Consensus Division in an Arbitrary Ratio

We consider the problem of partitioning a line segment into two subsets, so that n finite measures all have the same ratio of values for the subsets. Letting ? ? [0,1] denote the desired ratio, this generalises the PPA-complete consensus-halving problem, in which ? = 1/2. Stromquist and Woodall [Stromquist and Woodall, 1985] showed that for any ?, there exists a solution using 2n cuts of the segment. They also showed that if ? is irrational, that upper bound is almost optimal. In this work, we elaborate the bounds for rational values ?. For ? = ?/k, we show a lower bound of (k-1)/k ? 2n - O(1) cuts; we also obtain almost matching upper bounds for a large subset of rational ?. On the computational side, we explore its dependence on the number of cuts available. More specifically, 1) when using the minimal number of cuts for each instance is required, the problem is NP-hard for any ?; 2) for a large subset of rational ? = ?/k, when (k-1)/k ? 2n cuts are available, the problem is in PPA-k under Turing reduction; 3) when 2n cuts are allowed, the problem belongs to PPA for any ?; more generally, the problem belong to PPA-p for any prime p if 2(p-1)??p/2?/?p/2? ? n cuts are available

Two's Company, Three's a Crowd:Consensus-Halving for a Constant Number of Agents

We consider the $\varepsilon$-Consensus-Halving problem, in which a set of heterogeneous agents aim at dividing a continuous resource into two (not necessarily contiguous) portions that all of them simultaneously consider to be of approximately the same value (up to $\varepsilon$). This problem was recently shown to be PPA-complete, for $n$ agents and $n$ cuts, even for very simple valuation functions. In a quest to understand the root of the complexity of the problem, we consider the setting where there is only a constant number of agents, and we consider both the computational complexity and the query complexity of the problem. For agents with monotone valuation functions, we show a dichotomy: for two agents the problem is polynomial-time solvable, whereas for three or more agents it becomes PPA-complete. Similarly, we show that for two monotone agents the problem can be solved with polynomially-many queries, whereas for three or more agents, we provide exponential query complexity lower bounds. These results are enabled via an interesting connection to a monotone Borsuk-Ulam problem, which may be of independent interest. For agents with general valuations, we show that the problem is PPA-complete and admits exponential query complexity lower bounds, even for two agents

A topological characterization of modulo-p arguments and implications for necklace splitting

The classes PPA-p have attracted attention lately, because they are the main candidates for capturing the complexity of Necklace Splitting with p thieves, for prime p. However, these classes were not known to have complete problems of a topological nature, which impedes any progress towards settling the complexity of the Necklace Splitting problem. On the contrary, topological problems have been pivotal in obtaining completeness results for PPAD and PPA, such as the PPAD-completeness of finding a Nash equilibrium [18, 15] and the PPA-completeness of Necklace Splitting with 2 thieves [24]. In this paper, we provide the first topological characterization of the classes PPA-p. First, we show that the computational problem associated with a simple generalization of Tucker's Lemma, termed p-polygon-Tucker, as well as the associated Borsuk-Ulam-type theorem, p-polygon-Borsuk-Ulam, are PPA-p-complete. Then, we show that the computational version of the well-known BSS Theorem [8], as well as the associated BSS-Tucker problem are PPA-p-complete. Finally, using a different generalization of Tucker's Lemma (termed Zp-star-Tucker), which we prove to be PPA-p-complete, we prove that p-thief Necklace Splitting is in PPA-p. This latter result gives a new combinatorial proof for the Necklace Splitting theorem, the only proof of this nature other than that of Meunier [42]. All of our containment results are obtained through a new combinatorial proof for Zp-versions of Tucker's lemma that is a natural generalization of the standard combinatorial proof of Tucker's lemma by Freund and Todd [27]. We believe that this new proof technique is of independent interest

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Efficient Splitting of Measures and Necklaces

We provide approximation algorithms for two problems, known as NECKLACE SPLITTING and $\epsilon$-CONSENSUS SPLITTING. In the problem $\epsilon$-CONSENSUS SPLITTING, there are $n$ measures, represented by valuation functions over the interval $[0, 1]$ and $k$ agents. The goal is to divide the interval, via at most $n (k-1)$ cuts, into pieces and distribute them to the $k$ agents in an approximately equitable way, so that each agent receives $1/k$ of each type of measure, with an error up to $\epsilon / 2$. It is known that this is possible even for $\epsilon = 0$. NECKLACE SPLITTING is a discrete version of $\epsilon$-CONSENSUS SPLITTING. For $k = 2$ and some absolute positive constant $\epsilon$, both of these problems are PPAD-hard. We consider two types of approximation. The first one has the aim of providing every agent with a positive amount of measure of each type under the constraint of making at most $n (k - 1)$ cuts. The second one has the aim of providing an approximately equitable split with as few cuts as possible. Apart from the offline model, we consider the Online model as well, where the interval (or necklace) is presented as a stream, and decisions about cutting and distributing must be made on the spot. For the first type of approximation, we provide an efficient algorithm that gives every agent at least $\frac{1}{nk}$ of each type of measure and works for the Online model as well. For the second type of approximation, we provide an efficient online algorithm that makes $\text{poly}(n, k, \epsilon)$ cuts and an offline algorithm making $O(nk \log \frac{1}{\epsilon})$ cuts. We also provide unconditional hardness results in the Online model for both problems even in the case of $k=2$ agents. Our lower bounds show that our online algorithm for the second type of approximation is optimal up to a factor of $\Theta(\log n)$ in terms of the number of cuts made

Efficient Splitting of Measures and Necklaces

We provide approximation algorithms for two problems, known as NECKLACE SPLITTING and $\epsilon$-CONSENSUS SPLITTING. In the problem $\epsilon$-CONSENSUS SPLITTING, there are $n$ measures, represented by valuation functions over the interval $[0, 1]$ and $k$ agents. The goal is to divide the interval, via at most $n (k-1)$ cuts, into pieces and distribute them to the $k$ agents in an approximately equitable way, so that each agent receives $1/k$ of each type of measure, with an error up to $\epsilon / 2$. It is known that this is possible even for $\epsilon = 0$. NECKLACE SPLITTING is a discrete version of $\epsilon$-CONSENSUS SPLITTING. For $k = 2$ and some absolute positive constant $\epsilon$, both of these problems are PPAD-hard. We consider two types of approximation. The first one has the aim of providing every agent with a positive amount of measure of each type under the constraint of making at most $n (k - 1)$ cuts. The second one has the aim of providing an approximately equitable split with as few cuts as possible. Apart from the offline model, we consider the Online model as well, where the interval (or necklace) is presented as a stream, and decisions about cutting and distributing must be made on the spot. For the first type of approximation, we provide an efficient algorithm that gives every agent at least $\frac{1}{nk}$ of each type of measure and works for the Online model as well. For the second type of approximation, we provide an efficient online algorithm that makes $\text{poly}(n, k, \epsilon)$ cuts and an offline algorithm making $O(nk \log \frac{1}{\epsilon})$ cuts. We also provide unconditional hardness results in the Online model for both problems even in the case of $k=2$ agents. Our lower bounds show that our online algorithm for the second type of approximation is optimal up to a factor of $\Theta(\log n)$ in terms of the number of cuts made

Efficient Splitting of Measures and Necklaces

We provide approximation algorithms for two problems, known as NECKLACE SPLITTING and $\epsilon$-CONSENSUS SPLITTING. In the problem $\epsilon$-CONSENSUS SPLITTING, there are $n$ measures, represented by valuation functions over the interval $[0, 1]$ and $k$ agents. The goal is to divide the interval, via at most $n (k-1)$ cuts, into pieces and distribute them to the $k$ agents in an approximately equitable way, so that each agent receives $1/k$ of each type of measure, with an error up to $\epsilon / 2$. It is known that this is possible even for $\epsilon = 0$. NECKLACE SPLITTING is a discrete version of $\epsilon$-CONSENSUS SPLITTING. For $k = 2$ and some absolute positive constant $\epsilon$, both of these problems are PPAD-hard. We consider two types of approximation. The first one has the aim of providing every agent with a positive amount of measure of each type under the constraint of making at most $n (k - 1)$ cuts. The second one has the aim of providing an approximately equitable split with as few cuts as possible. Apart from the offline model, we consider the Online model as well, where the interval (or necklace) is presented as a stream, and decisions about cutting and distributing must be made on the spot. For the first type of approximation, we provide an efficient algorithm that gives every agent at least $\frac{1}{nk}$ of each type of measure and works for the Online model as well. For the second type of approximation, we provide an efficient online algorithm that makes $\text{poly}(n, k, \epsilon)$ cuts and an offline algorithm making $O(nk \log \frac{1}{\epsilon})$ cuts. We also provide unconditional hardness results in the Online model for both problems even in the case of $k=2$ agents. Our lower bounds show that our online algorithm for the second type of approximation is optimal up to a factor of $\Theta(\log n)$ in terms of the number of cuts made