routes, each starting and ending at the root and each being of length at most B. To this end, we analyze the following natural greedy tree traversal process that is based on decomposing a depth first search traversal into a sequence of limited length routes. Given any arbitrary depth first search traversal R of the tree T, we cover R with routes (R_1,ldots ,R_l), each of length at most B such that: (R_i) starts at the root, reaches directly the farthest point of R visited by (R_{i-1}), then (R_i) continues along the path R as far as possible, and finally (R_i) returns to the root. We call the above algorithm piecemeal-DFS and we prove that it achieves the asymptotically minimal number of routes l, regardless of the choice of R. Our analysis also shows that the total length of the traversal (and thus the traversal time) of piecemeal-DFS is asymptotically minimum over all energy-constrained exploration strategies. The fact that R can be chosen arbitrarily means that the exploration strategy can be constructed in an online fashion when the input tree T is not known in advance. Each route (R_i) can be constructed without any knowledge of the yet unvisited part of T. Surprisingly, our results show that depth first search is efficient for energy constrained exploration of trees, even though it is known that the same does not hold for energy constrained exploration of arbitrary graphs.

The problem of scheduling unrelated machines has been studied since the inception of algorithmic mechanism design (Nisan and Ronen, Algorithmic mechanism design(extended abstract). In: Proceedings of the Thirty First Annual ACM Symposium on Theory of Computing (STOC), pp. 129–140, 1999. It is a resource allocation problem that entails assigning m tasks to n machines for execution. Machines are regarded as strategic agents who may lie about their execution costs so as to minimize their time cost. To address the situation when monetary payment is not an option to compensate the machines’ costs, Koutsoupias (Theory Comput Syst 54:375–387, 2014) devised two truthful mechanisms, K and P respectively, that achieves an approximation ratio of (frac{n+1}{2}) and n, for social cost minimization. In addition, no truthful mechanism can achieve an approximation ratio better than (frac{n+1}{2}). Hence, mechanism K is optimal. While the approximation ratio provides a strong worst-case guarantee, it also limits us to a comprehensive understanding of mechanism performance on various inputs. This paper investigates these two scheduling mechanisms beyond the worst case. We first show that mechanism K achieves a smaller social cost than mechanism P on every input. That is, mechanism K is pointwise better than mechanism P. Next, for each task, when machines’ execution costs are independent and identically drawn from a task-specific distribution, we show that the average-case approximation ratio of mechanism K converges to a constant determined by the task-specific distribution. This bound is tight for mechanism K. For a better understanding of this distribution-dependent constant, on the one hand, we estimate its value by plugging in a few common distributions; on the other, we show that this converging bound improves a known bound (Zhang in Algorithmica 83(6):1638–1652, 2021)) which only captures the single-task setting. Last, we find that the average-case approximation ratio of mechanism P converges to the same constant.

This work revisits quantum algorithms for the well-known welded tree problem, proposing a succinct quantum algorithm based on the simple coined quantum walks. It iterates the naturally defined coined quantum walk operator for a classically precomputed number of iterations, and measures. The number of iterations is linear in the depth of the tree. The success probability of this procedure is inversely linear in the depth of the tree. Moreover, it is the same for all instances of the problem of a fixed size, therefore, we can use the exact quantum amplitude amplification subroutine to answer with probability 1. This gives an exponential speedup over any classical algorithm for the same problem. The significance of the results may be seen as follows. (i) Our algorithm is rather simple compared with the one in (Jeffery and Zur, STOC’2023), which not only breaks the stereotype that coined quantum walks can only achieve quadratic speedups over classical algorithms, but also demonstrates the power of the simplest quantum walk model. (ii) Our algorithm achieves certainty of success for the first time. Thus, it becomes one of the few examples that exhibit exponential separation between exact quantum and randomized query complexities.

We study a new combinatorial problem based on the famous Minimum Common String Partition problem, which we call Permutation-constrained Common String Partition (PCSP for short). In PCSP, we are given two sequences/genomes s and t with the same length and a permutation (pi ) on ([ell ]), the question is to decide whether it is possible to decompose s and t into (ell ) blocks that can be matched according to some specified requirements, and that conform with the permutation (pi ). Our main result is that PCSP is FPT in parameter (ell + d), where d is the maximum number of occurrences that any symbol may have in s or t. We also study a variant where the input specifies whether each matched pair of block needs to be preserved as is, or reversed. With this result on PCSP, we show that a series of genome rearrangement problems are FPT (k + d), where k is the rearrangement distance between two genomes of interest.

In the allocation of indivisible goods, a prominent fairness notion is envy-freeness up to one good (EF1). We initiate the study of reachability problems in fair division by investigating the problem of whether one EF1 allocation can be reached from another EF1 allocation via a sequence of exchanges such that every intermediate allocation is also EF1. We show that two EF1 allocations may not be reachable from each other even in the case of two agents, and deciding their reachability is PSPACE-complete in general. On the other hand, we prove that reachability is guaranteed for two agents with identical or binary utilities as well as for any number of agents with identical binary utilities. We also examine the complexity of deciding whether there is an EF1 exchange sequence that is optimal in the number of exchanges required.

The classical and extensively-studied Fréchet distance between two curves is defined as an inf max, where the infimum is over all traversals of the curves, and the maximum is over all concurrent positions of the two agents. In this article we investigate a “flipped” Fréchet measure defined by a sup min – the supremum is over all traversals of the curves, and the minimum is over all concurrent positions of the two agents. This measure produces a notion of “social distance” between two curves (or general domains), where agents traverse curves while trying to stay as far apart as possible. We first study the flipped Fréchet measure between two polygonal curves in one and two dimensions, providing conditional lower bounds and matching algorithms. We then consider this measure on polygons, where it denotes the minimum distance that two agents can maintain while restricted to travel in or on the boundary of the same polygon. We investigate several variants of the problem in this setting, for some of which we provide linear-time algorithms. We draw connections between our proposed flipped Fréchet measure and existing related work in computational geometry, hoping that our new measure may spawn investigations akin to those performed for the Fréchet distance, and into further interesting problems that arise.