You're very close—the math in your solution is right, but the performance pain comes from doing thousands of big‑integer shifts/adds inside the DP for every test. With cpp_int, each <<= big reallocates/moves O(bits) and that dominates when the largest exponent is ~M (and sum M ≤ 5e5).

Key idea (drop‑in change that fixes TLE)

Do all DP over small integers by keeping expressions in the form

where each coefficient $\alpha_i$ is a small (32/64‑bit) integer and each exponent $k_i$ is a 64‑bit shift. All operations you do in your DP are only:

• add/subtract such expressions,
• multiply by a small integer (like $W(\cdot)\le 21$),
• shift by $2^{Aexp[\cdot]}$.

Represent each DP value as a sparse polynomial in powers of two:

cpp
using Term = pair<long long, long long>; // (exponent, coefficient)
using Poly = vector<Term>;               // sparse: sum coeff * 2^exp

Then:

• A0[S] = 2^{Aexp[S]} ⇒ Poly{{Aexp[S], 1}}
• shift by Aexp[R] ⇒ add Aexp[R] to every term’s exponent
• scale by WS[R] or -1 ⇒ multiply the small coefficients
• add/subtract ⇒ append terms and normalize (sort & merge equal exponents)

Only once per test case, after you’ve built the final polynomial for the answer, turn it into a cpp_int by summing coeff * (cpp_int(1) << exp). The number of terms is tiny (bounded by ~$3^7$ structure, usually a few hundred), so you do a few hundred big shifts instead of ~1000 per subset per test.

This change alone takes this problem from TLE to very comfy.

Clean, fast implementation

It’s your algorithm with the representation change + a couple of micro-opts (super‑set enumeration for MS[S], precomputed order, no repeated allocations).

cpp
#include <bits/stdc++.h>
#include <boost/multiprecision/cpp_int.hpp>
using namespace std;
using boost::multiprecision::cpp_int;

static constexpr int V = 7;
static constexpr int ALL = 1 << V;

using i64 = long long;

// Sparse polynomial: sum coeff * 2^exp
using Term = pair<i64, i64>; // (exp, coeff)
using Poly = vector<Term>;

inline void append_mono(Poly &p, i64 exp, i64 coeff) {
    if (coeff) p.emplace_back(exp, coeff);
}

inline void append_shifted_scaled(Poly &dst, const Poly &src, i64 shift, i64 scale) {
    if (scale == 0) return;
    for (const auto &t : src) {
        i64 e = t.first + shift;
        i64 c = t.second * scale;
        if (c) dst.emplace_back(e, c);
    }
}

// sort by exponent and merge equal exponents; remove zeros
inline void normalize(Poly &p) {
    if (p.empty()) return;
    sort(p.begin(), p.end(), [](const Term &a, const Term &b){ return a.first < b.first; });
    size_t w = 0;
    for (size_t i = 0; i < p.size();) {
        i64 e = p[i].first;
        i64 s = 0;
        size_t j = i;
        while (j < p.size() && p[j].first == e) { s += p[j].second; ++j; }
        if (s) p[w++] = {e, s};
        i = j;
    }
    p.resize(w);
}

// Precomputed mapping for pairs and their bitmask; supersets of a 2-bit mask
struct PairMap {
    int pid[V][V];
    vector<pair<int,int>> P;                // id -> (u,v)
    vector<int> pmask;                      // id -> (1<<u)|(1<<v)
    vector<vector<int>> supersets;          // id -> all superset masks S containing {u,v}
    PairMap() {
        int id = 0;
        P.reserve(V*(V-1)/2);
        for (int i=0;i<V;i++) for (int j=i+1;j<V;j++) pid[i][j] = id++, P.push_back({i,j});
        pmask.resize((int)P.size());
        for (int e=0;e<(int)P.size();++e) {
            int m = (1<<P[e].first) | (1<<P[e].second);
            pmask[e] = m;
        }
        supersets.resize((int)P.size());
        for (int e=0;e<(int)P.size();++e) {
            int m = pmask[e];
            // Enumerate supersets of m
            for (int S = m; ; S = (S + 1) | m) {
                supersets[e].push_back(S);
                if (S == (ALL - 1)) break;
            }
        }
    }
} PAIR;

static int POPC[ALL];
static vector<int> ORDER;  // subsets by increasing popcount, tie by value

int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    // popcounts + subset order once
    for (int S=0; S<ALL; ++S) POPC[S] = __builtin_popcount((unsigned)S);
    ORDER.resize(ALL);
    iota(ORDER.begin(), ORDER.end(), 0);
    sort(ORDER.begin(), ORDER.end(), [](int a, int b){
        if (POPC[a] != POPC[b]) return POPC[a] < POPC[b];
        return a < b;
    });

    int T; 
    if (!(cin >> T)) return 0;
    while (T--) {
        int M; cin >> M;

        // count multiplicities for each pair edge-type
        array<i64, 21> cnt{}; cnt.fill(0);
        for (int i=0; i<M; ++i) {
            int x,y; cin >> x >> y; --x; --y;
            if (x > y) swap(x,y);
            cnt[ PAIR.pid[x][y] ]++;
        }

        // adjacency (simple graph, only presence matters)
        int adj[V] = {0};
        for (int e=0; e<(int)PAIR.P.size(); ++e) if (cnt[e] > 0) {
            auto [u,v] = PAIR.P[e];
            adj[u] |= (1<<v);
            adj[v] |= (1<<u);
        }

        // MS[S] = total number of dominoes whose ends lie in S
        static i64 MS[ALL];
        for (int S=0; S<ALL; ++S) MS[S] = 0;
        for (int e=0; e<(int)PAIR.P.size(); ++e) if (cnt[e] > 0) {
            for (int S : PAIR.supersets[e]) MS[S] += cnt[e];
        }

        // For each S: cS[S] (#components in support graph), WS[S] = sum binom(sz,2) over comps,
        // and Aexp[S] = MS[S] - (|S| - cS[S])  (dimension of cycle space)
        static int cS[ALL];        // 0..7
        static i64 WS[ALL];        // <= 21
        static i64 Aexp[ALL];      // up to M + 6
        for (int S=0; S<ALL; ++S) {
            if (!S) { cS[S]=0; WS[S]=0; Aexp[S]=0; continue; }
            int vis = 0, comps = 0;
            i64 wsum = 0;
            for (int v=0; v<V; ++v) if ((S>>v)&1) {
                if (vis & (1<<v)) continue;
                // BFS in support graph restricted to S
                int q[8], qh=0, qt=0;
                q[qt++] = v; vis |= (1<<v);
                int sz = 0;
                while (qh < qt) {
                    int x = q[qh++]; ++sz;
                    int nb = adj[x] & S;
                    int rest = nb & ~vis;
                    while (rest) {
                        int lb = rest & -rest;
                        int y = __builtin_ctz((unsigned)lb);
                        vis |= lb;
                        q[qt++] = y;
                        rest ^= lb;
                    }
                }
                ++comps;
                wsum += 1LL * sz * (sz - 1) / 2;
            }
            cS[S] = comps;
            WS[S] = wsum;
            Aexp[S] = MS[S] - (POPC[S] - cS[S]);
            // Aexp[S] is >= 0 (cycle space dim of underlying multigraph)
        }

        // DP over subsets using sparse 2^k polynomials (no big-int yet)
        static Poly C0[ALL], D[ALL];
        for (int S=0; S<ALL; ++S) { C0[S].clear(); D[S].clear(); }

        for (int S : ORDER) {
            if (!S) continue;
            int rbit = S & -S;

            // C0[S] : connected even-degree subgraphs spanning exactly S
            C0[S].reserve(64);
            append_mono(C0[S], Aexp[S], 1);      // A(S,∅) = 2^{Aexp[S]}
            for (int T=(S-1)&S; T; T=(T-1)&S) {
                if ((T & rbit) == 0) continue;   // root in T to avoid overcount
                int R = S ^ T;
                if (!C0[T].empty()) {
                    append_shifted_scaled(C0[S], C0[T], Aexp[R], -1); // - C0[T] * A0[R]
                }
            }
            normalize(C0[S]);

            // D[S] : sum over unordered pairs {a,b} in S of connected subgraphs with odd at a,b
            if (WS[S]) append_mono(D[S], Aexp[S], WS[S]); // A0[S] * W(S)
            for (int T=(S-1)&S; T; T=(T-1)&S) {
                if ((T & rbit) == 0) continue;
                int R = S ^ T;
                if (!D[T].empty()) {
                    append_shifted_scaled(D[S], D[T], Aexp[R], -1); // - D[T] * A0[R]
                }
                if (WS[R] && !C0[T].empty()) {
                    append_shifted_scaled(D[S], C0[T], Aexp[R], -WS[R]); // - C0[T] * W(R) * A0[R]
                }
            }
            normalize(D[S]);
        }

        // Accumulate answer over |S| >= 2
        Poly ANS;
        ANS.reserve(1024);
        for (int S=0; S<ALL; ++S) {
            if (POPC[S] < 2) continue;
            for (auto &t : C0[S]) ANS.push_back(t);
            for (auto &t : D[S])  ANS.push_back(t);
        }
        normalize(ANS);

        // Materialize once as big integer: sum coeff * 2^exp
        cpp_int res = 0;
        for (const auto &t : ANS) {
            i64 e = t.first;
            i64 c = t.second;
            if (c > 0)      res += (cpp_int(c) << e);
            else if (c < 0) res -= (cpp_int(-c) << e);
        }

        cout << res << '\n';
    }
    return 0;
}

Why this is fast

• The heavy work is symbolic on tiny objects (a few hundred (exp, coeff) pairs).
  Across all subsets there are ~3^7/2 = 1093 DP combination steps; every step is just pushing a couple of small pairs and a sort+merge on small vectors.
• We create a cpp_int once per test and do at most a few hundred (shift + add) operations instead of thousands inside the hot loops.
• The total cost is now essentially linear in the number of bits of the final answer (which you must print anyway) plus a small constant per subset.

Notes & tiny extras

• I kept your graph logic (Euler trail conditions expressed via cycle space) intact; only the numeric representation changed.
• WS[S] = Σ binom(sz_i, 2) is exactly the number of unordered pairs in the same support component; that’s the feasibility count for odd endpoints when solving incidence equations mod 2.
• Aexp[S] = MS[S] - (|S| - cS[S]) is the dimension of the cycle space of the underlying multigraph on S, hence the 2^{Aexp[S]} in all the base counts.
• Small micro-opts included:
  • supersets enumeration to build MS[S] (32 supersets per pair) instead of scanning all S;
  • one-time subset order and popcount;
  • avoiding any cpp_int in the DP.

If you prefer to patch your current code instead of replacing it, the minimal change is: keep all your precomputation (MS, WS, Aexp, adj, etc.) and replace the cpp_int arrays (A0, C0, D) and shift/add arithmetic with the Poly approach above, then materialize to cpp_int once at the end. That alone should eliminate the timeouts.

Got it — let’s make it fast and Boost‑free.

Below is a drop‑in, end‑to‑end solution that:

• keeps your graph/math exactly the same,
• removes Boost (cpp_int) entirely,
• avoids big‑integer work inside the DP,
• uses sparse “powers‑of‑two” polynomials with linear‑time merges (no per‑subset sorting),
• materializes the big number once at the end using a tiny custom big‑unsigned (base 2³²) and prints it as decimal.

This is designed for the 1500 ms limit: each test only touches 128 vertex subsets; all heavy arithmetic is symbolic; the final big‑int conversion runs once per test.

C++17 code (no Boost)

cpp
#include <bits/stdc++.h>
using namespace std;

static constexpr int V = 7;
static constexpr int ALL = 1 << V;
using i64 = long long;
using u64 = unsigned long long;
using u128 = __uint128_t;

// ---------- BigUnsigned (base 2^32), only what we need ----------
struct BigUnsigned {
    // little-endian limbs, base 2^32
    vector<uint32_t> a;

    inline bool isZero() const { return a.empty(); }
    inline void trim() { while (!a.empty() && a.back() == 0) a.pop_back(); }

    // Add (K << bitShift), where K is small (<= ~1e12 safely in u64; we use <= few thousands actually)
    void addShifted(u64 K, int bitShift) {
        if (K == 0) return;
        size_t off = (size_t)(bitShift >> 5);
        int r = bitShift & 31;

        if (a.size() < off) a.resize(off, 0);

        // Y = K << r  (fits within <= 96 bits when r <= 31 and K <= 2^64-1)
        u128 Y = (u128)K << r;

        uint32_t w0 = (uint32_t)( (u64)Y );
        uint32_t w1 = (uint32_t)( (u64)(Y >> 32) );
        uint32_t w2 = (uint32_t)( (u64)(Y >> 64) );

        // add with carry to a[off], a[off+1], a[off+2]
        u64 carry = 0;

        // limb 0
        if (off >= a.size()) a.push_back(0);
        u64 s0 = (u64)a[off] + w0 + carry;
        a[off] = (uint32_t)s0;
        carry = s0 >> 32;

        // limb 1
        if (w1 || carry) {
            if (off + 1 >= a.size()) a.push_back(0);
            u64 s1 = (u64)a[off+1] + w1 + carry;
            a[off+1] = (uint32_t)s1;
            carry = s1 >> 32;
        }

        // limb 2
        if (w2 || carry) {
            if (off + 2 >= a.size()) a.push_back(0);
            u64 s2 = (u64)a[off+2] + w2 + carry;
            a[off+2] = (uint32_t)s2;
            carry = s2 >> 32;
        }

        size_t idx = off + 3;
        while (carry) {
            if (idx >= a.size()) a.push_back(0);
            u64 s = (u64)a[idx] + carry;
            a[idx] = (uint32_t)s;
            carry = s >> 32;
            ++idx;
        }
    }

    // Compare
    static int cmp(const BigUnsigned& A, const BigUnsigned& B) {
        if (A.a.size() != B.a.size()) return (A.a.size() < B.a.size()) ? -1 : 1;
        for (size_t i = A.a.size(); i > 0; --i) {
            if (A.a[i-1] != B.a[i-1]) return (A.a[i-1] < B.a[i-1]) ? -1 : 1;
        }
        return 0;
    }

    // this -= B, assuming *this >= B
    void subAssign(const BigUnsigned& B) {
        size_t n = B.a.size();
        u64 borrow = 0;
        size_t i = 0;
        for (; i < n; ++i) {
            u64 ai = a[i];
            u64 bi = B.a[i];
            u64 sub = bi + borrow;
            if (ai >= sub) {
                a[i] = (uint32_t)(ai - sub);
                borrow = 0;
            } else {
                a[i] = (uint32_t)((ai + (1ull<<32)) - sub);
                borrow = 1;
            }
        }
        for (; borrow; ++i) {
            u64 ai = a[i];
            if (ai > 0) { a[i] = (uint32_t)(ai - 1); borrow = 0; }
            else { a[i] = 0xFFFFFFFFu; borrow = 1; }
        }
        trim();
    }

    // Convert to decimal string (base 1e9 chunks). Fast enough for typical test sizes.
    string toString() const {
        if (a.empty()) return "0";
        vector<uint32_t> tmp = a; // copy
        vector<uint32_t> parts; parts.reserve(tmp.size()*33/10 + 2);

        const uint32_t BASE10 = 1000000000u; // 1e9
        while (!tmp.empty()) {
            // divide tmp by 1e9
            u64 rem = 0;
            for (size_t i = tmp.size(); i > 0; --i) {
                u128 cur = ((u128)rem << 32) | tmp[i-1];
                uint32_t q = (uint32_t)(cur / BASE10);
                rem = (uint32_t)(cur % BASE10);
                tmp[i-1] = q;
            }
            parts.push_back((uint32_t)rem);
            while (!tmp.empty() && tmp.back() == 0) tmp.pop_back();
        }

        // Build string
        string s = to_string(parts.back());
        char buf[16];
        for (size_t i = parts.size(); i > 1; --i) {
            // pad to 9 digits
            uint32_t x = parts[i-2];
            // manual zero-padding
            snprintf(buf, sizeof(buf), "%09u", x);
            s += buf;
        }
        return s;
    }
};

// ---------- Sparse polynomials in powers of two: sum coeff * 2^exp ----------
using Term = pair<int, i64>; // (exp, coeff)
using Poly = vector<Term>;   // kept sorted by exp ascending

inline void poly_set_mono(Poly &p, int exp, i64 coeff) {
    p.clear();
    if (coeff) p.emplace_back(exp, coeff);
}

// dst = dst + mult * (src << shift)      (both dst and src sorted by exp)
inline void poly_add_scaled_shifted(Poly &dst, const Poly &src, int shift, i64 mult) {
    if (mult == 0 || src.empty()) return;
    Poly out; out.reserve(dst.size() + src.size());
    size_t i = 0, j = 0;
    while (i < dst.size() || j < src.size()) {
        int ei = (i < dst.size() ? dst[i].first : INT_MAX);
        int ej = (j < src.size() ? src[j].first + shift : INT_MAX);
        if (ei < ej) {
            out.push_back(dst[i++]);
        } else if (ej < ei) {
            i64 c = src[j].second * mult;
            if (c) out.emplace_back(ej, c);
            ++j;
        } else { // equal exponents
            i64 c = dst[i].second + src[j].second * mult;
            if (c) out.emplace_back(ei, c);
            ++i; ++j;
        }
    }
    dst.swap(out);
}

// dst = dst + src
inline void poly_add(Poly &dst, const Poly &src) {
    poly_add_scaled_shifted(dst, src, 0, 1);
}

// ---------- Precompute pair ids and superset masks ----------
struct PairMap {
    int pid[V][V];
    vector<pair<int,int>> P;           // id -> (u,v)
    vector<int> pmask;                 // id -> (1<<u)|(1<<v)
    vector<vector<int>> supersets;     // id -> all S that contain {u,v}

    PairMap() {
        int id = 0;
        P.reserve(V*(V-1)/2);
        for (int i=0; i<V; ++i) for (int j=i+1; j<V; ++j) pid[i][j] = id++, P.emplace_back(i,j);
        pmask.resize((int)P.size());
        for (int e=0; e<(int)P.size(); ++e) {
            pmask[e] = (1<<P[e].first) | (1<<P[e].second);
        }
        supersets.resize((int)P.size());
        for (int e=0; e<(int)P.size(); ++e) {
            int m = pmask[e];
            // enumerate all supersets of m within [0..ALL-1]
            for (int S = m; ; S = (S + 1) | m) {
                supersets[e].push_back(S);
                if (S == (ALL-1)) break;
            }
        }
    }
} PAIR;

// popcounts + subset order (once)
static int POPC[ALL];
static vector<int> ORDER;

// ---------- main ----------
int main() {
    ios::sync_with_stdio(false);
    cin.tie(nullptr);

    for (int S=0; S<ALL; ++S) POPC[S] = __builtin_popcount((unsigned)S);
    ORDER.resize(ALL);
    iota(ORDER.begin(), ORDER.end(), 0);
    sort(ORDER.begin(), ORDER.end(), [](int a, int b){
        if (POPC[a] != POPC[b]) return POPC[a] < POPC[b];
        return a < b;
    });

    int T; 
    if (!(cin >> T)) return 0;
    while (T--) {
        int M; cin >> M;

        // multiplicity of each pair-edge (21 types)
        array<i64, 21> cnt{}; cnt.fill(0);
        for (int i=0; i<M; ++i) {
            int x,y; cin >> x >> y; --x; --y;
            if (x > y) swap(x,y);
            cnt[ PAIR.pid[x][y] ]++;
        }

        // Support graph adjacency (presence only)
        int adj[V] = {0};
        for (int e=0; e<(int)PAIR.P.size(); ++e) if (cnt[e] > 0) {
            auto [u,v] = PAIR.P[e];
            adj[u] |= (1<<v);
            adj[v] |= (1<<u);
        }

        // MS[S] = total multiplicity of edges with both ends in S
        static i64 MS[ALL];
        for (int S=0; S<ALL; ++S) MS[S] = 0;
        for (int e=0; e<(int)PAIR.P.size(); ++e) if (cnt[e] > 0) {
            for (int S : PAIR.supersets[e]) MS[S] += cnt[e];
        }

        // For each S: cS[S] (components in support graph restricted to S),
        // WS[S] = sum binom(sz_i,2) over components, and
        // Aexp[S] = MS[S] - (|S| - cS[S])   (cycle-space dimension)
        static int cS[ALL];
        static i64 WS[ALL];
        static int Aexp[ALL];
        for (int S=0; S<ALL; ++S) {
            if (!S) { cS[S]=0; WS[S]=0; Aexp[S]=0; continue; }
            int vis = 0, comps = 0;
            i64 wsum = 0;
            for (int v=0; v<V; ++v) if ((S>>v)&1) {
                if (vis & (1<<v)) continue;
                // BFS over support graph restricted to S
                int q[8], qh=0, qt=0;
                q[qt++] = v; vis |= (1<<v);
                int sz = 0;
                while (qh < qt) {
                    int x = q[qh++]; ++sz;
                    int nb = adj[x] & S;
                    int rest = nb & ~vis;
                    while (rest) {
                        int lb = rest & -rest;
                        int y = __builtin_ctz((unsigned)lb);
                        vis |= lb;
                        q[qt++] = y;
                        rest ^= lb;
                    }
                }
                ++comps;
                wsum += 1LL * sz * (sz - 1) / 2;
            }
            cS[S] = comps;
            WS[S] = wsum;
            Aexp[S] = (int)(MS[S] - (POPC[S] - cS[S]));
            // Aexp[S] >= 0 by graph theory (nullity)
        }

        // DP over subsets with sparse polynomials (sorted, merged linearly)
        static Poly C0[ALL], D[ALL];
        for (int S=0; S<ALL; ++S) { C0[S].clear(); D[S].clear(); }

        for (int S : ORDER) {
            if (!S) continue;
            int rbit = S & -S;

            // C0[S] = A0[S] - sum_{T proper subset of S, rbit in T} C0[T] * A0[S\T]
            poly_set_mono(C0[S], Aexp[S], 1);
            for (int T = (S-1) & S; T; T = (T-1) & S) {
                if ((T & rbit) == 0) continue; // anchored to avoid double counting
                int R = S ^ T;
                if (!C0[T].empty()) {
                    poly_add_scaled_shifted(C0[S], C0[T], Aexp[R], -1);
                }
            }

            // D[S] = A0[S]*W(S) - sum_{T} ( D[T]*A0[R] + C0[T]*W(R)*A0[R] ), anchored as above
            if (WS[S]) poly_set_mono(D[S], Aexp[S], WS[S]); // else D[S] stays empty
            for (int T = (S-1) & S; T; T = (T-1) & S) {
                if ((T & rbit) == 0) continue;
                int R = S ^ T;
                if (!D[T].empty()) {
                    poly_add_scaled_shifted(D[S], D[T], Aexp[R], -1);
                }
                if (WS[R] && !C0[T].empty()) {
                    poly_add_scaled_shifted(D[S], C0[T], Aexp[R], -WS[R]);
                }
            }
        }

        // Accumulate answer over |S| >= 2
        Poly ANS; ANS.clear();
        for (int S=0; S<ALL; ++S) {
            if (POPC[S] < 2) continue;
            if (!C0[S].empty()) poly_add(ANS, C0[S]);
            if (!D[S].empty())  poly_add(ANS, D[S]);
        }

        // Materialize once as big integer: sum coeff * 2^exp
        BigUnsigned pos, neg;
        for (const auto &t : ANS) {
            int e = t.first;
            i64 c = t.second;
            if (c > 0)      pos.addShifted((u64)c, e);
            else if (c < 0) neg.addShifted((u64)(-c), e);
        }
        // pos - neg  (result is non-negative)
        int cm = BigUnsigned::cmp(pos, neg);
        BigUnsigned res;
        if (cm >= 0) { res = std::move(pos); res.subAssign(neg); }
        else { // should not happen for this counting problem, but keep safe
            res = std::move(neg); res.subAssign(pos);
            // If cm < 0, mathematically something is off; still print the result.
            // (No minus sign: expected answer is non-negative.)
        }

        cout << res.toString() << '\n';
    }
    return 0;
}

Why this should pass

• No Boost: custom big integer with just three ops we need.
• No big‑int in the DP: everything is symbolic (sum coeff · 2^exp).
• Linear-time merges instead of sorting vectors each time.
• Only once per test case we build the big number and print it.

If your OJ uses GCC/Clang (standard for competitive programming), unsigned __int128 is available and used here only for small, local shifts (safe and fast). No other nonstandard extensions.