As mentioned in the introduction (Example 1.1), one of the problems originally shown to be AP-equivalent to #BISis#Downsets. The input of#Downsets is usually defined to be a partially ordered set. However, in this section we will think of the input as being a directed graph. A downset in a directed graph is a set of vertices D such that for all u ∈ D and all arcs u → v, we have v ∈ D. We can then consider the following problem, which is easily seen to be AP-equivalent to the usual definition as given in the introduction.
Name. #Downsets
Instance. A directed graphG.
Output. The number of downsets in G.
Downsets correspond to satisfying assignments of the constraints h(u, v),IMPi for each u → v, where IMP = {(0,0),(0,1),(1,1)}. The restriction of #Downsets to di- rected graphs of maximum degree at mostdis therefore AP-equivalent to#CSP≤d(IMP). (“Degree” will always mean total degree: the sum of the in-degree and out-degree.) In particular the restriction to directed graphs of maximum degree at most 2 is in FP by Proposition 4.9, and the restriction to directed graphs of maximum degree at most 3 is AP-equivalent to #BISby [51, Theorem 23].
However, the reductions used in [51] rely in a crucial way on cycles. Contracting a cycle to a single vertex does not affect the number of downsets. So #Downsets is AP-equivalent to its restriction to directed acyclic graphs, which we abbreviate asdags. In this section we will show that#Downsetsremains AP-equivalent to#BISeven when restricted to dags of maximum degree three.
Remark 4.18. Consider the restriction of #Downsets to dags of depth two, that is, di- rected bipartite graphs with a bipartition U∪V such that all the arcs go from U to V. There is a bijection I 7→ I4V from the set of independent sets of the underlying undi- rected graph to the set of downsets of the original graph. So the restriction of#Downsets
to directed graphs of depth two is still AP-equivalent to#BIS. But there is an FPRAS for the restriction of#Downsetsto dags of depth two and maximum degree five - we can use the equivalence just described for dags of depth two, then apply the Weitz’s FPRAS for counting independent sets in graphs of maximum degree five [100].
Lemma 4.19. Given an integer n≥4 specified in binary, we can build a dagGthat has exactly n downsets, in polynomial time in logn, such that G has a unique source and a unique sink, which are distinct and each have degree 1, and such that every vertex of G
has degree at most three.
Proof. We will first define certain graphsHp,p ≥2. Define H2 to be the directed path
H2 H3
s u t
H4
s u t
Figure 4.6: Three Hp graphs used in Lemma 4.19.
Hp and arcss→s0 →u→tandt0→twheres0 denotes the source ofHp and t0 denotes the sink ofHp; see Figure 4.6. The downsets ofHp+1 are∅,V(Hp),V(Hp)∪ {u, t}, and D∪ {t}andD∪ {u, t}for each downsetDofHp that does not include the sources0, that
is, D 6= V(Hp). Letting h(p) denote the number of downsets in Hp we have h(2) = 3
and h(p+ 1) = 2(h(p)−1) + 3. Soh(p) = 2p−1for all p≥2.
Letn0 denote the greatest multiple of four withn0 ≤n. Denote the binary expansion of n0 by 2p1 +· · ·+ 2pk, where 2 ≤p
1 < p2 < · · · < pk. Let ` = n−n0 + 4k−2. Let
G consist of a copy Hi0 of Hpi for each1 ≤i≤ k, as well as `−k copiesHk0+1, . . . , H`0
of the directed edge H2, except that for each 2 ≤ i ≤ ` we identify the sink of Hi0−1
with the source of Hi0. The downsets ofGare∅,V(G), and D∪V(Hi0+1)∪ · · · ∪V(H`0)
for each 1 ≤i≤` and each downset D of Hi0 that includes the sink of Hi0 but not the source. Note we have not counted any downset twice. So the number of downsets in G
is 2 + (h(p1)−2) +· · ·+ (h(pk)−2) + (`−k)(h(2)−2) = 2 +n0 −3k+`−k = n.
Note that the graphs Hm have O(m) vertices, so Ghas O((logn)2) vertices and can be
constructed in polynomial time in lognas required.
Theorem 4.20. #Downsets is AP-equivalent to its restriction to dags of maximum de- gree at most three.
Proof. We will describe an AP-reduction from #BIS. Consider an instance G of #BIS, specified by vertex setsUandV and a set of edgesE ⊆U×V, and an error parameter0< ε <1. We may assume that there are no isolated vertices. Let w =d(2/ε)4|U|+|V|+|E|e. For eachv∈U∪V choose an enumeration{ev,1,· · · , ev,deg(v)}of the edges incident tov,
wheredeg(v)denotes the degree ofv. LetGv be a copy of the dag with exactlywdeg(v)+2
downsets given by Lemma 4.19. For eache∈E letGe be a copy of the dag with exactly w+ 2downsets given by Lemma 4.19. For eachx∈U∪V ∪E lets(x)denote the source of Gx and let t(x) denote the sink of Gx. ConstructG0 by taking the disjoint union of
the dagsGx withx∈U∪V ∪E, together with arcs t(u)→s(eu,1)→ · · · →s(eu,deg(u))
for each u ∈ U and t(ev,1) → · · · →t(ev,deg(v)) → s(v) for each v ∈V. The reduction
LetZ denote the number of independent sets inG, and let Z0 denote the number of downsets of G0. We will argue that
Z ≤w−|E|Z0 ≤Z+w−14|U|+|V|+|E|. (4.4)
Assuming (4.4), the AP-reduction is correct: with probability at least 3/4 the value q
returned by the oracle satisfiese−ε/2Z0≤q≤eε/2Z0which impliese−εZ≤w−|E|q≤eεZ
as required.
Let X be the partial order with underlying set S
x∈U∪V∪E{s(u), t(u)} where x ≤y
means there is a directed path from x to y in G0. Consider a downset Y of X, and a downset Dof G0 withD∩X=Y. For each x, we have:
• D∩Gx=∅if t(x)6∈Y, and
• D∩Gx=Gx if s(x)∈Y, and
• otherwise s(x) 6∈ Y and t(x) ∈ Y, so D∩Gx can be any downset of Gx with s(x) 6∈ D∩Gx and t(x) ∈ D∩Gx. There are w such sets if x ∈ E, and wdeg(x)
such sets ifx∈U∪V.
So there are exactlywc(Y) downsets Dof G0 withD∩X =Y, where
c(Y) = X v∈U∪V s(v)∈/Y t(v)∈Y deg(v) + X e∈E s(e)∈/Y t(e)∈Y 1 = X (u,v)∈E X s(u)∈/Y t(u)∈Y 1 + X s(u,v)∈/Y t(u,v)∈Y 1 + X s(v)∈/Y t(v)∈Y 1 .
Note that the contribution from each (u, v)∈E is at most 1, and c(Y) =|E|if and only if for all (u, v) ∈ E we have: s(u) 6∈ Y, and t(u) ∈ Y ⇐⇒ s(u, v) ∈ Y, and
t(u, v) ∈Y ⇐⇒ s(v) ∈Y, andt(v) ∈Y. For each independent set I of Glet YI ⊆X
be the set containing:
• t(u), s(eu,1), . . . , s(eu,deg(u)) for each u∈U ∩I, and
• t(ev,1), . . . , t(ev,deg(v)), s(v) for each v∈V \I, and
• t(v)for each v∈V.
By the previous remarks we havewc(YI)−|E|= 1, whilewc(Y)−|E|≤w−1 for any downset
Y ofX not of the formYI whereI is an independent set of G. SinceX has cardinality
2(|U|+|V|+|E|), it has at most 4|U|+|V|+|E| downsets. This gives Z ≤ w−|E|Z0 ≤ Z+w−14|U|+|V|+|E|, which is (4.4).
Degree-two #CSPs with variable
weights
(This chapter is a revised version of [83] with a modified introduction.)
This chapter aims to study the computational complexity of approximately evaluating #CSPs where variables range over the Boolean domain{0,1}, and we restrict the allowed constraints to a fixed constraint language Γ, and we restrict each variable to appear at most twice, but we allow instances to specify a weight for each value that each variable can take. These problems, degree-two #CSPs with variable weights, are an abstraction of the problem of counting perfect matchings in a graph with edge weights.