213 research outputs found
A Linear-Time Algorithm for Integral Multiterminal Flows in Trees
In this paper, we study the problem of finding an integral multiflow which maximizes the sum of flow values between every two terminals in an undirected tree with a nonnegative integer edge capacity and a set of terminals. In general, it is known that the flow value of an integral multiflow is bounded by the cut value of a cut-system which consists of disjoint subsets each of which contains exactly one terminal or has an odd cut value, and there exists a pair of an integral multiflow and a cut-system whose flow value and cut value are equal; i.e., a pair of a maximum integral multiflow and a minimum cut. In this paper, we propose an O(n)-time algorithm that finds such a pair of an integral multiflow and a cut-system in a given tree instance with n vertices. This improves the best previous results by a factor of Omega(n). Regarding a given tree in an instance as a rooted tree, we define O(n) rooted tree instances taking each vertex as a root, and establish a recursive formula on maximum integral multiflow values of these instances to design a dynamic programming that computes the maximum integral multiflow values of all O(n) rooted instances in linear time. We can prove that the algorithm implicitly maintains a cut-system so that not only a maximum integral multiflow but also a minimum cut-system can be constructed in linear time for any rooted instance whenever it is necessary. The resulting algorithm is rather compact and succinct
An Improved Approximation Algorithm for the Traveling Tournament Problem with Maximum Trip Length Two
The Traveling Tournament Problem is a complex combinatorial optimization problem in tournament timetabling, which asks a schedule of home/away games meeting specific feasibility requirements, while also minimizing the total distance traveled by all the n teams (n is even). Despite intensive algorithmic research on this problem over the last decade, most instances with more than 10 teams in well-known benchmarks are still unsolved. In this paper, we give a practical approximation algorithm for the problem with constraints such that at most two consecutive home games or away games are allowed. Our algorithm, that generates feasible schedules based on minimum perfect matchings in the underlying graph, not only improves the previous approximation ratio from (1+16/n) to about (1+4/n) but also has very good experimental performances. By applying our schedules on known benchmark sets, we can beat all previously-known results of instances with n being a multiple of 4 by 3% to 10%
Brief Announcement: Bounded-Degree Cut is Fixed-Parameter Tractable
In the bounded-degree cut problem, we are given a multigraph G=(V,E), two disjoint vertex subsets A,B subseteq V, two functions u_A, u_B:V -> {0,1,...,|E|} on V, and an integer k >= 0. The task is to determine whether there is a minimal (A,B)-cut (V_A,V_B) of size at most k such that the degree of each vertex v in V_A in the induced subgraph G[V_A] is at most u_A(v) and the degree of each vertex v in V_B in the induced subgraph G[V_B] is at most u_B(v). In this paper, we show that the bounded-degree cut problem is fixed-parameter tractable by giving a 2^{18k}|G|^{O(1)}-time algorithm. This is the first single exponential FPT algorithm for this problem. The core of the algorithm lies two new lemmas based on important cuts, which give some upper bounds on the number of candidates for vertex subsets in one part of a minimal cut satisfying some properties. These lemmas can be used to design fixed-parameter tractable algorithms for more related problems
Algorithms for Manipulating Sequential Allocation
Sequential allocation is a simple and widely studied mechanism to allocate
indivisible items in turns to agents according to a pre-specified picking
sequence of agents. At each turn, the current agent in the picking sequence
picks its most preferred item among all items having not been allocated yet.
This problem is well-known to be not strategyproof, i.e., an agent may get more
utility by reporting an untruthful preference ranking of items. It arises the
problem: how to find the best response of an agent?
It is known that this problem is polynomially solvable for only two agents
and NP-complete for arbitrary number of agents.
The computational complexity of this problem with three agents was left as an
open problem. In this paper, we give a novel algorithm that solves the problem
in polynomial time for each fixed number of agents. We also show that an agent
can always get at least half of its optimal utility by simply using its
truthful preference as the response
Breaking the Barrier for Subset Feedback Vertex Set in Chordal Graphs
The Subset Feedback Vertex Set problem (SFVS), to delete vertices from a
given graph such that any vertex in a vertex subset (called a terminal set) is
not in a cycle in the remaining graph, generalizes the famous Feedback Vertex
Set problem and Multiway Cut problem. SFVS remains -hard even in
split and chordal graphs, and SFVS in Chordal Graphs can be considered as a
special case of the 3-Hitting Set problem. However, it is not easy to solve
SFVS in Chordal Graphs faster than 3-Hitting Set. In 2019, Philip, Rajan,
Saurabh, and Tale (Algorithmica 2019) proved that SFVS in Chordal Graphs can be
solved in , slightly improving the best result for 3-Hitting Set. In this paper, we break the
"-barrier" for SFVS in Chordal Graphs by giving a -time algorithm. Our algorithm uses reduction and branching
rules based on the Dulmage-Mendelsohn decomposition and a divide-and-conquer
method.Comment: 27 pages, 8 figures. Full versio
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