10 research outputs found
A Fixed-Parameter Algorithm for the Schrijver Problem
The Schrijver graph is defined for integers and with as the graph whose vertices are all the -subsets of
that do not include two consecutive elements modulo , where two such sets
are adjacent if they are disjoint. A result of Schrijver asserts that the
chromatic number of is (Nieuw Arch. Wiskd., 1978). In the
computational Schrijver problem, we are given an access to a coloring of the
vertices of with colors, and the goal is to find a
monochromatic edge. The Schrijver problem is known to be complete in the
complexity class . We prove that it can be solved by a randomized
algorithm with running time , hence it is
fixed-parameter tractable with respect to the parameter .Comment: 19 pages. arXiv admin note: substantial text overlap with
arXiv:2204.0676
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
Pure-Circuit: Strong Inapproximability for PPAD
The current state-of-the-art methods for showing inapproximability in PPAD
arise from the -Generalized-Circuit (-GCircuit)
problem. Rubinstein (2018) showed that there exists a small unknown constant
for which -GCircuit is PPAD-hard, and subsequent
work has shown hardness results for other problems in PPAD by using
-GCircuit as an intermediate problem.
We introduce Pure-Circuit, a new intermediate problem for PPAD, which can be
thought of as -GCircuit pushed to the limit as , and we show that the problem is PPAD-complete. We then prove
that -GCircuit is PPAD-hard for all by a
reduction from Pure-Circuit, and thus strengthen all prior work that has used
GCircuit as an intermediate problem from the existential-constant regime to the
large-constant regime.
We show that stronger inapproximability results can be derived by reducing
directly from Pure-Circuit. In particular, we prove tight inapproximability
results for computing -well-supported Nash equilibria in
two-action polymatrix games, as well as for finding approximate equilibria in
threshold games
Constant Inapproximability for PPA
In the -Consensus-Halving problem, we are given probability measures on the interval , and the goal is to partition into two parts and using at most cuts, so that for all . This fundamental fair division problem was the first natural problem shown to be complete for the class PPA, and all subsequent PPA-completeness results for other natural problems have been obtained by reducing from it. We show that -Consensus-Halving is PPA-complete even when the parameter is a constant. In fact, we prove that this holds for any constant . As a result, we obtain constant inapproximability results for all known natural PPA-complete problems, including Necklace-Splitting, the Discrete-Ham-Sandwich problem, two variants of the pizza sharing problem, and for finding fair independent sets in cycles and paths
Pure-Circuit: Strong Inapproximability for PPAD
The current state-of-the-art methods for showing inapproximability in PPAD arise from the ε-Generalized-Circuit (ε-GCircuit) problem. Rubinstein (2018) showed that there exists a small unknown constant ε for which ε-GCircuit is PPAD-hard, and subsequent work has shown hardness results for other problems in PPAD by using ε-GCircuit as an intermediate problem.
We introduce Pure-Circuit, a new intermediate problem for PPAD, which can be thought of as ε-GCircuit pushed to the limit as ε→1, and we show that the problem is PPAD-complete. We then prove that ε-GCircuit is PPAD-hard for all ε<0.1 by a reduction from Pure-Circuit, and thus strengthen all prior work that has used GCircuit as an intermediate problem from the existential-constant regime to the large-constant regime.
We show that stronger inapproximability results can be derived by reducing directly from Pure-Circuit. In particular, we prove tight inapproximability results for computing ε-well-supported Nash equilibria in two-action polymatrix games, as well as for finding approximate equilibria in threshold games
Pure-Circuit: Strong Inapproximability for PPAD
The current state-of-the-art methods for showing inapproximability in PPAD arise from the ε-Generalized-Circuit (ε-GCircuit) problem. Rubinstein (2018) showed that there exists a small unknown constant ε for which ε-GCircuit is PPAD-hard, and subsequent work has shown hardness results for other problems in PPAD by using ε-GCircuit as an intermediate problem.
We introduce Pure-Circuit, a new intermediate problem for PPAD, which can be thought of as ε-GCircuit pushed to the limit as ε→1, and we show that the problem is PPAD-complete. We then prove that ε-GCircuit is PPAD-hard for all ε<0.1 by a reduction from Pure-Circuit, and thus strengthen all prior work that has used GCircuit as an intermediate problem from the existential-constant regime to the large-constant regime.
We show that stronger inapproximability results can be derived by reducing directly from Pure-Circuit. In particular, we prove tight inapproximability results for computing ε-well-supported Nash equilibria in two-action polymatrix games, as well as for finding approximate equilibria in threshold games
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Consensus halving for sets of items
Consensus halving refers to the problem of dividing a resource into two parts so that every agent values both parts equally. Prior work shows that, when the resource is represented by an interval, a consensus halving with at most n cuts always exists but is hard to compute even for agents with simple valuation functions. In this paper, we study consensus halving in a natural setting in which the resource consists of a set of items without a linear ordering. For agents with linear and additively separable utilities, we present a polynomial-time algorithm that computes a consensus halving with at most n cuts and show that n cuts are almost surely necessary when the agents’ utilities are randomly generated. On the other hand, we show that, for a simple class of monotonic utilities, the problem already becomes polynomial parity argument, directed version–hard. Furthermore, we compare and contrast consensus halving with the more general problem of consensus k-splitting, with which we wish to divide the resource into k parts in possibly unequal ratios and provide some consequences of our results on the problem of computing small agreeable sets
Consensus halving for sets of items
Consensus halving refers to the problem of dividing a resource into two parts so that every agent values both parts equally. Prior work shows that, when the resource is represented by an interval, a consensus halving with at most n cuts always exists but is hard to compute even for agents with simple valuation functions. In this paper, we study consensus halving in a natural setting in which the resource consists of a set of items without a linear ordering. For agents with linear and additively separable utilities, we present a polynomial-time algorithm that computes a consensus halving with at most n cuts and show that n cuts are almost surely necessary when the agents’ utilities are randomly generated. On the other hand, we show that, for a simple class of monotonic utilities, the problem already becomes polynomial parity argument, directed version–hard. Furthermore, we compare and contrast consensus halving with the more general problem of consensus k-splitting, with which we wish to divide the resource into k parts in possibly unequal ratios and provide some consequences of our results on the problem of computing small agreeable sets